Brownian Entanglement.
A.E. Allahverdyan$^{1,2)}$
A. Khrennikov$^{3)}$ and Th.M. Nieuwenhuizen$^{1)}$
We show that for two classical brownian particles there exists an analog of continuous-variable quantum entanglement: The common probability distribution of the two coordinates and the corresponding coarse-grained velocities cannot be prepared via mixing of any factorized distributions referring to the two particles in separate. This is possible for particles which interacted in the past, but do not interact in the present. Three factors are crucial for the effect: 1) separation of time-scales of coordinate and momentum which motivates the definition of coarse-grained velocities; 2) the resulting uncertainty relations between the coordinate of the brownian particle and the change of its coarse-grained velocity; 3) the fact that the coarse-grained velocity, though pertaining to a single brownian particle, is defined on a common context of two particles. The brownian entanglement is a consequence of a coarse-grained description and disappears for a finer resolution of the brownian motion. We discuss possibilities of its experimental realizations in examples of macroscopic brownian motion.
quant-ph0412132.tex
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\begin{document} 
\draft
\title
{Brownian Entanglement.}
\date{Today:~\today}

\author{A.E. Allahverdyan$^{1,2)}$, A. Khrennikov$^{3)}$
and Th.M. Nieuwenhuizen$^{1)}$}
\affiliation{$^{1)}$ Institute for Theoretical Physics,
Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands}
\affiliation{$^{2)}$Yerevan Physics Institute,
Alikhanian Brothers Street 2, Yerevan 375036, Armenia}
\affiliation{$^{3)}$International Centre for Mathematical Modeling
in Physics and Cognitive Sciences,\\
MSI, University of V\"axj\"o, S-35195, Sweden}


\begin{abstract} 

We show that for two classical brownian particles there exists an
analog of continuous-variable quantum entanglement: The common
probability distribution of the two coordinates and the corresponding
coarse-grained velocities cannot be prepared via mixing of any
factorized distributions referring to the two particles in
separate. This is possible for particles which interacted in the
past, but do not interact in the present.  Three factors are crucial
for the effect: 1) separation of time-scales of coordinate and momentum
which motivates the definition of coarse-grained velocities; 2) the
resulting uncertainty relations between the coordinate of the brownian
particle and the change of its coarse-grained velocity; 3) the fact
that the coarse-grained velocity, though pertaining to a single
brownian particle, is defined on a common context of two
particles. The brownian entanglement is a consequence of a
coarse-grained description and disappears for a finer resolution of
the brownian motion. We discuss possibilities of its experimental
realizations in examples of macroscopic brownian motion.

\end{abstract}
\pacs{05.40.Jc, 03.65.Ta, 05.70.Ln}

%% 05.30.-d quantum statistical mechanics
%% 05.70.Ln non-equilibrium thermodynamics
%% 03.65.Ta Foundations of quantum mechanics; measurement theory
%% 05.30.Ch Quantum ensemble theory
%% 03.65.Yz Decoherence; open systems; quantum statistical methods 
%% 05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin)
%% 05.40.Jc Brownian motion
%% 82.60.Qr Thermodynamics of nanoparticles
%% 65.40.Gr Entropy and other thermodynamical quantities

\maketitle

\section{Introduction.}

There is a long tradition of seeking connections between quantum
mechanics and classical statistical physics. A list of known
examples includes {\it i)} analogies 
between the notion of complementarity in quantum mechanics and statistical
thermodynamics, in particular, between quantum mechanical
uncertainty relations and the energy-temperature  
uncertainty relation in statistical thermodynamics
\cite{bohruncertain,sch,rosenfeld,landau}; {\it ii)} mathematical relations
between Schr\"odinger and Fokker-Planck equations \cite{risken}, which
makes quantum intuition very useful, e.g., for polymers
\cite{gros}; {\it iii)}
attempts to derive Schr\"odinger equations from the classical kinetic
picture \cite{nelson} or from classical stochastic electrodynamics
\cite{timo}.

More generally, both quantum mechanics and statistical physics are
essentially probabilistic theories and already at this level one
expects to find certain analogies between their concepts.  The
progress in this direction has been for a long time plagued by
statements on incompatibility between quantum mechanics and classical
probability theory, which lies in the basis of classical statistical
physics. As an example see Ref.~\cite{feynman} for statements that
quantum interference phenomena contradict to classical probability
theory.  It was, however, noted that such opinions are not warranted
\cite{ball}, and recently it was shown explicitly that several basic
relations of quantum mechanics can be derived from the
classical probability theory provided the contexts of physical
conditions (measurements) are properly taken into account \cite{and}.

Once the basic probabilistic ground of quantum mechanics and classical
statistical physics is recognized to be the same, one wonders whether
some unusual aspects of quantum mechanics, such as entanglement, can
find analogies in classical statistical physics. We see at least two
reasons for seeking such analogies. First, it is going to refresh our
understanding of classical statistical physics, and may imply in
future that advantages offered by quantum mechanics in certain tasks
of information processing and transfer are not unique to quantum
mechanics, and can be looked for in classical areas of physics as well
\cite{and}.  Second, it is useful for understanding quantum mechanics
that certain concepts believed to be purely quantum |that is,
incomprehensible in classical terms| can find natural classical
analogies. In fact the works and ideas mentioned in the above point
{\it i)--iii)} were partially directed toward this goal.  More recent
results along these lines are classical analogies to quantum
entanglement found in optics \cite{spr,italo} and classical
information theory \cite{pop} (secret classical correlations), and a
classical probabilistic model for certain aspects of (multi-time)
quantum measurements \cite{ki}.

The purpose of the present paper is to show that quantum entanglement
can have a natural analogy in the physics of brownian particles
(brownian entanglement).  The reason for the existence of this analogy
can be qualitatively explained as follows. It is known that the
dynamics of a brownian particle can be observed at two levels
\cite{risken}. Within the first, more fundamental level the brownian
particle coupled to a thermal bath at temperature $T$ is described via
definite coordinate $x$ and momentum $p$ and moves under influence of
external potential, friction force and an external random force.  The
latter two forces are generated by the bath. The second, overdamped
regime applies when the characteristic relaxation time of the
coordinate $\tau_x$ is much larger than that of the momentum $\tau_p$,
$\tau_x\gg\tau_p$ (overdamped regime). On times much larger than
$\tau_p$ one is interested in the change of the coordinate and defines
the {\it coarse-grained} velocity as $v=\Delta x/\Delta t$ for
$\tau_x\gg \Delta t\gg \tau_p$.  This definition of $v$ is the only
operationally meaningful one for the (effective) velocity within the
overdamped regime.  It appears that the coarse-grained velocity,
though pertaining to single particles, is defined in the context of
the whole systems of coupled brownian particles. Together with
uncertainty relations between the coordinate and the change of the
coarse-grained velocity |the role of Planck's constant is being played
by the temperature of the bath,| this contextuality feature will be
shown to cause a phenomenon similar to quantum entanglement: The
common probability distribution of the two coordinates and the
corresponding coarse-grained velocities |for two brownian particles
which interacted in the past, but need not interact in the present||
cannot be prepared via mixing of any factorized distributions
referring to the two particles separately.  This brownian entanglement
is a consequence of a coarse-grained description and disappears within
the first (more fundamental) level of description, simply because
entanglement is absent in classical mechanics.

The paper is organized as follows.  In section \ref{qm} we recall the
phenomenon of (continuous-variable) quantum entanglement, focusing
especially on the relations between the entanglement and the
uncertainty relations. Section \ref{III} discusses coarse-grained
velocities and uncertainty relations for the classical brownian
motion.  Next two sections define and study the phenomenon of brownian
entanglement. Sections \ref{artashir} and
\ref{samson} offers detailed comparison between the features of
quantum entanglement and those of its brownian counterpart. In section
\ref{expo} we discuss possibilities of experimental realization of
brownian entanglement. Our conclusions are presented in the last
section. Some technical questions are worked out in Appendix.



\section{Quantum entanglement}
\label{qm}

\subsection{Statistical interpretation.}

This section recalls the phenomenon of entanglement in quantum
mechanics, and especially underlines its connections with the
uncertainty relations. 

Before starting, it is useful to stress that in the present paper
we adhere to the statistical (ensemble) interpretation of
quantum mechanics,
where a quantum `state' is described by a density matrix $\hat\rho$,
and any
state, including a pure state $|\psi\rangle\langle\psi|$, refers to an
ensemble $\E(\hrho)$ of identically prepared systems; see, e.g.,
\cite{blokh,ballentine,newton,balian,home,espagnat,willi}
~\footnote{
The minimal statistical interpretation should of course be
distinguished from various hidden-variable theories and assumptions,
in particular, from the pre-assigned initial value assumptions, where
quantum measurements are viewed as merely revealing pre-existing
values of all observables. Unfortunately, some proponents
of the statistical interpretation were
unclear at this point, a fact that for a while discredited this
interpretation. For a well-balanced discussions on this
and related points, see \cite{espagnat,willi}.}. To put it succinctly:
quantum mechanics makes\footnote{This way of putting the message of
statistical interpretation is adopted from \cite{kruger}.}
\begin{itemize}

\item statistical statements on

\item the results of measurements done

\item on ensembles of identically prepared systems.

\end{itemize}

As was stressed repeatedly
\cite{blokh,ballentine,newton,balian,home,espagnat,willi},
in particular by experimentalists \cite{itano}, 
the experimentally relevant statements of quantum mechanics do not
require more than the minimal statistical interpretation
\footnote{Still, as correctly pointed out in \cite{ocohen,jaynes},
the choice of interpretation can and does influence one's estimates of
importance for various scientific problems.}.
Moreover,
this interpretation deals more successfully with the measurement
problem, as instanced by a recent exactly solvable model
\cite{cw}, and allows to reconcile quantum mechanics with
classical probability theory \cite{and}.  The fact that discussions on
quantum entanglement (and on Bell inequalities and related matters) do
not employ the statistical interpretation is a mere prejudice; see
Refs.
\cite{ballentine,ocohen,kruger,cw} for examples of such discussions.

\subsection{Definition of entanglement.}

Consider a quantum system S consisting of two subsystems $\1$ and $\2$. A
state $\hrho$ of S is called entangled, see e.g. \cite{peres},
(or non-separable) with respect to $\1$ and $\2$, if it {\it cannot} be 
represented as
\BEA
\label{1}
\hrho=\sum_{k=1}^np_k\,\hrho^{(1)}_k\otimes\hrho^{(2)}_k,\qquad
\sum_{k=1}^np_k=1, \qquad p_k\ge 0,
\EEA
where $\hrho^{(1)}_i$ and $\hrho^{(2)}_i$ are arbitrary density matrices
living in the Hilbert spaces of $\1$ and $\2$, respectively, 
$n$ is an integer, and where $\{p_k\}_{k=1}^n$ is a probability distribution. 

According to definition (\ref{1}), a separable quantum state can
always be prepared by means of mixing \footnote{Mixing ensembles
$\E(\hrho_1)$ and $\E(\hrho_2)$ with probabilities $p_1$ and $p_2$,
respectively, means that one throws a dice with probabilities of
outcomes equal to $p_1$ and $p_2$, and depending on the outcome one
picks up a system from $\E(\hrho_1)$ or $\E(\hrho_2)$, keeping no
information on where the system came from. Alternatively, one can join
together $Np_1$ systems from $\E(\hrho_1)$ and $Np_2$ systems from
$\E(\hrho_2)$ ($N\gg 1$), so that no information information is kept
on where a single system came from. } 
non-correlated states $\hrho^{(1)}_k\otimes\hrho^{(2)}_k$
of the two subsystems. For a pure state $\hrho
=|\psi\rangle\langle\psi|$ we return to the more known definition of
entanglement: $|\psi\rangle$ cannot be represented as
$|\psi\rangle=|\psi\rangle_1\otimes|\psi\rangle_2$. For this
particular case the absence of entanglement implies the absence of any
correlation.  In contrast, for the more general situation given by
(\ref{1}), a non-entangled state can still posses certain (classical)
correlations, since the totally uncorrelated situations will be given
as $\hrho=\hrho^{(1)}\otimes\hrho^{(2)}$.

Two important features of quantum entanglement should be noted.  First,
it can be tested only through measuring some correlations between the
subsystems $\1$ and $\2$. Observables pertaining to $\1$ or to $\2$ alone
are obviously insensitive to entanglement.  Second, if the state
(\ref{1}) is prepared by two different observers 1 and 2, then this
preparation is seen to involve correlated actions of them, and thus
the corresponding observers have to communicate classically.

\subsection{Classical systems.}
\label{claa}

In classics the representation (\ref{1}) is apparently
always possible. Indeed,
let us have a probability distribution $P(x_1,x_2)$ of two classical
systems $\1$ and $\2$ represented by
random variables $x_1$ and $x_2$. Assume for simplicity that
$x_1$ can take values $a_1,..,a_n$, 
while $x_2$ can take values $b_1,..,b_n$.
Then $P(x_1,x_2)$ can be written as
\BEA
P(x_1,x_2)=\sum_{\alpha=a_1,...,a_n}~\sum_{\beta=b_1,...,b_n}
P(\alpha,\beta)\,
\delta_{\alpha x_1}\, \delta_{x_2\beta},
\label{111}
\EEA
where $\delta_{\alpha\beta}=1$ if $\alpha=\beta$, and
$\delta_{\alpha\beta}=0$ otherwise. {\it Provided} that
$\delta_{\alpha x_1}$ and $\delta_{x_2\beta}$ are two legitimate
probability distributions belonging to $\1$ and $\2$, that is,
provided there are no mechanisms prohibiting the realization of
$\delta_{\alpha x_1}$ and $\delta_{x_2\beta}$ as physically acceptable
distributions usable in the actual preparation, the representation
(\ref{1}) is realized for the classical situation: there is no
entanglement.

In fact the quantum situation goes to the classical one right in (\ref{1})
if we assume that the involved density matrices are always
diagonal. So it is the presence of non-diagonal elements of the given
density matrix that makes the situations different.  

\subsection{Semiclassical systems.}

We shall now localize the cause of quantum entanglement for systems
which behave (semi)classically in several other respects.  This will
help us to understand the way of searching for entanglement in
non-quantum situations.

Let $\1$ and $\2$ be two quantum (non-interacting) harmonic
oscillators, and the overall density matrix (in the coordinate
representation) $\rho(x_1,x'_1;x_2,x'_2)$ be a gaussian function of
its variables.  The corresponding coordinate and momenta operators are
denoted by $\x_1,\,\p_1$ and $\x_2,\,\p_2$, respectively. As
well-known, gaussian states are conveniently dealt with help of Wigner
functions $W(x,p)$, which are functions of $x$ and $p$
\footnote{For good reviews on the
properties of Wigner function see
\cite{wignerreview}.}. The
Wigner function is equivalent to the density matrix, that is, it
allows to calculate all averages, but in contrast to the density
matrix it has several properties expected for the common probability
distribution of the coordinate and momentum. In particular, the
analysis via Wigner functions will provide us below with a richer
intuition on the relations between entanglement and uncertainty relations;
thus, between entanglement and non-commutativity.
 
Since the overall Hamiltonian of $\1$ and $\2$ is assumed to be
harmonic, the initially gaussian state remains gaussian for all times
and the corresponding Wigner function will be positive and thus will
{\it partially} admit a classical interpretation in terms of common
probability distribution of the coordinate and momentum
\cite{wignerreview}. A related
fact is that for two harmonic oscillators the Ehrenfest equations of
motion can be recast into a classical form
\cite{wignerreview}. So many aspects of this system can be accounted
for in classical terms.

In order to see why in spite of these classical features the system
of two oscillators can be entangled, note that for the Wigner functions one
can rewrite the condition (\ref{1}) as
\BEA
W(x_1,p_1;x_2,p_2)=\int \d \lambda \,{\cal P}(\lambda)\,
W_1(x_1,p_1|\lambda)\,W_2(x_2,p_2|\lambda),
\qquad {\cal P}(\lambda)\geq 0,\qquad
\int\d \lambda\,{\cal P}(\lambda)=1.
\EEA
where $W_1$ and $W_2$ are separate Wigner functions for $\1$ and
$\2$ respectively, and ${\cal P}(\lambda)$ is some probability 
distribution. If $\1$ and $\2$ were classical oscillators,
then instead of Wigner functions we would have distribution functions,
and one can always
write down the analog of (\ref{111}) for the common distribution function
$P(x_1,p_1;x_2,p_2)$ and 
$\lambda=(\alpha_1\,\beta_1,\alpha_2,\beta_2)$:
\BEA
\label{222}
P(x_1,p_1;x_2,p_2)=\int \d \alpha_1\,\d \beta_1\,\d \alpha_2\,
\d \beta_2\,P(\alpha_1,\beta_1;\alpha_2,\beta_2)\,
\delta(\alpha_1-x_1)\,\delta(\beta_1-p_1)\,
\delta(\alpha_2-x_2)\,\delta(\beta_2-p_2).
\EEA
In the classical situation this means that there is no
entanglement. The same formula (\ref{222}) can formally be written
down also for the positive Wigner functions $W(x_1,p_1;x_2,p_2)$.
However, in the quantum situation
$\delta(\alpha_1-x_1)\,\delta(\beta_1-p_1)$ and
$\delta(\alpha_2-x_2)\,\delta(\beta_2-p_2)$ are {\it not} legitimate
Wigner functions, since they prescribe definite values to both
coordinate
and momentum and thus
do not respect the
uncertainty relations.

We conclude that the uncertainty relations are necessary |but
not sufficient| for the existence of entanglement in
semiclassical systems. 


\subsection{A simple sufficient condition for quantum entanglement}

The definition of the entanglement as given by Eq.~(\ref{1}) is not
practical (except for a pure density matrix $\hrho$, when no
entanglement means no correlations). It is therefore useful to have
certain sufficient conditions for the presence of entanglement which
will be easy to handle in applications and which will have a
transparent physical meaning.

We choose the units in such a way that the coordinate and the
momentum have the same dimension as $\sqrt{\hbar}$. For a harmonic
oscillator with mass $m$ and frequency $\omega$ it will suffice to
make the following canonical transformation: $\x\to\sqrt{m\omega}\,\x$,
$\p\to\p/\sqrt{m\omega}$. 

For a two-particle system S with coordinate and momenta operators
$\hat{x}_1$, $\hat{x}_2$ and $\hat{p}_1$, $\hat{p}_2$, respectively, one
can propose the following sufficient condition for the entanglement
\cite{hoffmann}. 

Let us first note that the standard uncertainty relation 
\begin{gather}
\label{dub1}
\langle \Delta\hat{x}^2\rangle\,\langle \Delta\hat{p}^2\rangle
\geq \frac{\hbar^2}{4},\\
\Delta \x\equiv\x-\langle\x\rangle,\qquad
\Delta \p\equiv\p-\langle\p\rangle,
\end{gather}
implies
\BEA
\label{dalal}
&&\langle \Delta\hat{x}^2\rangle+\langle \Delta\hat{p}^2\rangle
\geq\langle \Delta\hat{x}^2\rangle+\frac{\hbar^2}
{4~\langle \Delta\hat{x}^2\rangle},
\EEA
and then one gets via minimizing the RHS of (\ref{dalal}) over $\langle
\Delta\hat{x}^2\rangle$ (which produces
$\langle\Delta\hat{x}^2\rangle\to\hbar/2$, to be
put in the RHS of (\ref{dalal})):
\BEA
\langle \Delta\hat{x}^2\rangle+\langle \Delta\hat{p}^2\rangle
\geq \hbar.
\label{dub2}
\EEA
%%Note that the converse is not true: (\ref{dub2}) does not imply (\ref{dub1}).

Now assume that the two-particle system is described by a factorized
density matrix:
\BEA
\hrho=\hrho^{(1)}\otimes\hrho^{(2)}.
\label{gambar}
\EEA
Then due to (\ref{dub2}) one can write:
\BEA
\label{dub3}
\langle (\Delta\hat{x}_1-\Delta\hat{x}_2)^2\rangle+
\langle (\Delta\hat{p}_1+\Delta\hat{p}_2)^2\rangle
=\langle \Delta\hat{x}_1^2\rangle+\langle \Delta\hat{p}_1^2\rangle
+\langle \Delta\hat{x}_2^2\rangle+\langle \Delta\hat{p}_2^2\rangle
\geq 2\hbar,
\EEA
just because for non-correlated systems the corresponding
variances add up.  If the overall system is in a separable state, then
(\ref{dub3}) is even strengthened, since the variance of {\it any} observable
$\langle\Delta\hat{A}^2\rangle$
increases under mixing, i.e. under a transformation:
\BEA
\{p_k,\hrho_k\}
\to \sum_kp_k\hrho_k,\qquad p_k\geq 0, \qquad\sum_kp_k=1,
\EEA
where $\hrho_k$ are normalized (${\rm tr}\,\hrho_k=1$)
density matrices. Indeed,
\BEA
\langle\Delta\hat{A}^2\rangle=\sum_kp_k{\rm tr}\,[\,
\hrho_k\,(\hat{A}-\langle\hat{A}\rangle)^2 \,]&&=
\sum_kp_k\left\{{\rm tr}\,[\,\hrho_k\,\hat{A}^2\,]
-{\rm tr}\,[\,\hrho_k\,\hat{A}\,]^2\right\}+
\sum_kp_k\left\{{\rm tr}\,[\,\hrho_k\,\hat{A}\,]
-\langle\hat{A}\rangle\right\}^2\nonumber\\
&&=\sum_kp_k\langle\Delta\hat{A}^2\rangle_k
+\sum_kp_k\left\{{\rm tr}\,[\,\hrho_k\,\hat{A}\,]
-\langle\hat{A}\rangle\right\}^2\geq
\sum_kp_k\langle\Delta\hat{A}^2\rangle_k.
\label{lulu}
\EEA
Employing (\ref{lulu}) for 
$\hat{A}=\Delta\hat{x}_1-\Delta\hat{x}_2$ and
for $\hat{A}=\Delta\hat{p}_1-\Delta\hat{p}_2$, we get that
though (\ref{dub3}) was obtained for 
the factorized state (\ref{gambar}), it remains valid 
for an arbitrary non-entangled state 
$\sum_kp_k\hrho^{(1)}_k\otimes\hrho^{(2)}_k$. Thus the violation
\BEA
\langle (\Delta\hat{x}_1-\Delta\hat{x}_2)^2\rangle+
\langle (\Delta\hat{p}_1+\Delta\hat{p}_2)^2\rangle
\leq 2\hbar
\label{buenosaires}
\label{klm}
\EEA
of (\ref{dub3}) 
is a sufficient condition for entanglement. 

Eq.~(\ref{buenosaires}) has a transparent physical meaning:

\begin{itemize}

\item
entanglement is present, if fluctuations do not sum up additively,
i.e., if the changes $\Delta\hat{x}_1$ and $\Delta\hat{x}_2$ of the
coordinates tend to correlate with each other, while those of the
momenta tend to anticorrelate. 
\end{itemize}

Alternative sufficient conditions for entanglement 
can be built based on $\langle
(\Delta\hat{x}_1+\Delta\hat{x}_2)^2\rangle+ \langle
(\Delta\hat{p}_1-\Delta\hat{p}_2)^2\rangle$ or $\langle
(\Delta\hat{x}_1+\Delta\hat{x}_2)^2\rangle+ \langle
(\Delta\hat{p}_1+\Delta\hat{p}_2)^2\rangle$, etc. Various conditions
obtained in this way are obviously not equivalent to each other. 
An obvious way to strengthen it is to demand that
\BEA
\label{lapas}
\langle (\Delta\hat{x}_1+\epsilon
\Delta\hat{x}_2)^2\rangle+ \langle
(\Delta\hat{p}_1+\zeta\Delta\hat{p}_2)^2\rangle\leq 2\hbar,
\EEA
at least
for one of four independent choices $\epsilon=\pm 1$, $\zeta= \pm 1$.

For some special states (e.g. gaussian states) there exist
in literature necessary and sufficient conditions for entanglement
\cite{wolf}.  However, they are technically involved and do not have
such a straightforward physical meaning as (\ref{buenosaires}).
For our purposes conditions (\ref{buenosaires}, \ref{lapas})
are sufficient.

\subsection{Operational issues.}
\label{oper}

Let us finally recall how the condition (\ref{buenosaires})
is checked operationally. To this end, we re-write this inequality
as
\BEA
\label{ap1}
&&\langle \Delta\hat{x}_1^2\rangle +\langle\Delta\hat{x}_2^2\rangle+
\langle \Delta\hat{p}_1^2\rangle+\langle\Delta\hat{p}_2^2\rangle\\
&&-2\langle \Delta\hat{x}_1\,\Delta\hat{x}_2\rangle+
2\langle \Delta\hat{p}_1\,\Delta\hat{p}_2\rangle
\label{ap2}\\
&&\leq 2\hbar.\nonumber
\EEA
One now needs two ensembles of systems S, each one
consisting of identically prepared
correlated sub-systems $\1$ and $\2$. Assuming that $\1$
and $\2$ are in possession of observers 1 and 2, respectively, the
observer 1 measures on the first (second) ensemble $\hat{x}_1$ ($\hat{p}_1$),
while the observer 2
measures on the first (second) ensemble $\hat{x}_2$ ($\hat{p}_2$).
Note that the above two ensembles are necessary, since 
$\hat{x}_1$ and $\hat{p}_1$ 
(respectively, $\hat{x}_2$ and $\hat{p}_2$) do not commute:
$[\hat{x}_k,\hat{p}_l]=i\hbar\delta_{kl}$, with $\delta_{kl}$
being Kronecker's delta, and thus cannot be measured simultaneously.  
Now the quantities in (\ref{ap1}) can be
estimated by each observer separately, while for the quantities in
(\ref{ap2}) the observers need to put the results of their
measurements together (or to communicate in any other classical way)
and to count the coinciding events.


\section{Coarse-grained velocities and
uncertainty relations for brownian particles.}
\label{III}

Consider $N$ identical brownian particles with coordinates
$\xb=(x_1,...,x_N)$ and mass $m$ interacting with $N$ independent
thermal baths at temperatures $T_i$ and subjected to a potential
$U(x_1,...,x_N)$.  The overdamped limit is defined by the following
two conditions \cite{risken} (for a more detailed discussion see
Appendix):

\begin{itemize} 

\item The characteristic relaxation time 
\footnote{The characteristic relaxation time 
$\tau_\theta$ of a random time-dependent variable $\theta$
(e.g. coordinate or momentum) is an {\it ensemble notion} and is
defined as the time necessary for the conditional distribution of
$\theta$ to become memoryless: $P(\theta,t'+\tau_\theta |\theta',t')
=P(\theta,t'+\tau_\theta )$.  For the system of brownian particles
sufficiently strongly coupled to their thermal baths, the
arbitrarinesses of the above definition (e.g., the precise choice of
$\theta'$ and $t'$) do not change the physical content of the
definition \cite{risken}. } of the (real) momenta $m\dot{x}_i$ is much
smaller than the one of the coordinates.

\item One is interested in times which are much larger than the
relaxation time of the momenta, but which can be much smaller than
or comparable
to the relaxation time of the coordinates.

\end{itemize}

Under these conditions the dynamics of the system is described
by the following Langevin equations \cite{risken}:
\BEA
\label{langevin}
\dot{x}_i=f_i(\xb)+\eta_i(t),\qquad f_i(\xb)=-\partial_{x_i} U(\xb),
\qquad \langle\eta_i(t)\,\eta_j(t')\rangle=2\,T_i\,\delta_{ij}\,\delta(t-t'),
\EEA
where for our convenience 
both the mass $m$ and the damping constant (coupling constant of
the particles to the bath) are in the main text taken equal to one
(they are recorded in the Appendix)
\footnote{Eq.~(\ref{langevin}) can be obtained from
the complete Langevin equations of motion: [with the same definition
of $\eta_i(t)$ as in (\ref{langevin})]
$m\ddot{x}_i+\dot{x}_i=f_i(\xb)+\eta_i(t)$, by disregarding the first
term $m\ddot{x}_i$ corresponding to acceleration. One might make this
heuristically by requiring that the friction force $\dot{x}_i$
dominates, and that the second-derivative $\ddot{x}_i$ is small for
long times. More rigorous derivation of this classical problem is
presented in \cite{risken}, and is recalled in Appendix for a simple
model.  In their turn the above complete Langevin equations can be
rigorously derived from the Newton equations of motion for the
brownian particles and the baths. Here are the basic conditions for
this derivation. 1) The thermodynamical (macroscopic) limit for the
baths. 2) A reasonable model for the particle-bath interaction. 3) The
initially equilibrium state of the baths, described by the
corresponding Gibbs distributions at temperature $T_i$ (this is how
the random noises $\eta_i(t)$ come into existence).  The details of
the derivation can be looked up, e.g., in \cite{cl,gardiner}.  }.


The conditional probability $P(\xb,t|\xb ',t')$ is known to satisfy the 
following Fokker-Planck equation \cite{risken}:
\BEA
\label{fokkerplanck}
\label{d1}
\partial_t P(\xb,t|\xb ',t')
=-\sum_i\partial_{x_i}\,[\, f_i(\xb)\,P(\xb,t|\xb ',t') \,]+
\sum_iT_i\,\partial^2_{x_ix_i}P(\xb,t|\xb ',t'),\qquad
t\geq t'.
\EEA
Eq.~(\ref{d1}) is associated with the following initial condition:
\BEA
\label{d2}
P(\xb,t|\xb ',t)=\delta(\xb-\xb ')\equiv\prod_{i=1}^N\delta(x_i-x'_i),
\EEA
as follows from the very definition of the conditional probability.

Consider an ensemble $\Sigma(\xb,t)$ of all realizations of the whole
$N$-particle system which at time $t$ have a coordinate vector
$\xb$. Such an ensemble can be selected out of all possible realizations
by measuring $\xb=(x_1,...,x_N)$.  For this ensemble the average coarse-grained
velocity for the brownian particle with index $j$ might naively be
defined as:
\BEA
v_{j}(\xb,t)={\rm lim}_{\ep\to 0}
\,\int\d \yb\,\frac{y_j-x_j}{\ep}\,P(\yb,t+\ep|\xb,t).
\EEA
However, the absence of regular trajectories
enforces us to define different velocities for different directions of
time \cite{nelson}:
\BEA
\label{dish1}
v_{+,j}(\xb,t)={\rm lim}_{\ep\to +0}
\,\int\d y_j\,\frac{y_j-x_j}{\ep}\,P(y_j,t+\ep|\xb,t),\\
v_{-,j}(\xb,t)={\rm lim}_{\ep\to +0}
\,\int\d y_j\,\frac{x_j-y_j}{\ep}\,P(y_j,t-\ep|\xb,t).
\label{dish2}
\EEA

The physical meaning of these expressions can be explained as follows.

\begin{enumerate}

\item As seen from the definitions, $v_{+,j}(\xb,t)$ is the average 
velocity to move anywhere starting from $(\xb,t)$, whereas
$v_{-,j}(\xb,t)$ is the average velocity to come from anywhere and to
arrive at $\xb$ at the moment $t$.

Since these velocities are defined already in the overdamped limit,
$\ep$ is assumed to be much larger than the characteristic relaxation
time of the (real) momentum which is small in the overdamped limit.
Therefore, we call (\ref{dish1}, \ref{dish2}) coarse-grained
velocities. It is known that for the overdamped brownian motion almost
all trajectories are not smooth. This is connected to the chaotic
influences of the bath(s) which randomize the real momenta on much
smaller times, and this is also the reason for $v_{+,j}(\xb,t)\not
=v_{-,j}(\xb,t)$. The difference $v_{+,j}(\xb,t)- v_{-,j}(\xb,t)$
thus characterizes the degree of the above non-smoothness.

Recall that would one take $\ep$ much smaller than the
characteristic relaxation time of the momentum |which will amount to
applying definitions (\ref{dish1}) and (\ref{dish2}) to a smoother
trajectory| then $v_{+,j}(\xb,t)$ and $v_{-,j}(\xb,t)$ will be equal
to each other and equal to the average momentum.  These points are
discussed in detail in Appendix.

\item Here is an operational procedure for measuring 
$v_{+,j}(\xb,t)$ and $v_{-,j}(\xb,t)$.
Consider the overall ensemble of the brownian particles.
The single members of this ensemble are $N$ coupled brownian
particles. For each such single member one measures

{\it i)} the coordinate of the brownian particle with index $j$
at the moment $t-\varepsilon$;

{\it ii)} the coordinates of {\it all} brownian particles 
at the moment $t$;

{\it iii)} the coordinate of the same brownian particle with index $j$
at the moment $t+\varepsilon$.

Repeating these points many times on various members of the above
overall ensemble, ignoring all the results from {\it i)} and {\it
iii)}, and selecting events from the second step, we reconstruct the
ensemble $\Sigma(\xb,t)$. Employing the results from {\it i)} and {\it
iii)} and conditioning repeatedly upon a single member from
$\Sigma(\xb,t)$, we estimate the conditional probabilities
$P(y_j,t\pm\ep|\xb,t)$ and then finally calculate $v_{+,j}(\xb,t)$ and
$v_{-,j}(\xb,t)$ via Eqs.~(\ref{dish1}, \ref{dish2}).


\item The above ensemble $\Sigma(\xb,t)$ is different
from an ensemble $\Sigma_j(x_j,t)$ obtained by measuring $x_j$ (at
time $t$) {\it irrespective} of other coordinates. Indeed,
$\Sigma(\xb,t)$ is defined globally, and if there are different
observers in possession of each brownian particle, they have to
communicate to each other in order to be able to construct
$\Sigma(\xb,t)$. In contrast, $\Sigma_j(x_j,t)$ is defined exclusively
with respect to the brownian particle with index $j$.  This point will
be discussed in more details later on in section
\ref{artashir}.

\end{enumerate}

The calculation of $v_{+,i}(x,t)$ and $v_{-,i}(x,t)$
is straightforward upon using the following three things: relation
\BEA
P(\xb,t+\ep|\yb,t)=\delta(\xb-\yb)+\ep\sum_i\left[
-f_i(\yb)\partial_{x_i}\,\delta(\xb-\yb)+
T_i\,\partial^2_{x_ix_i}\delta(\xb-\yb)
\right],
\EEA
which follows from (\ref{d1}, \ref{d2}), Bayes formula, and partial
integration assuming natural boundary conditions at infinity: $P(x,t)\to
0$ if $x_j\to \pm\infty$. 
Starting from the definitions (\ref{dish1}, \ref{dish2}), we obtain
\BEA
\label{shun}
v_{+,j}(\xb,t)&=&
{\rm lim}_{\ep\to +0}
\,\int\d \yb\,\frac{y_j-x_j}{\ep}\,P(\yb,t+\ep|\xb,t)\nonumber\\
&=&\,\int\d \yb\,(y_j-x_j)\sum_i
\left[\,
- f_i(\xb)\partial_{y_i}\delta(\xb-\yb) 
+T_i\partial^2_{y_iy_i}\delta(\xb-\yb) \,\right]\nonumber\\
&=&f_j(\xb),
\EEA
\BEA
v_{-,j}(\xb,t)&=&{\rm lim}_{\ep\to +0}
\,\int\d \yb\,\frac{x_j-y_j}{\ep}\,P(\yb,t-\ep|\xb,t)\nonumber\\
&=&{\rm lim}_{\ep\to +0}
\,\int\d \yb\,\frac{x_j-y_j}{\ep}\,P(\xb,t|\yb,t-\ep)\,
\frac{P(\yb,t-\ep)}{P(\xb,t)}\nonumber\\
&=&\int\d \yb\,(x_j-y_j)
\,\frac{P(\yb,t)}{P(\xb,t)}
\sum_i\left[-f(\yb)\,\partial_{x_i}\delta(\xb-\yb)
+T_i\partial_{x_ix_i}\delta(\xb-\yb)\right]\,
\nonumber\\
&=&f_j(\xb)-2T_j\,\partial _{x_j}\ln P(\xb,t).
\label{katu}
\EEA


The difference between the coarse-grained
velocities $v_{+,j}(\xb,t)$ and $v_{-,j}(\xb,t)$ is 
\BEA
\label{babek}
u_j(\xb,t)=\frac{v_{-,j}(\xb,t)
-v_{+,j}(\xb,t)}{2}=-T_j\partial_{x_j}\ln P(\xb,t).
\EEA
Recall that this quantity is non-zero due to the action of the thermal
bath (the factor $T_j$ in (\ref{babek})), and due to the fact that at
the coarse-grained level of description almost all trajectories of the
brownian particles are not smooth. Sometimes $u_j(\xb,t)$ is referred
to as osmotic velocity of the brownian particle with index $j$
\cite{nelson}. We shall use this word as a useful short-hand for the
more precise term ``change of the coarse-grained velocity''.

Note that once the interactions between the brownian particles are
absent: $U(\xb)=\sum_kU(x_k)$ |and consequently correlations are absent
as well, $P(\xb,t)=\prod_kP(x_k,t)$, if they were absent initially| the
coarse-grained velocities $v_{\pm,j}$ depend only the corresponding
coordinate: $v_{\pm,j}(\xb,t)=v_{\pm,j}(x_j,t)$. 


The regular (or mechanical) counterpart of the coarse-grained velocity
for the brownian particle with unit mass and index $j$ can be
naturally associated with the newtonian force $f_j(\xb)$ as defined by
(\ref{langevin}). When the bath(s) are absent, this is the only
contribution to the coarse-grained velocity.  The fact that velocity
appears to be proportional to force reminds us that we are in the
overdamped regime of description, where, in particular, friction
dominates over acceleration.


\subsection{Uncertainty relations and their interpretation.}
\label{ma}

Uncertainty relations exist not only in quantum mechanics but also in
physics of overdamped brownian motion \cite{ur}. As the very subject
of statistical physics, they arise out of ignorance reasons, or more
precisely due to separation of time-scales: though the real classical
particles involved in the classical brownian motion certainly do have
sharply defined coordinates and momenta at the microscopical level, at
the coarse-grained (overdamped) level of description a brownian
particle does not have a well-defined trajectory, and cannot
posses sharply defined coordinate and coarse-grained velocity, as 
seen below.


We saw above that the coarse-grained velocities $v_{+,j}(\xb,t)$ and
$v_{-,j}(\xb,t)$ are defined with respect to the ensemble
$\Sigma(\xb,t)$ of all brownian particles which at the moment $t$ pass
via coordinate vector $\xb$. On the other hand, $\Sigma(\xb,t)$ is by
itself a subensemble embedded with probability (weight) $P(\xb,t)$ into
the ensemble of all realizations of the random coordinate vector $\xb$
at the time $t$. Thus, both $v_{+,j}(\xb,t)$ and $v_{-,j}(\xb,t)$, as
well as the osmotic velocity $u_j(\xb,t)$ are random quantities as
functions of of the random configuration of $N$ brownian particles ${\bf
x}=(x_1,\cdots, x_N)$.  In other words, this randomness enters via the
context $(\xb,t)$ which was chosen to define the ensemble
$\Sigma(\xb,t)$. 

More specifically, let us focus on the common distribution function 
$P(\xb,\ub;t)$ of
\BEA
\xb=(x_1,...,x_N)\quad {\rm and}\quad \ub=(u_1,...,u_N),
\EEA
\BEA
P(\xb,\ub;t)=P(\xb,t)\prod_{j=1}^N\delta(\,u_j-u_j(\xb,t)\,),
\label{darder}
\EEA
where $u_j(\xb,t)$ is defined by (\ref{babek}).

One has after averaging over $P(\xb,\ub;t)$
\BEA
\label{toto}
\langle u_j\rangle=
\int \d u_j\,\d \xb\, u_j\,P(\xb,t)\,
\delta(\,u_j-u_j(\xb,t)\,)=-T_j\int \d \xb\,
\partial_{x_j}P(\xb,t)
=0,
\EEA
\BEA
\label{gogo}
\langle\,(\,x_k-\langle x_k\rangle\,)\,(\,u_j-\langle u_j\rangle\,)\rangle
=\int \d \xb\, u_j(\xb,t)\,P(\xb,t)\,(x_k-\langle x_k\rangle)
=-T_j\int \d \xb\,(x_k-\langle x_k\rangle)\,\partial_{x_j}P(\xb,t)
=T_j\delta_{kj}.
\EEA
The fact of $\langle u_j\rangle=0$ is natural, since $u_j$ is the
(coarse-grained) velocity difference generated due to the interaction
to the bath.  It is also seen from (\ref{gogo}) that $u_j$ of the
corresponding brownian particle correlates only with its own
coordinate. This is related to the independence of the thermal baths
which act on different brownian particles.

Applying to (\ref{gogo}) the standard Cauchy-Schwartz inequality one
deduces:
\BEA
\langle\,(\,x_k-\langle x_k\rangle\,)^2\,\rangle\,
\langle\,(\,u_j-\langle u_j\rangle\,)^2\,\rangle\,\geq
|\langle\,(\,x_k-\langle x_k\rangle\,)\,(\,u_j-\langle u_j\rangle\,)\rangle|^2
=T^2_j\delta_{kj},
\label{sq}
\EEA 
which implies an uncertainty relation 
between the uncertainty of the coordinate 
and that of the osmotic velocity 
\cite{ur}.

Obviously, the regular counterpart $f_j(x,t)$ of the coarse-grained velocity
does not (and should not) have any uncertainty relation with the coordinate.

\subsection{Discussion.}

Let us discuss in more detail the difference between the physical
meaning of the brownian uncertainty relation (\ref{sq}) as compared to the
standard quantum mechanical uncertainty relation (\ref{dub1}). The
latter refers to two identical quantum ensembles $\E(\,\hrho(t)\,)$
such that on the first (second) ensemble one measures $\hat{x}$
($\hat{p}$), and then combines into the single relation (\ref{dub1})
the obtained dispersions. Note that the frequent interpretation of
(\ref{dub1}) as referring to disturbance of the momentum when measuring
the coordinate (and {\it vice versa}) is in general incorrect
\cite{willi,ozawa}, since, in particular, the quantities $\langle
\Delta\hat{x}^2\rangle$ and $\langle
\Delta\hat{p}^2\rangle$ clearly refer to different measurements done on
different ensembles, rather than to a simultaneous measurement of the
coordinate and momentum done on a single ensemble. More refined arguments
and explicit counterexamples for this point can be found in 
\cite{willi,ozawa}.

In contrast to this, the quantities entering into Eq.~(\ref{sq}) refer
to consecutive measurements done on the same ensemble.  The involved
coarse-grained velocity is operationally well-defined and characterizes
the change of the coordinate.  This is contrasting to the situation with
the quantum mechanical observable of the momentum which (in the
Heisenberg representation ) in general does {\it not fully} characterize the
change of the coordinate for any initial state; see Footnote \ref{get}
for more details. 



\section{Brownian entanglement.}

Consider now two brownian particles.  An analog of entanglement can be
introduced in the following way.  Call the state of two particles
non-entangled (separable) if the common distribution function
(\ref{darder}) of the coordinates and the osmotic velocities can be
prepared by mixing non-correlated distributions
\footnote{We consider only the common distribution function
of the coordinates and the osmotic velocities, and not, e.g.,  the
common distribution of the coordinates and forward velocities
$v_{+,j}(\xb,t)$, simply because due to the brownian uncertainty
relation (\ref{gogo}) only the former can lead to a non-trivial
definition of entanglement.  Recall in this context that the quantum
uncertainty relations are necessary for the existence of the quantum
entanglement.}
\BEA
P(x_1,u_1,x_2,u_2)=\int \d \lambda\,
{\cal P}(\lambda)\,
P_1(x_1,u_1|\lambda)\,P_2(x_2,u_2|\lambda),
\qquad {\cal P}(\lambda)\geq 0,\qquad
\int\d \lambda\,{\cal P}(\lambda)=1.
\label{tartar}
\EEA

We shall naturally require that the separate distributions
$P_1(x_1,u_1|\lambda)$ and $P_2(x_2,u_2|\lambda)$ ensure the basic
properties (\ref{toto}, \ref{gogo}) of the osmotic velocity $u_j$. For
the rest they can be completely arbitrary.  The requirements
(\ref{toto}, \ref{gogo}) make the basic difference as compared to the
(naive) classical discussion in section \ref{claa}, where any
distribution was admissible.  More detailed discussion on the physical
meaning of (\ref{tartar}), and, in particular, on its similarities and
differences with the definition (\ref{1}) of quantum entanglement, is
postponed till section \ref{samson}.

For simplicity we choose units such that $x$ and $u$ have the same
dimension as $\sqrt{T}$ (e.g., we measure time in units of square root
of the damping constant; see Appendix). We take also $T_1=T_2=T$ again
for simplicity. In the same way as we derived (\ref{dub1},
\ref{buenosaires}), one can write the brownian uncertainty relation
(\ref{sq}) as
\BEA
\langle\,\Delta u_1^2\,\rangle+\langle\,\Delta x_1^2\,\rangle\geq 2T,
\qquad
\langle\,\Delta u_2^2\,\rangle+\langle\,\Delta x_2^2\,\rangle\geq 2T,
\EEA
and proceed to derive
\BEA
\label{dodo}
\langle\,(\,\Delta u_1+\Delta u_2\,)^2\,\rangle+
\langle\,(\,\Delta x_1-\Delta x_2\,)^2\,\rangle< 4T,
\EEA
as a sufficient condition for the entanglement. More general relations 
(\ref{lapas}) can also be obviously
transferred to the brownian situation:
\BEA
\label{dodo1}
\langle\,(\,\Delta u_1+\zeta \Delta u_2\,)^2\,\rangle+
\langle\,(\,\Delta x_1+\epsilon \Delta x_2\,)^2\,\rangle< 4T,
\EEA
where $\zeta$ and $\epsilon$ can independently assume values
$\pm 1$.


\subsection{Gaussian state of two interacting brownian particles.}
\label{utkonos}

It will be useful to work out a simple model for two coupled brownian particle,
where the above general concepts can be visualized and studied in detail.

Consider now two harmonically interacting brownian particles
with an overall potential energy
\BEA
U(x_1,x_2)=\frac{ax^2_1}{2}+\frac{ax^2_2}{2}+gx_1x_2,
\label{po}
\EEA
where $a>0$ and $g$ characterizes the interaction between the
particles \footnote{Note that the positivity of 
$U(x_1,x_2)$ is implied by $g^2<a^2$.}. 
The particles interact with independent bath at the same temperature
$T$. 


Fokker-Planck equation (\ref{d1})
for the dynamics of these brownian particles
can be solved directly and the outcome is known to be 
given by the following two-dimensional gaussian
(provided the initial distribution was gaussian):
\BEA
\label{zarzand}
P(x_1,x_2;t)
=&&\frac{\sqrt{{\rm det}\,C}}{2\pi}\,
\exp\left[-\half \sum_{i,j=1}^2C_{i,j}x_ix_j
\right],\nonumber\\
C^{-1}=&&
\left(\begin{array}{rr}
\langle x_1^2(t)\rangle~~&~~\langle x_1(t)x_2(t)\rangle \\
\\
\langle x_1(t)x_2(t)\rangle&~~\langle x_2^2(t)\rangle~~ \\
\end{array}\right)\equiv 
\left(\begin{array}{rr}
\sigma_{11}(t)~~&~~\sigma_{12}(t) \\
\\
\sigma_{12}(t)~~&~~\sigma_{22}(t) \\
\end{array}\right),
\EEA
where for simplicity we assumed that 
\BEA
\langle x_1(t)\rangle=\langle x_2(t)\rangle=0,
\EEA
for all times. It is also useful to note the marginal distributions
of $P(x_1,x_2;t)$:
\BEA
\label{taratuj}
P_j(x_j;t)=\exp\left[
-\frac{x_j^2}{2\sigma_{jj}(t)}
\right],\qquad j=1,2.
\EEA

The easiest way to obtain $\sigma_{11}(t)$, $\sigma_{22}(t)$ and
$\sigma_{12}(t)$ is to look directly at the Langevin equations
(\ref{langevin}) with the potential (\ref{po}):
\BEA
\label{lacot}
\dot{x}_1(t)+ax_1(t)+gx_2(t)=\eta_1(t),
\qquad
\dot{x}_2(t)+ax_2(t)+gx_1(t)=\eta_2(t),
\EEA
which can be more conveniently re-written in terms of the relative
coordinate $r_-=(x_1-x_2)/2$ and the center of mass coordinate 
$r_+=(x_1+x_2)/2$:
\BEA
\dot{r}_{\pm}(t)=-(a\pm g)r_\pm(t)+\frac{1}{2}(\eta_1(t)\pm\eta_2(t)),
\EEA
and then solved directly:
\BEA
\label{boa}
r_\pm(t)=e^{-(a\pm g)t}r_\pm(0)+\frac{1}{2}\int_0^t\d s\,e^{-(a\pm g)s}
(\eta_1(t-s)\pm \eta_2(t-s)).
\EEA

Eqs.~(\ref{babek}, \ref{zarzand}) 
produce for the coarse-grained velocity difference
(osmotic velocity) of the first and the second particle, respectively,
\BEA
\label{bakht}
u_1(\xb,t)=\frac{T\,(\sigma_{22}x_1-\sigma_{12}x_2)}
{\sigma_{11}\sigma_{22}-\sigma^2_{12}},\qquad
u_2(\xb,t)=\frac{T\,(\sigma_{11}x_2-\sigma_{12}x_1)}
{\sigma_{11}\sigma_{22}-\sigma^2_{12}}.
\EEA

The sufficient condition (\ref{dodo1}) reduces to
\BEA
\langle\,(\,\Delta u_1+\zeta \Delta u_2\,)^2\,\rangle+
\langle\,(\,\Delta x_1+\epsilon \Delta x_2\,)^2\,\rangle=
\frac{T^2\,(\sigma_{22}+\sigma_{11}- 2\zeta\sigma_{12})}
{\sigma_{11}\sigma_{22}-\sigma^2_{12}}+
\sigma_{22}+\sigma_{11}+ 2\epsilon\sigma_{12}
<4T.
\label{dard}
\EEA
A simpler expression is gotten for 
$\sigma_{11}=\sigma_{22}$~
\footnote{This equality will be satisfied for two identical brownian
particles if their initial conditions are the same.}:
\BEA
\label{burunduk}
\frac{T^2}{\sigma_{11}+\zeta\sigma_{12}}+\sigma_{11}
+\epsilon\sigma_{12}<2T.
\EEA

This condition is naturally not satisfied for $\sigma_{12}=0$, as the
minimal value of $\sigma_{11}+T^2/\sigma_{11}$ over $\sigma_{11}$ is
just equal to $2T$. For the correlated situation $\sigma_{12}>0$ and
we apply (\ref{burunduk}) for $\zeta=1$, $\epsilon=-1$, while for the
anti-correlated situation $\sigma_{12}<0$ we use in the same way
(\ref{burunduk}) for $\zeta=-1$, $\epsilon=1$. With these we get from
(\ref{burunduk}) the following condition
\BEA
(\sigma_{11}-T)^2<\sigma_{12}^2+2T|\sigma_{12}|.
\label{konkord}
\EEA
It is seen that this sufficient condition for the brownian
entanglement is symmetric with respect to correlation and
anticorrelation, and it can be satisfied if $\sigma_{11}$ is
sufficiently close to $T$ and $|\sigma_{12}|$ is finite.
Eq.~(\ref{konkord}) can be also satisfied when the particles are
either correlated or anti-correlated sufficiently strongly, e.g., when
$|\sigma_{12}|$ is sufficiently large and sufficiently close to its
upper bound $\sigma_{11}$. Note that (\ref{konkord}) is not satisfied
for $T\to 0$. This is natural, since there are no brownian uncertainty
relations in this limit.

Thus the brownian entanglement can exist for sufficiently strongly
fluctuating and/or sufficiently strongly interacting brownian particles. 

Let us check (\ref{burunduk}) more specifically with the stationary
(equilibrium) state of the brownian particles
which is established for long times provided the potential energy 
$U(x_1,x_2)$ is (strictly) positive, and where 
\BEA
\sigma_{11}=\sigma_{22}=\frac{Ta}{a^2-g^2},\qquad
\sigma_{12}=-\frac{Tg}{a^2-g^2}.
\EEA
These relations can be obtained either by directly solving (\ref{boa})
and then taking the limit $t\to \infty$, or directly via the stationary
distribution for the particles which is known to be Gibbsian: 
$P(x_1,x_2)\propto \exp[-U(x_1,x_2)/T]$.

Eq.~(\ref{konkord}) 
reduces to \footnote{Conditions (\ref{ka1}) are compatible with the stability
requirement $a>|g|$, which comes from demanding positivity of $U(x_1,x_2)$.
Indeed, once $|g|$ satisfies (\ref{ka1}), $a>|g|$
becomes $a(2+a)>(a-1)^2$ and is satisfied provided $a$ is not very small. }
\BEA
\label{ka1}
|g|>-1+\sqrt{1+(a-1)^2}.
\EEA
Thus if $|g|$ is large enough, 
there can be entanglement in the equilibrium state.



\subsection{Entanglement for two non-interacting brownian particles.}

In the quantum case two subsystems can be entangled even if they
interacted in the past, and they do not interact at the moment when
the entanglement is tested. This is also the case with the brownian
particles, as we show now. 

Return to
the above system of two harmonically interacting brownian
particles and assume that they did interact for $t<0$ but are not
interacting for positive times\footnote{
To avoid possible misunderstanding, recall that
the thermal baths acting on two particles were taken to be completely
independent of each other, so that there is no influence
via them. }. One has from (\ref{lacot})
(or from (\ref{boa})~):
\BEA
\sigma_{jj}(t)=e^{-2at}\sigma_{jj}(0)+\frac{T}{a}(1-e^{-2at}),\qquad
\sigma_{12}(t)=e^{-2at}\sigma_{12}(0),
\qquad j=1,2.
\EEA

We assume that the condition (\ref{burunduk}) (with
$\sigma_{11}=\sigma_{22}$ at all times) was satisfied at the initial
time, so that the brownian entanglement was present.  For positive
times and $a>0$, $\sigma_{12}(t)$ gradually disappears. This means
that for long times the sufficient condition (\ref{burunduk}) for the
brownian entanglement will not be valid.  Clearly, no brownian
entanglement can persists for very long times, since the particles
become in this limit completely non-correlated, i.e., the common
probaility distribution for $t\to\infty$ factorizes
$P(x_1,x_2,t)=P(x_1,t)P(x_2,t)$ (compare with our discussion after
(\ref{babek})).

It is however to be stressed that for certain not very long times we
shall surely have the condition (\ref{burunduk}) satisfied, at least
once it was satisfied initially.  Moreover, if even this condition was
not satisfied at $t=0$, it can curiously get valid for some (not very
long) times.  Here is an example.  For free brownian particles with
$a=0$, the correlation $\sigma_{12}(t)$ does not change in time at all,
but instead the dispersion of the coordinates increases linearly with
time: $\sigma_{jj}(t)=\sigma_{jj}(0)+2Tt$.  Condition (\ref{konkord})
for the presence of entanglement now reads:
\BEA
\label{geliogabal}
|\sigma_{11}(0)-T+2Tt|<\sqrt{\sigma_{12}^2(0)+2T|\sigma_{12}(0)|}.
\EEA
Even if this condition was not valid at $t=0$,
$|\sigma_{11}(0)-T|>\sqrt{\sigma_{12}^2(0)+2T|\sigma_{12}(0)|}$,
and additionally $\sigma_{11}(0)<T$, Eq.~(\ref{geliogabal})
can still be satisfied for 
\BEA
t_-<t<t_+,
\EEA
where
\BEA
t_\pm=\frac{T-\sigma_{11}(0)\pm\sqrt{\sigma^2_{12}+2T|\sigma_{12}|}}{2T}.
\EEA

Our general conclusion is that the brownian entanglement is
possible if the two brownian particles do interact or had interacted
strongly enough. The absence of interactions at present need not always
to destroy the brownian entanglement, for some finite times it may even
facilitate the sufficient conditions for its existence, as we saw above. 


\section{Local osmotic velocities.}
\label{artashir}

In section \ref{III} we have seen that the definition of the coarse-grained
velocities (\ref{dish1}, \ref{dish2}, \ref{babek}) is given 
via the ensemble $\Sigma(x_1,x_2,t)$
in the common context of the two brownian particles. 

Let us now turn to the local velocities determined
with respect to the ensemble $\Sigma_1(x_1,t)$, which is obtained
by measuring at time $t$ only the coordinate $x_1$ of the first particle.

There are two completely equivalent ways for
determining the coarse-grained velocities over this ensemble. The
first way amounts to repeating definitions (\ref{dish1}, \ref{dish2})
for the ensemble $\Sigma_1(x_1,t)$
\BEA
\nu_{\pm,1}(x_1,t)=\pm \,{\rm lim}_{\ep\to +0}
\,\int\d y_1\,\frac{y_1-x_1}{\ep}\,P(y_1,t\pm \ep|x_1,t),
\label{papakan}
\EEA
where all the involved probability distributions contain no references
on the second brownian particle. The corresponding osmotic
velocity reads in complete analogy to (\ref{babek}):
\BEA
\label{ga1}
\mu_1(x_1,t)=\nu_{-,1}(x_1,t)-\nu_{+,1}(x_1,t).
\EEA

The second way is to note that if observer 1 has
no information at all from observer 2, then he effectively sums the ensemble
$\Sigma(x_1,x_2,t)$ over all possible results of 
the second coordinate $x_2$ at time $t$ with
his result $x_1$ being fixed. Each value of $x_2$ during this summation
is then met with probability $P(x_2,t|x_1,t)$, and this results in
\BEA
\label{ga2}
\nu_{\pm,1}(x_1,t)=\int\d x_2\,v_{\pm,1}(x_1,x_2,t)\,P(x_2,t|x_1,t).
\EEA
The equivalence between definitions (\ref{papakan}) and (\ref{ga2})
can be established via (\ref{shun}, \ref{katu}) and the Bayes formula
\footnote{Note that for obtaining the local 
coarse-grained velocities $\nu_{\pm,1}(x_1,t)$ we had to average
$v_{\pm,1}(x_1,x_2,t)$ over the conditional distribution
$P(x_2,t|x_1,t)$, and {\it not} over the unconstrained probability of
the second coordinate $P(x_2,t)$. This in-equivalence can be illustrated
with help of the following fact of probability theory concerning three
random variables $A,B,C$:
$\sum_CP(C)\,P(A|BC)\not=\sum_CP(C|B)\,P(A|BC)=P(A|B)$.  
Would we adopt the second possibility, e.g.,
$\tilde{u}(x_1,t)=
\int\d x_2\,u(x_1,x_2,t)\,P(x_2,t)$, this
would lead us to an explicitly common-context (non-local) quantity,
e.g., for the above example of harmonic oscillators we would get
$\tilde{u}(x_1,t)=\frac{T\sigma_{22}(t)x_1}
{\sigma_{11}(t)\sigma_{22}(t)-\sigma^2_{12}(t)}$, which explicitly
depends on the dispersion $\sigma_{22}(t)$ of the second particle. If
the two particles do not interact for positive times, but did interact
in the past (for $t<0$), $\tilde{u}(x_1,t)$ can still be influenced,
e.g., by external fields which apply on the second particle for
positive times (apparent or false non-locality).  }.

Completely similar definitions can be given for the second particle,
employing the ensemble $\Sigma(x_2)$ obtained by measuring the second
particle's coordinate only.  For the corresponding osmotic component one
now has from (\ref{ga1}, \ref{ga2}, \ref{babek}):
\BEA
\mu_j(x_j,t)
=-T_j\partial_{x_j}P(x_j,t), \qquad j=1,2.
\EEA

For the brownian particle in the harmonic potential, as described 
e.g. by (\ref{taratuj}), one has
\BEA
\label{barbos}
\mu_j(x_j,t)=T_j\frac{x_j}{\sigma_{jj}(t)}.
\EEA

If now the definition (\ref{tartar}) of non-entanglement is applied to the
common distribution function
\BEA
P(x_1,\mu_1,x_2,\mu_2)=
P_1(x_1)\,\delta(\,\mu_1-\mu_1(x_1)\,)\,
P_2(x_2)\,\delta(\,\mu_2-\mu_2(x_2)\,),
\EEA
then it is seen to be satisfied trivially: No entanglement occurs
with locally defined osmotic velocities


\section{Discussion.}
\label{samson}

Let us compare in more detail the physical meaning of the quantum
mechanical entanglement versus its brownian analog.

\begin{itemize}

\item
In analogy to continuous-variable quantum entanglement, brownian
entanglement is defined as a type of correlation between the
coordinates and the changes of the coarse-grained velocities (osmotic
velocities) of two brownian particles, which is impossible to
reproduce by mixing non-correlated |that is, referring to each particle
separately| distributions.

\item The brownian uncertainty relation (\ref{gogo}) between the
coordinate and the change of the coarse-grained velocity is necessary
for the very existence of the brownian entanglement.  This is again
similar to the quantum situation, where the analogous role of a
necessary condition is being played by the quantum uncertainty
relations.  

\item
For checking quantum entanglement via sufficient condition (\ref{klm})
one needs to make measurements of non-commuting coordinate $\hat{x}_k$
and momentum $\hat{p}_k$ for each quantum particle ($k=1,2$). To this
end one has to have two different ensembles of the particle $\1$ and
$\2$ (compare with our discussion in section \ref{oper}). In contrast,
the brownian entanglement involves only consecutive coordinate
measurements done on a single ensemble of the brownian pairs.

\item
The momentum operator $\hat{p}$ in quantum mechanics |though being
equal in Heisenberg representation to the time-derivative of the
coordinate operator $\hat{p} =m\frac{\d}{\d t}\hat{x}$|
does not in general fully characterize the intuitive notion of ``change of the
coordinate for an infinitesimal time''
~\footnote{\label{get} This is a general
point. A difference $\hat{A}(t)-\hat{A}(0) =\int_0^t\d t\frac{\d}{\d
t}\hat{A}$ of Heisenberg operators does not fully
characterize the change in
time of the observable $\hat{A}$, because there are ensembles
described by (time-independent in the Heisenberg representation)
states $|\psi\rangle\langle\psi|$ for which $(\hat{A}(t)-\hat{A}(0))
|\psi\rangle=0$. This seems to imply that the value of $\hat{A}$ did
not change at all, but this is not correct, since the above eigenvalue
relation may be still compatible |due to non-commutativity
$[\hat{A}(t),\hat{A}(0)]\not =0$| with different statistics of
$\hat{A}(t)$ and $\hat{A}(0)$, e.g.,
$\langle\psi|\hat{A}^3(t)|\psi\rangle
\not =\langle\psi|\hat{A}^3(0)|\psi\rangle$. For more elaborated
discussion and concrete examples see \cite{work}. For unbound
operators, such $\hat{x}$ and $\hat{p}$, the above reasonings may need
to be technically modified, since eigenstates of unbound
operators are not normalizable. In this context it may suffice to require
$(\hat{A}(t)-\hat{A}(0))|\psi\rangle\approx 0 $.  }.

In contrast, the brownian entanglement is about the coordinates and
the coarse-grained velocities which do characterize the change of the
coordinates on the coarse-grained scale of time.

\item Both quantum entanglement and its brownian analog can exist for
subsystems which interacted in the past, but do not interact at the
present.

\item In quantum mechanics the sufficient condition 
(\ref{klm}) for entanglement involves correlations between the
coordinate operators $\hat{x}_1$ and $\hat{x}_2$, and the momentum
operators $\hat{p}_1$, $\hat{p}_2$ of the subsystems $\1$ and $\2$.
Quantum entanglement can only be found after the results got via
measurements on the corresponding subensembles for the subsystems $\1$
and $\2$ are put together.  The situation with the brownian
entanglement is similar, since it also requires correlation
experiments.

\item Here is finally
the main conceptual difference between the quantum
entanglement and its brownian analog.  In quantum mechanics the above
operators of coordinate and momentum pertain to the corresponding
subsystems $\1$ and $\2$ {\it independently} of the full system context.  By
this we mean that all the statistics of, e.g., $\hat{p}_1$ can be
collected via local measurements on the corresponding quantum
subensemble, whether or not this subensemble forms a
part of any larger ensemble.

In contrast, the very definition of the coarse-grained velocities
(\ref{dish1}, \ref{dish2}, \ref{babek}) involves a global (that is,
depending on the two subsystems) ensemble $\Sigma(\xb,t)$. As seen in
section \ref{artashir}, the purely local definition of coarse-grained
velocities can also be given, but there will not be any entanglement
on that level, for the same reason as there is no entanglement in
other classical systems (see section
\ref{claa}).

This conclusion on the main difference is close to the analogous
conclusion of Ref.~\cite{spr}, which discusses similarities between
quantum entanglement and certain correlations in classical optics.

\end{itemize}

\section{Possibilities of experimental realization.}
\label{expo}

One hopes that an experimental verification of the brownian
entanglement is going to be easier than that of its quantum analog:
since Brown's discovery in 1828, various examples of brownian motion
are routinely observed in many systems. On the other hand, the basic
conditions needed for observation of the brownian entanglement amount
to two coupled brownian particles and a resolution of the brownian
motion sufficient for observing the osmotic velocities.  Recall that
on the coarse-grained time scale the osmotic velocities can be
visualized as average kicks got by the brownian particle due to its
interaction with bath particles.

Here we discuss two experimentally studied examples of brownian
motion. The pecularity of these examples is that the brownian motion
can be detected with human eyes (without microscopes).  These are thus
examples of macroscopic brownian motion. Recall that the typical
examples of brownian motion involve much smaller scales: pollen
molecules in water observed originally by Brown had a size $\approx
10^{-3}$ cm, and even smaller lengths are typical for many other
realizations of the brownian motion.


The first experiment \cite{e1} considers a two-dimensional circular
container with elastic walls. Inside of the container there are
motorized balls with a mass of 120 g and a diameter of 8 cm. Each ball
is driven by a battery-powered motor inside, and moves chaotically due
to elastic collisions with the walls of the container and with other
motorized balls. The set of balls (with concentration 50 balls/m$^2$)
models the bath particles. The brownian particles are modelled with a
set of ping-pong balls (with a mass 2 g and diameter 4 cm) which
undergo random collisions with the motorized balls.  The primary
purpose of \cite{e1} was to study polymer statistics, so that the
ping-pong balls were jointed into a long chain by means of
springs. The resulting coupled brownian motion produced a probability
distribution for the end-to-end distance of the ping-pong chain in a
close agreement with the existing theories of two-dimensional
(self-avoiding) random walk.  In this experiment Langevin equation as
such was not tested directly, but there were several indirect reasons
supporting its validity \cite{e1}. This system would fit for observing
the brownian entanglement, because almost all needed ingredients of
this phenomenon are present.

A curious pedagogical analog of the above experiment is the motion of
a child toy called Bumble Ball \cite{e0}: a plastic sphere about 11 cm
in diameter with a symmetrical pattern of rubber knobs extending about
3 cm from its surface. An internal motor rotates an off-axis mass such
that when activated and placed on a hard surface, a Bumble Ball
simulates a random walk \cite{e0}. Joining two such balls with a
spring and placing them into a container creates a toy model which
might provide at least the most rough indications of the brownian
entanglement.

The second experiment \cite{e2} studied a single sphere (a ping-pong
ball of a mass 2.5 g and diameter 4 cm) rolling stochastically in an
upflow of gas under flow speed 280 cm/c. The sphere rolls
stochastically due to the turbulence it generates in the gas stream
(the corresponding Reynolds number is $\approx$40). The study focussed
on the full time-dependent dynamics of the sphere, as captured by
video imaging.  It appears that the dynamics is that of the
underdamped brownian particle in a harmonic potential. This
correspondence was thoroughly checked from various perspectives. When
looking for the brownian entanglement in such a situation, it would be
necessary to go to the overdamped motion regime (e.g., by decreasing
the mass of the sphere and by increasing the flow speed of the stream)
and to add there a second sphere.


\section{Conclusion.}

We have uncovered the phenomenon of brownian entanglement: a
correlation effect between the coordinates and the coarse-grained
velocities of two classical brownian particles, which resembles the
quantum entanglement. In contrast to the latter, which is presently
given a fundamental status, the brownian entanglement |as the very
subject of statistical physics| arises out of coarse-graining
(incomplete description) reasons. In that respect it is similar to
other basic relations of the statistical physics, such as the second
law \cite{work}.  In the present situation the coarse-graining comes
due to the time-scale separation: the evolution of the momenta of the
brownian particles is very fast and cannot be resolved on the
time-scales available to the experiment. The idea of time-scale
separations is one of the most pertinent ones in non-equilibrium
statistical physics. In a qualitative form it appears already in good
textbooks on this subject
\cite{landau,ma}, and has been since then formalized in various
contexts and on various levels of generality
\cite{landauer,kurchan,theo,at,shu}.

Once there is time-scale separation between the (stochastic) motion of
the coordinate and the momentum of the brownian particles, the
operational definition of velocity via the rate of the coordinate
leads to the coarse-grained velocity which is not equal to the real
momentum. The change of the coarse-grained velocity is controlled by
the brownian uncertainty relation. Moreover, the coarse-grained
velocities appear to be contextual random quantities, i.e., they
depend on the concrete setting of the coordinate measurement used to
define them.  These two aspects (uncertainty relations and
contextuality) suffice to define brownian entanglement, similarly to
quantum entanglement, as impossibility to prepare the common
distribution of the coordinates and the coarse-grained velocities of
two brownian particles via mixing locally independent (non-correlated)
distributions referring to the two particles separately. Again in
analogy to quantum entanglement, brownian entanglement can be
witnessed via the uncertainty relations.

In this paper we demonstrated the brownian entanglement on the simplest,
exactly solvable, one-dimensional models of two interacting brownian
particles. It should be however kept in mind that interacting brownian
particles (random walks) are basic for several fields of modern
statistical physics, such as colloids \cite{collo} or polymers
\cite{gros}. One of the most transparent examples from the polymer
science is a DNA macromolecule which consists of two interacting random
walks (strands) \cite{gros}.  As we argued, the experimental observation
of the brownian entanglement might be easier than those of the quantum
entanglement, since there are realizations of macroscopic brownian
motion that are visible with the human eye. 

\acknowledgments 
A.E. A thanks E. Mamasakhlisov for useful discussions.
The work of A.E. A is part of the research programme of the Stichting voor 
Fundamenteel Onderzoek der Materie (FOM, financially supported by 
the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)). 



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\appendix

\section{}

This Appendix has two closely related purposes. First, we shall apply
the definition (\ref{dish1}, \ref{dish2}) of the coarse-grained
velocities in a non-overdamped situation, that is, we shall take
$\eps$ in those definitions much smaller than the characteristic
relaxation time of the (real) momenta. We shall convince ourselves
that the expected answer is obtained, relating the coarse-grained
velocities to the average (real) momentum. In addition, we shall see
that the the coarse-grained velocity difference (osmotic velocity)
(\ref{babek}) disappears in this situation, i.e., the definitions
(\ref{dish1}, \ref{dish2}) become equivalent. Second, based on an
exactly solvable situation, we shall follow in detail to the behavior
of the coarse-grained velocity as a function of $\eps$. For simplicity
we operate only with one brownian particle.

\subsection{}

The evolution of the common probability distribution 
of the coordinate
$x$ and the momentum $p$ of the brownian particle is described by the
following Fokker-Planck-Kramers-Klein equation \cite{risken}
\BEA
\partial_t{\cal P}=-\frac{p}{m}\partial _x{\cal P}+\partial_p
(\frac{\gamma}{m}p+V'(x)+\gamma T\partial_p){\cal P},
\label{t1}
\EEA
where $m$, $\gamma$ and $V(x)$ are mass, damping constant and potential, 
respectively, and where
\BEA
\label{t2}
{\cal P}(x,p,t|x',p',t'),\qquad t\geq t',\qquad
{\cal P}(x,p,t|x',p',t)=\delta(x-x')\,\delta(p-p'),
\EEA
is the conditional probability to move from $(x',p')$ at time $t'$ 
to  $(x,p)$ at time $t$. 
This equation corresponds to the Langevin equation
\BEA
m\ddot{x}=-ax-\gamma\dot{x}+\eta(t),\qquad
\langle\eta(t)\rangle=0,\qquad
\langle\eta(t)\eta(t')\rangle=2\gamma T\delta(t-t'),
\label{t5}
\EEA
where $-\gamma\dot{x}$ is the friction force, and where $\eta(t)$
is the random gaussian white noise.

Applying the definition (\ref{dish1}) of the coarse-grained velocity
and using (\ref{t1},\ref{t2}) one has
\BEA
\label{dish101}
\nu_{+}(x,t)&&={\rm lim}_{\epsil\to +0}
\,\int\d y\,\frac{y-x}{\epsil}\,{\cal P}(y,t+\ep|x,t)\nonumber
\\
&&={\rm lim}_{\epsil\to +0}
\,\int\d y\,\d p\,\d p'\,
\frac{y-x}{\epsil}\,{\cal P}(y,p,t+\ep|x,p',t)\,
{\cal P}(p',t|x,t)\nonumber\\
&&=-\int\d y\,\d p\,(y-x)\,{\cal P}(p,t|x,t)\,
\frac{p}{m}\,\partial_y\delta(x-y)\nonumber\\
&&=\int\d p\,{\cal P}(p,t|x,t)\,
\frac{p}{m},
\EEA
where ${\cal P}(p,t|x,t)$ is the conditional probability for having the
momentum equal to $p$ at time $t$, provided the coordinate was equal to 
$x$ at the same time. 


Likewise,

\BEA
\nu_{-}(x,t)&&={\rm lim}_{\epsil\to +0}
\,\int\d y\,\frac{x-y}{\ep}\,P(y,t-\ep|x,t)\nonumber\\
&&={\rm lim}_{\epsil\to +0}
\,\int\d y\,\d p\,\d p'\,
\frac{x-y}{\epsil}\,
{\cal P}(x,p,t|y,p',t-\ep)\,
\frac{  {\cal P}(y,p',t-\ep)}{{\cal P}(x,t)}\,
\nonumber\\
&&=\int\d y\,\d p\,(y-x)\,
\frac{ {\cal P} (y,p,t) }{ {\cal P}(x,t) }\,
\frac{p}{m}\,\partial_x\delta(x-y)\nonumber\\
&&=\int\d p\,{\cal P}(p,t|x,t)\,
\frac{p}{m}=\nu_{+}(x,t).
\label{dish200}
\EEA

The fact that $\nu_{-}(x,t)=\nu_{+}(x,t)$ can be easily generalized 
to any number of interacting brownian particles. 

Now if there are no correlations between the momentum and coordinate
${\cal P}(p,t|x,t)={\cal P}(p,t)$, then $\nu_{-}(x,t)=\nu_{+}(x,t)$
reduce to the usual average momentum.  This is the case, in
particular, for the deterministic situation, where ${\cal
P}(x,p,t)=\delta(x-x(t)\,)\,\delta(p-p(t)\,)$.

Thus provided $\eps$ has been taken much smaller than any relevant
time-scale related to the coordinate and/or the momentum, the
coarse-grained velocity reduces to the average momentum as it should.
In this situation the osmotic velocity is zero.

\subsection{}

Now let us show on the exactly solvable situation with a harmonic
potential
\BEA
V(x)=\frac{ax^2}{2},
\EEA
that if the characteristic 
relaxation times of the momentum 
\BEA
\tau_p=\frac{m}{\gamma},
\EEA
and the
coordinate 
\BEA
\tau_x=\frac{\gamma}{a},
\EEA
are well separated: 
\BEA
\tau_p\ll\tau_x, 
\EEA
and if additionally
\BEA
\tau_x\gg
\epsil \gg
\tau_p,
\label{dada}
\EEA
one gets back the values for osmotic velocity which were obtained
in the main text by means of the overdamped Fokker-Planck equation
(\ref{d1}).


The case with $V(x)=ax^2/2$ can be solved either directly from 
(\ref{t1}), or using the equivalent Langevin equation (\ref{t5}).
The solution of the latter
|obtained, e.g., via Laplace transformation| reads:
\BEA
\label{t7}
x(t)=x(0)g(t)
+\frac{1}{m}p(0)f(t)+\frac{1}{m}\int_0^{t}\d t'
f(t-t')\eta (t'),
\EEA
where 
\BEA
\label{t77}
f(t)=\frac{e^{-\omega_2t}-e^{-\omega_1t}}{\omega_1-\omega_2},
\qquad
g(t)=\frac{\omega_1e^{-\omega_2t}-
\omega_2e^{-\omega_1t}}{\omega_1-\omega_2},
\EEA
\BEA
\label{brut1}
\omega _{1,2}=\frac{\gamma}{2m}
\left (1\pm \sqrt{1-\frac{4am}{\gamma ^2}}\,\, \right ).
\EEA
For simplicity reasons we shall put $x(0)=p(0)=0$. This can be done, since
these quantities are assumed to be independent of the noise by the very
definition of the considered stochastic process. 

As the process (\ref{t7}) is gaussian, the 
two-time probability distribution of the coordinate
can be written down as:
\BEA
{\cal P}(y,s;x,t)=\frac{\sqrt{d}}{2\pi}
\exp-\frac{1}{2}\left[
A_{11}y^2+A_{22}x^2+2A_{12}xy
\right],
\EEA
where
\begin{gather}
d\equiv\sigma(t,t)\sigma(s,s)-\sigma^2(s,t),\\
\left (\begin{array}{rr}
A_{11} & A_{12}\\
A_{12} & A_{22}
\end{array}\right )^{-1}=
\left (\begin{array}{rr}
\sigma(s,s) & \sigma(s,t)\\
\sigma(s,t) & \sigma(t,t)
\end{array}\right )=
\frac{1}{d}
\left (\begin{array}{rr}
\sigma(t,t) & -\sigma(s,t)\\
-\sigma(s,t) & \sigma(s,s)
\end{array}\right ),
\end{gather}
and where $\sigma(s,t)$ is the correlation function of the coordinate,
\BEA
\sigma(s,t)
=\langle x(s)x(t)\rangle
=\frac{2\gamma T}{m^2}\int_0^s\int_0^t\d t_1\d t_2\,
f(s-t_2)f(t-t_2)\delta(t_1-t_2)
=\frac{2\gamma T}{m^2}\int_0^{{\rm min}(s,t)}\d t'
f(t')f(t'+|s-t|).
\label{t10}
\EEA

Using:
\BEA
\int\d y\,y\,{\cal P}(y,s|x,t)=
\int\d y\,y\,\frac{{\cal P}(y,s;x,t)}{{\cal P}(x,t)}=
-x\,\frac{A_{12}}{A_{11}}=x\,\frac{\sigma(s,t)}{\sigma(t,t)},
\EEA
one gets keeping $\ep$ finite and fixed
\BEA
\label{mob1}
\nu_{+}(x,t)&&=\,\int\d y\,\frac{y-x}{\epsil}\,{\cal P}(y,t+\ep|x,t)
=\frac{x}{\epsil}\,
\left(\frac{\sigma(t+\ep,t)}{\sigma(t,t)}-1\right),\\
\nu_{-}(x,t)&&=
\,\int\d y\,\frac{x-y}{\ep}\,P(y,t-\ep|x,t)
=\frac{x}{\epsil}\,
\left(1-\frac{\sigma(t-\ep,t)}{\sigma(t,t)}\right),
\label{mob2}
\EEA
for any $\epsil\geq 0$ smaller than $t$: $t-\epsil>0$.

If now $\ep$ is smallest time-scale,
\BEA
\nu_{-}(x,t)-\nu_{+}(x,t)
=\frac{2\gamma T}{m^2}\,\frac{x\ep}{\sigma(t,t)}
\left[
\int_0^t\d t'\,\dot{f}^2(t')-f(t)\dot{f}(t)
\right]+{\cal O}(\ep^2)
\EEA
goes to zero with $\epsil\to 0$ as was predicted above.

As seen from (\ref{brut1}), the overdamped limit is realized with the
dimensionless parameter $4am/\gamma^2$ being small:
\BEA
\frac{4am}{\gamma^2}\ll 1,
\EEA
and then the charactersitc relaxation times of the momentum
and the coordinate [obtained from (\ref{t7}, \ref{t77}, \ref{brut1})]
are well-separated
\BEA
\label{esher}
\tau_p=\frac{m}{\gamma}\,\approx \,
\frac{1}{\om_1}\,\ll\, \tau_x=\frac{\gamma}{a}
\,\approx \,\frac{1}{\om_2}.
\EEA
Now if 
\begin{gather}
\label{dd}
\omega_1 t\gg 1,\qquad
\omega_1 s\gg 1,\qquad
\omega_1|s-t|\gg 1,
\end{gather}
one gets from (\ref{t10})
\BEA
\label{kaban}
\sigma(s,t)=
\frac{T}{a}\left(e^{-\omega_2|s-t|}-e^{-\omega_2(t+s)}
\right),
\EEA
which coincides with the corresponding correlator obtained from the
overdamped Fokker-Planck equation directly. Taking now $s=t\pm \ep$ in
(\ref{mob1}, \ref{mob2}) and employing (\ref{kaban}) with the limit
$\ep\to 0$ |which now should be understood in the context of conditions
(\ref{esher}, \ref{dd})| one can get the expressions for
$\nu_{\pm}(x,t)$ which can alternatively be derived from (\ref{shun},
\ref{katu}), if we put there $N=1$ (only one brownian particle),
$f(x)=-ax/\gamma$ (the linear force divided over the friction
constant),
and $P(x,t)\propto \exp[-\frac{x^2}{2\sigma(t,t)}]$ (gaussian
distribution
function).

\end{document}

\subsection{Random-context osmotic velocties and false non-locality.}

Let us now again consider the situation, where the two brownian
particles interacted for $t>0$, but do not interact for positive
times.  In view of this fact, one can think 
there is no influence of one particle on the other, and thus
the local osmotic velocity, e.g. for the first particle, can be defined simply
by averaging the global osmotic velocity over the coordinate of the
second brownian particle, that is, over all possible values of $x_2$
at the time $t>0$:
\BEA
\label{ba1}
\tilde{u}_1(x_1,t)=\int\d x_2\,P_2(x_2,t)\,u_1(x_1,x_2,t).
\EEA
The similar construction can
be obviously designed by interchanging 1 and 2, and will result to:
\BEA
\label{ba2}
\tilde{u}_2(x_2,t)=\int\d x_1\,P_1(x_1,t)\,u_2(x_1,x_2,t).
\EEA

It is seen from (\ref{ba1}, \ref{ba2}) that
\BEA
\tilde{u}_1(x_1,t)=\frac{T\sigma_{22}x_1}
{\sigma_{11}\sigma_{22}-\sigma^2_{12}},\qquad
\tilde{u}_2(x_2,t)=\frac{T\sigma_{11}x_2}
{\sigma_{11}\sigma_{22}-\sigma^2_{12}}.
\EEA


However, $\tilde{u}_{1}(x_1)$ is not equal to the truly local velocity
and $\mu_{1}(x_1)$, as the former one was obtained in the random
common context, while the latter one is defined without any common
context, that is without any connection or communication between the
observers 1 and 2.

This inequivalence can be illustrated with help of the
following fact of probability theory concerning three events $A,B,C$:
\BEA
\sum_CP(C)\,P(A|BC)\not=
\sum_CP(C|B)\,P(A|BC)=P(A|B):
\EEA
random context $C$ s still different as compared to having no context
at all.


It remains to stress that no any non-locality arises when operating with 
the truly local coarse-graned velocities.

\section{}
To see how  one can get incorrect conclusions by insisting on 
separability of non-interacting objects, let us apply independent external
fields on each brownian particle. The total potential energy is now
\BEA
U(x_1,x_2)=\frac{a_1(t)x^2_1}{2}+\frac{a_2(t)x^2_2}{2}+gx_1x_2,
\EEA
and the corresponding variances read from the solution of the
Langevin equation (\ref{langevin}):
\BEA
\label{muftar}
\sigma_{jj}(t)=e^{-2\bar{a}_j(t)}\sigma_{jj}(0)+2T
e^{-\bar{a}_j(t)}\int_0^t\d s\,e^{\bar{a}_j(s)},\qquad
\bar{a}_j(t)=\int_0^t\d s \, a_j(s), \qquad j=1,2,
\EEA
\BEA
\label{kuftar}
\sigma_{12}(t)=e^{-\bar{a}_1(t)-\bar{a}_2(t)}\sigma_{12}(0).
\EEA
It is seen from (\ref{ba1}, \ref{ba2}) that
\BEA
\tilde{u}_1(x_1,t)=\frac{T\sigma_{22}x_1}
{\sigma_{11}\sigma_{22}-\sigma^2_{12}},\qquad
\tilde{u}_2(x_2,t)=\frac{T\sigma_{11}x_2}
{\sigma_{11}\sigma_{22}-\sigma^2_{12}}.
\EEA
Now, e.g., the average square of $\tilde{u}_1(x_1,t)$ 
for the first particle (not interacting with the second one)
does depend on the field $a_2(t)$
exerted on the second particle for $t>0$:
\BEA
\langle \tilde{u}^2_1\rangle=\frac{T^2}{\sigma_{11}}
\left[1-\frac{\sigma^2_{12}}{\sigma_{11}\sigma_{22}}
\right]^{-2},
\EEA
provided the parameter
\BEA
\frac{\sigma^2_{12}(t)}{\sigma_{11}(t)\sigma_{22}(t)}=
\frac{\sigma^2_{12}(0)}{\sigma_{11}(0)\sigma_{22}(0)}\,
\left[
1+\frac{2T}{\sigma_{11}(0)}\int_0^t\d s\, e^{a_1(s)+a_1(t)}
\right]^{-1}\,
\left[
1+\frac{2T}{\sigma_{22}(0)}\int_0^t\d s\, e^{a_2(s)+a_2(t)}
\right]^{-1}
\EEA
is not very close to zero. This is a ``falce non-locality'', since
$\tilde{u}_1$, though refering to the first particle, it is not truly
local quantity, since random context differs from having no context at all.
Put differently, the absence of direct interactions cannot be a reason for
neglecting correlations between the two particles. 

\section{        EMERGENCE}




        Stephen C. Pepper

        Univeristy of California Journal of Philosophy 23 (1926): 241-245.

            Emergence signifies a kind change. There seem to be three
            important kinds of change considered possible in modern
            metaphysical discussion. First, there is chance
            occurrence, the assertion of a cosmic irregularity, an
            occurrence about which no law could be
            stated. Second. there is what we map call a "shift," a
            change in which one characteristic replaces another, the
            sort of change traditionally described as invariable
            succession and when more refined described as a functional
            relation. Thirdly, there is emergence, which is a
            cumulative change, a change in which certain
            characteristics supervene upon other characteristics,
            these characteristics being adequate to explain the
            occurrence on their level. The important points here are
            first, that in discussing emergence we are not discussing
            the possibility of cosmic chance. The emergent
            evolutionists admit a thorough-going regularity in
            nature. And secondly, we are not discussing the legitimacy
            of shifts. These also are admitted. The issue is whether
            in addition to shifts there are emergent changes.

            The theory of emergence involves three propositions:

               1. that there are levels of existence defined in terms
               of degrees of integration;

2. that there are marks which distinguish these levels from one
another over and above the degrees of integration;

3. that it is impossible to deduce the marks of a higher level from
those of a lower level, and perhaps also (though this is not clear)
impossible to deduce marks of a lower level from those of a higher.

            The first proposition, that there are degrees of
            integration in nature, is not controversial. The specific
            issue arises from the second and third propositions. The
            second states that there is cumulative change, the third
            that such change is not predictable.

            What I wish to show is that each of these propositions is
            subject to a dilemma:

               1. either the alleged emergent change is not cumulative
               or it is epiphenomenal;

2. either the alleged emergent change is predictable like any physical
change, or it is epiphenomenal.

            I assume that a theory of wholesale epiphenomenalism is
            metaphysically unsatisfactory. I feel the more justified
            in making this assumption because I have been led to
            understand that the theory of emergent evolution has been
            largely developed as a corrective of mechanistic theories
            with their attendant psycho-physical dualisms and
            epiphenomenalisms.

            Let us now exhibit the first dilemma. There seem to be two
            theories as to the nature of the marks which emerge and
            distinguish levels of integration. One theory states that
            these are qualities like sensory qualities, metaphysical
            simples. Alexander's system contains a theory of emergent
            qualities. The term "quality" is ambiguous and frequently
            made to cover both qualities like Alexander's and laws of
            activity. Bearing this ambiguity in mind, one will find
            that most emergent evolutionists have theories of emergent
            laws -- emergent laws being the other sort of marks
            thought of as emerging.

            Now, a theory of emergent qualities is palpably a theory
            of epiphenomena. It is not so obvious that a theory of
            emergent laws must also be such -- or else cease to be a
            theory of emergence.

            Accurately speaking, we must first observe, laws can not
            emerge. Emergence is supposed to be a cosmic affair and
            laws are descriptions. What emerge are not laws, but what
            the laws describe. This distinction is important, for I
            maintain that the same natural regularities can be
            described by a whole hierarchy of different laws and that
            this ladder of laws has been mistaken by the emergent
            evolutionists for a ladder of cosmic regularities. But I
            do not care to develop this matter here.

            What I wish to show is that all natural regularities are
            shifts and can not be otherwise described. Let us suppose
            a shift at level B is described as a function of four
            variables q, r, s, and t. Let us then suppose that r and s
            constitute an integration giving rise to level C at which
            level a new cosmic regularity emerges that can be
            described as a function of four variables r, s, a, and
            b. r and s must necessarily be variables in this emergent
            law even though they arc variables of level B, because
            they constitute part of the conditions umler which the
            emergent law is possible. Theoretically, to be sure, the
            emergent law may be thought of either as a function of new
            variables or as a new function of C-level variables. But
            actually only the former is possible. For if the new law
            were not f1 (q, r, s, t), but were f2 (q, r, s, t), then,
            of course, it would never be f1 (q, r, s, t), unless the
            event were a chance occurrence in which case no regularity
            could be described anyway. The point is, either f1
            adequately describes the interrelationships of (q, r, s,
            t) or f2 does; or if neither adequately describes the
            interrelationships there is some f3 that does, but there
            can not be two adequate descriptions of the same
            intcrrelationships among the same variables.

            An emergent law must, therefore, involve the emergence of
            new variables. But these new variables either have some
            functional relationship with the rest of the lower level
            variables or they haven't. If they haven't, they are sheer
            epiphenomena, and the view resolves itself into a theory
            of qualitative emergenee. If they have, they have to to be
            included among the total set of variables described by the
            lower level functional relation; they have to drop down
            and take their place among the lower level variables as
            elements in a lower level shift.

            Such being the case, our dilemma is established so far as
            concerns cumulative change -- either there is no such
            thing or it is epiphenomenal. Now for predictability.

            The assumption in theories of emergence is that mechanical
            changes, single level changes (call them shifts) are
            predictable, but that emergent changes are not. Now, it
            must be pointed out at once that in an ultimate sense, no
            changes of any kind are predictable. One event can never
            be deduced from another event. Even the first law of
            motion is not self-evident. From the state of rest or
            motion of an unmolested body at one moment, you can not
            deduce the state of rest or motion of that body at the
            next moment. And what will be the states of two bodies
            that do molest each other is equally not deducible from
            their states previous to molestation. Deduction from the
            states of bodies under certain conditions to their states
            under other conditions is possible only through the
            mediation of a descriptive law summarizing a set of shifts
            in which these bodies and their conditions are
            variables. It follows that events and qualities occurring
            on the same level are no more deducible from one another
            than if they occur on different levels. Cosmic events
            don't deduce or predict one another; they occur. It is
            only we who describe them by laws, who also make
            predictions concerning them by means of our
            laws. Cosmically speaking, nothing is deducible, and
            hypothetical emergent qualities or events would be no more
            peculiar in this respect than the qualities and events
            that occur on the bed rock level.

            It is only humanly speaking that anything is deducible. And what are strictly deducible are neither qualities nor events, but laws. There is a loose sense in which qualities and events are predictable. Thus by a process of interpolation we can and have predicted the properties of missing elements in the periodic table. And by a process of analogy we predict the states of consciousness of other people and of animals. But such loose predictions are not the sort meant by emergent evolutionists. For obviously emergent events and qualities could be predicted in this manner if there were any occasion to predict them. Just such a prediction Alexander makes about God. What the emergent evolutionists mean is that an emergent law could not be deduced from lower level laws as a mathematieal theorem can be deduced from other theorems. The condition under which such deduction is strictly possible is that by substitution of one sort or another, one law may be derived from another law or set of laws. It is a natura ideal of science to derive all laws from a certain limited number of primitive laws or principles -- not necessarily from one single law -- and so to convert science into a mathematics. If it could be assumed that there are no chance occurrences such a system of laws should be obtainable, though it might look very different from the traditional mechanics. The assumption of science appears to be that such a system is obtainable. I do not know what else the dissatisfaction of science with inconsistencies could mean.

            Now, there seems to be no intention on the part of emergent evolutionists to deny that such a system is possible or to assert that there are chance occurrences. If that is so, they seem to be faced with the following dilemma: either the emergent laws they are arguing for are ineffectual and epiphenomenal, or they are effectual and capable of being absorbed into the physical system. But apparently they want their laws to be both effectual and at the same time not part of the physical system. Professor Lovejoy, for instance, contrasts laws of behavior with physico-chemical laws. The laws of behavior imply that "some movements of certain complexes of protons and electrons are, in part, functions of other movements which are to occur afterwards. And this determination of some character of a present motion by some character of a yet unrealized future is a conception obviously foreign to the laws of physics and chemistry . . . and it is, in fact, incongruous with the most fundamental methodological assumptions of those sciences."1 But at the same time he affirms that there is an "assumption that intelligence is capable of being a factor in the control of human actions and thereby of man's physical environment. . . . If that postulate, at least, be not true, universities are absurdities and laboratories are costly monuments of delusion" (p. 208). Now clearly, if these laws of behavior are going to step down out of an epiphenomenal heaven and "be a factor in the control . . . of man's physical environment" they are bound soon to get into conflict with physico-chemical laws as now described. I am here, of course, not interested in whether the present movements of electrons and protons are functions of future electrons or prtons, or not. But if they are, that important relation would have to be stated among the primitive laws of the physical system or would have to be deducible from some of these laws. That is to say, if the laws of behavior enter into the physical system at any point they must constitute either primitive laws in that system or be deducible from the primitive laws. There is no other way out of it. These supposed emergent laws are either epiphenomena or physical laws. For if they represent irreconcilable inconsistencies in the physical system, they are not laws at all, but statements of chance occurrences.

            So, regarding the characteristics of unpredictability, the
            situation is the same as regarding the emergent marks; it
            is a possible charactcristic, but only as an epiphenomenal
            one.

            Notes

            1 Essays in Metaphysics; Univ. of Cal. Publications, V. 5,
            pp. 208-209.

            Commentary: Wilfrid Sellars and Paul Meehl, "The Concept
            of Emergence," in Minnesota Studies in The Philosophy of
            Science, Vol. I: The Foundations of Science and the
            Concepts of Psychology and Psychoanalysis, edited by
            Herbert Feigl and Michael Scriven (Minneapolis: University
            of Minnesota Press, 1956): 239-52.


\section{On Emergence and Explanation}



by

Nils A. Baas and Claus Emmeche

Abstract

Emergence is a universal phenomenon that can be defined mathematically
in a very general way. This is useful for the study of scientifically
legitimate explanations of complex systems, here defined as
hyperstructures. A requirement is that the observation mechanisms are
considered within the general framework. Two notions of emergence are
defined, and specific examples of these are discussed.  1. Introduction:
To explain Living as cognitive beings in a world of stability and
change, we permanently face known and unknown situations, old and new
phenomena - conversations, happenings, shifting perceptual patterns of
the world, but not often true chaos, nor strict regularity. We usually
cope with the complexity of ordinary life in a way so easy and by such
seemingly simple patterns of behaviour - when seen from the higher level
description of ordinary language (or `folk psychology') - that it is
astonishing to realize the underlying complexity of this `problem
solving behaviour' from a scientific point of view, i.e., when we try to
relate descriptions of the body as a complex system and its environment
(as given by physics, biology and cognitive science) to the life and
actions (as experienced) of our daily world.  What does it mean to
understand something? In the sciences, understanding is related to - or
even equated with - the notion of explanation. To claim that a
phenomenon is well understood, some explanation must have been offered
that is sufficiently precise, publicly communicable, and providing us
with some kind of generative mechanism for the phenomenon: If you
understand a particular solution to an abstract mathematical problem,
you can explain this by proving it to be true step by step or by giving
other kinds of logically convincing arguments for this solution. If you
claim fully to understand a phenomenon or pattern in Nature, you are
thought to possess an explanation of it in terms of some mechanism that
can be shown capable to generate - in the explanation - the phenomenon
in question. In the so-called sciences of complexity (e.g., non-linear
dynamics, theoretical biology, complex adaptive systems, artificial
life, artificial intelligence, cognitive science), "complex" phenomena,
such as the appearance of life on Earth, the evolution of new species,
or the structure of cognitive thought, are often considered as instances
of some emergent higher-order structure that may be explained by the
lower-level dynamics generating the collective behaviour or emergent
property of the system in question. Thus the old idea of emergent
evolution (cf. Blitz 1992) has recently attracted much attention and
created new interest in the level-structure of the universe.  Can the
emergence of real new properties in complex systems really be explained?
If the sciences of complexity offer important new insights, theories,
and methodologies for dealing with complex, higher-order phenomena (as
we think they do), and if the traditional view of explanation cannot
account for the explanatory strategies we find here, we should look for
other accounts of scientific explanation. Perhaps the very idea of
scientific explanation as a strictly deductive argument should be
reinterpreted and explanations seen in a more dynamic and
context-dependent setting, eventually themselves being emergent
structures, "emergent explanations". We will show below how this
intuitive idea can be made precise and explicated formally. Questioning
traditional notions of explanation may lead to a more general view of
what constitute genuine scientific understanding of complicated
phenomena. We suggest to pay attention to a new general framework for
description of higher-order structures (hyperstructures) which includes
the mechanisms of observation, and which eventually allows for
self-generation in such systems of new observational frames. New
observers may also emerge in the system (Baas 1996).[1] Within
philosophy of science, a lot of work has been done on giving detailed
accounts of what constitutes an explanation and what are the relations
between explanation and the concepts of reduction, cause, system,
determinism, probabilism, etc. We will not address this work here. We
feel, however, that when it comes to such questions as the very
generation of explanations, the discovery of new structures or
principles, the appearance of new insights to the inquiring mind - i.e.,
the emergence of structures that can be said to explain a phenomenon -
very little has been said. The feeling of having explained a phenomenon
cannot simply reduce to the fulfilment of some algorithmic procedure by
which - given a general law and some initial or boundary conditions -
you can arrive at the phenomenon to be explained by executing this
procedure. Of course, one can argue that it is the very laws of nature
that has the explanatory power in physical science, so any research
programme should attempt to discover such laws. However, in
understanding complex systems, the challenge is often to understand the
consequences of these laws on other levels of organization, i.e., to
understand the history and complexity of the boundary conditions
(compare Kuppers 1992). In a way, understanding a system means creating
some kind of generalized resonance!  Traditional deductive notions of
scientific explanation have frequently been critisized for being too
crude and reductive to account for complex mental phenomena. Hence, in a
somewhat romantic vein, the mathematician G. Spencer Brown (1969)
declared that "To explain, literally to lay out in a plane where
particulars can be readily seen. Thus to place or plan in flat land,
sacrificing other dimensions for the sake of appearance. Thus to expound
or put out at the cost of ignoring the reality or richness of what is so
put out. Thus to take a view away from its prime reality or royalty, or
to gain knowledge and lose the kingdom." We shall present a framework
that may help to circumvent a pure dilemma of reductionism and holism -
of gaining knowledge and retaining the richness of a world of emergent
structures.  2. Emergence and hyperstructures In the recent years the
notion of emergence has been studied extensively, but often without
making precise what emergence means. We will here use emergence in the
general sense defined by Baas (1994a). The crucial point in this
definition is the notion of an observer - in a very general sense, which
makes it very flexible. Let us here just recall the basic idea.  In the
study of complex systems one often sees that a collection of interacting
systems shows collective behaviour. This is intuitively what we
understand by emergence. In order to study the phenomenon further it may
be useful to introduce a more formal framework as follows: Let be a
family of general systems or "agents". Let be "observation" mechanisms
and be interactions between agents.  The observation mechanisms measures
the properties of the agents to be used in the interactions. The
interactions then generates a new kind of structure which is the result
of the interactions. This could be a stable pattern or a dynamically
interacting system. We call an emergent structure which may be subject
to new observational mechanisms . This leads to Definition: The
observational mechanism may be internal or external.  Examples: 1.
Coupling of dynamical systems.  2. Large collections of objects get new
behaviour like in phase transitions.  3. Functionality of biomolecules
in cells. The self-maintenance of a living cell is based on the
structure of the cell and the functionality of its molecules: Though in
general each type of macromolecule in virtue of its chemical properties
(characterizable by chemical analysis) can enter into reactions with an
infinite set of possible molecules, in the living cell each molecular
species is committed to one or a small number of reactions that defines
its specific function within the cell's metabolic system (e.g.,
catalysis of a part of a specific metabolic pathway).  4. The general
situation of a client (C) and a server (S). With the interactive help
from the server the client may perform tasks which none of them could do
separately. Hence we get a second order agent (CS) which again may serve
as a client in a new context (Baas, in press).  5. Consciousness is not
a property of individual neurones, it is a natural emergent property of
the interactions of the neurons in nervous system of the body in an
environment. It makes a structure that is related to lower level
interactions as well as higher level thoughts, and it represents a new
observational mechanism of the entire system (Baas 1996).  General
causes of emergence would be: non-linear interactions, large collections
of limits, open-ended evolution in heterogeneous environments,
context-dependence of properties in complex systems ...  Depending on
the case, the may have elements, or it may be an empty set if for
instance does not or cannot observe anything. E.g., if is some physical
properties of individual H2O molecules, is water, and is viscosity, then
would correspond to "the viscosity of an individual molecule", a
property we cannot measure, so is empty.  We think that the present
framework is quite useful for analysing the nature of emergence - in
particular the dependence on observational mechanisms. In order to put
this into a more mathematical framework category theory is very useful.
We let the systems be represented as objects in a category and the
interactions as morphisms. A complex system with interactions is then
represented by a diagram. Since the morphisms are represented by arrows,
this may be viewed as a process oriented representation.  The
observational mechanism may here be thought of as functors. In
categories we may form direct limits () of diagrams, and in this case
the may be thought as both an observation functor () and a
complexification process ().  For a further discussion of emergence, we
refer to Baas (1994a, 1994b, 1996) and for the use of category theory,
Ehresman & Vanbremeersch (1987, 1994).  As we see it here emergence is
just the same as holism. An emergent structure is a holistic structure.
We should emphasize, that from this refined notion of holism, it does
not follow that `the whole' cannot be analyzed, nor that it is always
impossible to deduce the properties of the whole from its constituents
and the observational mechanisms. Thus, within the general framework
proposed here, one must distinguish between two different kinds of
emergence: A. Deducible or computational emergence. There exists a
deductional or computational process or theory such that can be
determined by from .  B. Observational emergence. If P is an emergent
property, but cannot be deduced as in (A).  (Clearly, further
refinements are possible!). As argued in larger detail in Baas (1994a,
1996), examples of deducible emergence include some compositional
structures in engineering constructions, nonlinear dynamical systems,
phase transitions, nontriviality of complexity of manifolds in topology,
and the Scott model of the [[lambda]]-calculus. In these cases, the
various properties can be decided by well-defined procedures, so can be
seen as instantiating an algorithm leading to a set of properties.
Examples of observational emergence include Godel sentences in a formal
system (the is the truth function; cf. Godel theorem), and the property
of membership of the Mandelbrot set and most Julia sets. Furthermore, it
was indicated that the eventual semantic non-compositionality of a
language would imply that the meaning of sentences in such a language
was observationally emergent.[2] In spite of the existence in
mathematics of a large set of yet undecided statements or conjectures
about mathematical properties that may exemplify either deductible or
observational emergence, it is a general observation that the most
transparent and clear-cut cases of the two kinds of emergence are found
within the logical and mathematical domain. Nevertheless, the
distinction may also have considerable potential value in the empirical
sciences, for instance in the ongoing discussion of reductionism and the
explanatory relations between various theories. To give an example, it
is a sensible intuition that the autonomy of biology in relation to
physical science (as discussed with yet no definite conclusion by for
instance Mayr 1988, Rosenberg 1985, Kincaid 1990) is, so to speak,
grounded in the observational emergence of specific properties of
biosystems, such as the self-reproduction of living cells. For
theoretical biology it is of interest to determine in what sense such
observational emergent properties, as studied by Artificial Life models,
can or cannot be considered as real instantiations of living entities
(compare Pattee 1982, Kampis 1991, Emmeche 1994).  One may go even
further by using the concept of a living system as an autopoietic system
(Maturana and Varela 1980), i.e., a system organized as a bounded
network of processes of production, transformation and destruction of
components that regenerate and realize the very network and its boundary
and thereby constitute it as a unity in this new (autopoietic) space of
relations. This property can, we suggest, be understood as the
observational emergence in the physical space of systems that realize
their own self-production, boundary and self-observation (through the
boundary's distinctive or selective property, which is based on
molecular recognition reactions by the membrane-bound proteins), and
thus being `cognitive' in this primitive sense.[3] If this is true, an
autopoietic system is autonomous because it realizes observational
emergence of itself as an observer.  Hyperstructures are multi-level
emergent structures.  Definition:

A hyperstructure of order is given by

extending the construction in the definition of emergence. This is a cumulative structure, not necessarily purely recursive. For more details, see Baas (1994a, 1994b, 1996).

The client/server situation may easily be extended to a hyperstructure of the -th order clients or servers. Within the living realm, one can observe hyperstructures as realized by multicellular organisms and the community structure of ecosystems. Consciousness may be seen as a hyperstructure of mental representations embodied in the central nervous system and capable of self-observation and self-interaction.

Hyperstructures may be thought of as an organizational scheme or design principle. The role of - which may also be an interacting environment - may also allow for evolution to be incorporated.

The point is to combine the notion of emergence and hierarchy into the
notion of hyperstructure.

Our point of view - as we will try to argue - is that wherever a
problem, a situation, or a structure can be organized into a
hyperstructure, this has an explanatory power.

3. The observer

As already emphasized, the notions of emergence and hyperstructure
depend critically on an observer. We think that this is natural in as
far as phenomena has to be explained through observational facts (even
in the abstract sense of facts).

An example: When we model phenomena in thermodynamics by statistical
mechanics we may view temperature and pressure as emergent properties
and phase transitions as emergent phenomena. These are parts of
collective behaviour which in many cases are well understood in a
reductionistic sense in physics. However, temperature has been
introduced as a formal parameter ad hoc. A deeper level of explanation
- not yet reached - would be to see the laws of thermodynamics derived
from the laws of Hamiltonian mechanics, i.e., such that temperature
would emerge.

More informal examples may be the emergent properties of
hyperstructures of biological and social systems which are deeply
dependent of the existence of observers intrinsic to the system. As
argued above, the single cell with its membrane bound proteins
constitute an observer of some aspects of its immediate environment
(other molecules that can be recognized as signals or nutrition), and
multicellular organisms depend critically on inter-cell signalling. On
a higher level, most if not all institutions of a human society have
self-observing mechanisms (e.g., evaluation and assessment in research
and production), as well as mechanisms for observing other
institutions.

4. Mathematics

One may ask whether a phenomenon like emergence (or holism) can be
made mathematically respectable. Our answer is absolutely yes!

Take for example the ways complicated `surfaces' like manifolds are
being glued together from elementary pieces. New topological and
geometrical properties occur, but may often require sophisticated
"observational" functors like cohomology theories to be detected and
described.

Furthermore, in a knot where is the knottedness? It is a global
property, having no meaning locally.

Or in a Moebius band - where is the twist? Same thing again! (See
Penrose 1995).

Category theory is the mathematical language suited for discussing
structures in general. A category consists of

   1. Objects: X, Y, ...  (like spaces, groups, algebras, ...)
   2. Morphisms: represented by arrows (functions, interactions)

To each pair of objects we have a set of morphisms

Mor( X, Y ).

A category is closed under morphism composition

meaning that if and are morphisms, so is also .

We can form networks of morphisms as commutative diagrams and
construct their categorical limits.

D is an object which is a kind of synthesis of D and which is called
the direct (or co-)limit of D. Limits often play the role of emergent
structures, see Ehresman & Vanbremeersch (1987, 1994).

The limit construction may be viewed as a complexification process
leading to new emergent properties in many cases according to some
observational mechanisms. Specifically, if

F a functor:

say F(Di) = 0 trivial object in , but if F(D) is a non-trivial object
in , we may view F(D) as an emergent structure or property of the
diagram. Again: no mystery.

5. Mathematical reductionism and non-reductionism

Within the framework of category theory we can beautifully illustrate
the differences between reductionism and non-reductionism. We appeal
here to the figure:

The circles represent diagrams in a category, the horizontal arrows
are morphisms, and the cones are limits (Ehresman & Vanbremeersch
1994).

6. Explanation

Normally we mean by an explanation of something that there is a
deductional procedure, an algorithm, leading us from something basic,
well-known to the new thing. In essence this is reductionistic
explanation as formally illustrated in the figure 2. An observer just
singles out in a situation what is to be explained. Therefore
deductive emergence is basically a kind of traditional reductionistic
explanation, but with an observer as an additional enrichment (if not
wanted, it can just be ignored!).

Non-deducible or observational emergence is more subtle. In this case
there exists no algorithmic explanation. As we see in the figure in
the case of limits a two level structure is needed to "explain" a
structure. It cannot be explained or understood only from the
primitive level. At level 2 new phenomena emerge (in this case caused
by the composition properties in categories), which are needed at the
next level. Even if there is no algorithm from the bottom up, still we
would claim that this construction or figure is a perfectly good
explanation - a second order emergent explanation ( may be thought of
both as a construction and as an observation).

The iterated limit construction is just a special case of a
hyperstructure, which is a multilevel emergent structure. Therefore we
will say that we have higher order - -th order explanation of a
phenomenon- wherever it occurs as a result of a suitable -th order
structure.

In our discussion of hyperstructure we have suppressed time, which should be added in a dynamical picture. But the basic ideas remain the same.

7. Emergent deduction (or logic)

In Baas (1996) the notion of emergent deduction was introduced. We may
think of the scheme -

meaning that is an emergent structure - as an inference rule. A
property observed in , but not in the could then be said to have an
emergent explanation. This extends to more levels and hyperstructures
in general, and could lead to a dynamical process logic which we would
call emergent deduction, which would be useful in order to explain and
reason about emergent phenomena.

The important - and new - point is that emergent deduction is not only
a syntactic process, but combines syntax and semantics in the
reasoning and explanation. After all this is how much human reasoning
occurs - for example mathematical thinking.

8. Emergent laws

In science we often search for fundamental laws that cannot be reduced
as logical consequences of something else. But just as important as
fundamental laws are it is vital to understand their consequences or
what may be called emergent laws. These would be laws (or in general:
organizational and dynamical principles) that emerge from the
regularities in collective behaviour of structures - the laws that
emergent structures obey. A striking example would be
superconductivity. At the biological level, it could be natural
selection of genetically based self-reproducing entities.

In forming hyperstructures there will be several levels of emergent
laws, and they will provide explanatory tools in the sense of emergent
explanation (see also Wilczek 1993).

9. Further discussion of the idea

Emergence and supervenience:

The attempt presented here to make precise the notion of emergence can
be compared to a quite different though related approach, namely
efforts in analytical philosophy to explicate notions of emergence and
nonreducibility by refining the idea of supervenience (e.g., Teller
1984, Rosenberg 1985, Sober 1993, Savellos and Yalcin 1995). The basic
idea is, that an emergent property might be one that supervenes on,
without necessarily being definable in terms of (or reducible to) a
physical base, even though physical facts somehow fix or determine
biological and psychological facts. For example, one can (as Davidson
1970) deny that there are strict psychophysical laws even though
mental characteristics are in some sense dependent, or supervenient,
on physical characteristics. This means that there cannot be two
events alike in all physical respects but differing in some mental
respects, or that an object cannot alter in some mental respects
without altering in some physical respects. Recently Kim (1994) has
questioned if supervenient dependence signify a special type of
dependence relation, because the existence of a variety of ways in
which one could explain why the supervenience relation holds in a
given case. If so, it cannot constitute an explanatory account of the
mind-body problem, and one may think that (just as with some varieties
of the emergence thesis) it states the problem rather than offers a
solution to it. Kim suggests mereological supervenience - the
dependence of the properties of the whole on the properties and
relations characterizing its proper parts - to be a more promising
notion if one wants to explain psychological properties as higher
order properties of a whole organism that covary with its lower order
properties.[4] We suggest that mereological supervenience should be
understood as a special case of observational emergence, and suggest
the framework presented above as a useful tool to develop, within the
philosophy of mind, this concept further.

Explanation and insight:

On the subjective level, it is a common experience of mathematicians
to have sudden "flashes of insights". They involve the emergence in
consciousness of a possible solution to a hard problem that for long
have occupied the working mind (its conscious as well as subconscious
parts). The sudden "flash" is experienced as the appearance of the
solution as a new structure, that in a sense can be observed at once,
but nevertheless has to be worked out deductively in great detail and
tested formally before it can be trusted. This experience has been
vividly described by Poincare (1956). The appearance of a new idea or
a new orderly structure may often be similar to the detection of a
pattern - or a special "short description", rule or algorithm, that
may generate the more complex situation that was the starting point
(compare Chaitin 1987; Hofstadter 1995). Of course, such experiences
are not confined to mathematicians; a very common feature of
consciousness in general seems to be the property of "owerflow" (as
described by Searle 1992) in which whole and complex situations
experienced in personal life suddenly are seen in a new light, from a
new point of view as it were, as if a higher order observational
mechanism has accessed properties of new structures as generated
through the interactions of previous experiences. We propose that the
characteristic phenomenon that conscious states refer beyond their
immediate content (as seen in the owerflow phenomenon, in the
so-called "Aha! experience", and in the well-known Gestalt shifts
between figure and ground) is deeply related to the hyperstructured
constitution of consciousness. (See also Scott 1996).

The emergent mind explained?

Let us finally state some suggestions in order to put together what we
have said about explanation, emergent laws and consciousness. To
explain the mind without losing the very phenomenon, the explanation
must not, to quote Spencer Brown again, "take a view away from its
prime reality". The traditional conflict between objective
explanations of consciousness (the attempt to discover its lawlike
relations with other things) and subjective understanding of
consciousness (i.e., of its intrinsic qualities, phenomenal aspects,
"what it is like", etc.) may be transcended if we integrate the
following ideas into the general framework of hyperstructures:

(i) If laws of nature are a kind of algorithms giving us highly compressed descriptions of the phenomena we can observe in nature (Chaitin 1987), the existence of observationally emergent properties (that cannot be computed from the lower level algorithms) shows the necessity of reformulating the notion of lawful explanation so as to encompass the historical generation of hyperstructures.

(ii) Any scientific explanation has objective as well as subjective
aspects. As objective, the explanation can be publicly stated,
explicitly and precisely (eventually in a formal mode), it can be
critically discussed (and eventually tested), and it can generate
adequate representations of the phenomenon to be explained
("explanatory power"). As subjective, the explanation must be
understood and make sense to a competent observer; it must be fully or
partly accepted as a genuine explanation (i.e., as one that generates
the phenomenon); and it must thus mentally reflect the emergent
structure of the phenomenon to be explained.

(iii) Not all of reality is objective; some of it is subjective and
conscious (though still a completely natural phenomenon). For the
objective reality, one can often maintain the traditional view of
observation as something in which the observer is distinct from what
is observed. For some aspects of the subjective reality, this model of
observation cannot always be maintained.

(iv) Hyperstructures can be found existing in objective as well as in
subjective reality.

(v) The methodology of a science of consciousness must concern itself not only with objectively observable behaviour (from the third-person perspective), but also with what has been termed the "endo-view" (Kampis 1991), or the first-person point of view (Searle 1992), or what we will call the observer observing the emergence of new structures from within the system, as exemplified in mathematical discovery or in other kinds of sudden flashes of insight.

(vi) In certain living systems with mind, at a high level new
phenomena such as consciousness emerge as a construction process, in
which and occur at several levels, and in which (at least some)
observation mechanisms are intrinsic to the system. Even though there
is probably no algorithm from the bottom up, such a self-construction
is a perfectly good explanation - a high order emergent explanation,
open for further analysis.

The approach presented here does not view the mind as
self-transparent, nor does it invoke self-introspection as a
privileged observation mechanism; rather, it relaxes the exaggerated
expectations and quixotic hopes for some completely different sorts of
explanations for conscious phenomena. Even if some aspects of
consciousness, such as qualia, may constitute a particularly hard
problem, other aspects may be explained within a refined framework of
emergent hyperstructures. Without ignoring the richness of the
phenomena of life and consciousness, we can still continue to work out
better explanations.

Notes

[1] The problem of self-generation of new observational frames is discussed from a similar though different point of view in Kampis (1991).

[2] In formal string systems, compositionality can be defined as the
condition that there exists a computable function F such that M(s) =
F(M(s1), ..., M(sk)), where the s's are the syntactical relations and
M is a semantic "meaning function". If we take Obs as our M, then
compositionality amounts to deducible emergent meaning, whereas
non-compositionality would imply the existence of observationally
emergent meaning. As opposed to formal systems, the meaning of
sentences in natural language are often context-dependent, i.e., the
meaning of a sentence (such as "He saw the light") is not simply
deducible from the meaning of its components, but dependent of the
larger discourse in which it is placed. Thus, observationally emergent
meaning constitutes a problem for the semantic analysis in the Tarski
tradition (which is set theoretical, truth functional and
compositional), as well as for classical AI that follows `the
formalist motto' of deducing semantics from syntax (cf. Haugeland
1985): "If you [or the AI system] take care of the syntax, the
semantics will take care of itself".

[3] Cognitive in the sense of Maturana and Varela (1980) where a
cognitive domain is the entire domain of all interactions in which an
autopoietic system (an organism) can enter without loss if
identity. We think that the framework proposed in this paper allow for
an interpretation of the theory of autopoiesis that adequately
integrates evolution and emergence of autonomy into the theory.

[4] However, this will probably not solve "the hard problem" (Chalmers
1996) of phenomenological experience and qualia.  






\section{Miscellaneous}

In this paper I argue that -- contrary to the standard notion where
levels of description are usually seen as simply corresponding to
different degrees of generality or detail relative to an absolute
`bottom level' of `full causal description' -- there are in fact at
least five distinct ways of understanding and using a hierarchy of
levels of description, and, connectedly but separately, that the
`irreducibility' or `ineliminability' of higher levels of description
to lower ones comes in correspondingly different flavours.

The five kinds of hierarchies of levels of description which I
describe are:

1. Epistemologically Convenient Levels: levels which are of only
pragmatic, contingent use and which are not part of a full physical
description of the universe.

2. Modelling Levels: levels which take advantage of some model's
capacity to explain or describe the phenomena in a way not feasible at
lower levels.

3. Set Properties: levels motivated by the logical fact that sets can
have properties which are not possessed by their members.


4. Levels Motivated by the Pragmatics of Explanation: levels used to
secure the virtues of a good explanation, such as brevity, salience
and understandability.

5. Levels Motivated by Ontologically Emergent Properties: levels which
may be required to take account of ontologically significant
properties which simply do not exist at lower levels.

The interest here is three fold. First, writers who make fairly heavy
use of the notion of levels of description in dealing with the
phenomenon of consciousness, such as Dennett and Pylyshyn, often
appear to be making the tacit assumption that levels of description
are a fundamentally monolithic phenomenon: that one can speak about,
for example, the role or the reducibility of `levels of description'
simpliciter. I wish to correct that misconception here. Second, having
established the differences among breeds of levels of description, it
becomes important to be reflective about the way levels of
descriptions talk is used in the consciousness debate: disambiguating
levels talk can play a powerful role in correcting some mistakes and
solving some problems in the literature. Third, the last two of the
kinds of hierarchy of levels of description are especially interesting
and perhaps rather under-studied. I argue that the nature of
explanation is such that some higher levels of description are in
principle required if we are to construct `good explanations' of
common phenomena: this is a much stronger account than the standard
`epistemological convenience' view. I also try to make clear what
exactly it would be for a higher level property to be ontologically
irreducible to a lower level: in brief, I argue that it would have to
be the case that there is no relevant similarity or regularity
existing between, and only between, the set of token physical
instantiations of some upper level pattern.


Hierarchies are ordinations, as from smaller to larger, or from
simpler to more complex. Rank is canonical, not optional. In science
related discourses there have been two forms of hierarchy theory --
that based on scale (extension) and that based on descriptive
complexity (intension). This delivers two models of the material
world; the scalar hierarchy represents extensional complexity (defined
as the situation where dynamics of different scale contextualize each
other at one locale, resulting in non- linear dynamics), while the
specification hierarchy represents intensional complexity (defined as
the situation where more than a single discourse can be brought to
bear upon a phenomenon).

The scalar hierarchy represents subsystems (or components) nested
within supersystems (or wholes). In order not to lose the complexity,
models of this kind need to explicitly represent at least three levels
simultaneously. The lower level proposes possible dynamics at the
focal level, while the higher level regulates or interprets them, as
via boundary conditions. In these models small scale dynamics are
implicitly assimilated to large scale durations. Therefore, process
may be represented at different scalar levels as well as spatial
entities, but not change; the system is implicitly synchronic (moments
of different scale nest within each other). Signals will not simply
transit through scalar levels in amplification, but may be transduced
from one level to the next. A transitive downward plunge of signals
can be catastrophic. An operational model of the origin of scalar
levels is interpolation between primal highest and lowest, protecting
lower level organization.

The specification hierarchy too has a synchronic interpretation, as
when we overlay a physical description of an organism on a chemical
description. Each of these represents a different integrative
level. Higher levels here transitively integrate dynamics and
phenomena at lower levels, as when biology selects or harnesses a few
chemical pathways, or a few physical processes, from the many possible
to interact at the biological level. Any two levels are sufficient to
explore integration. The polarity of rank here is imposed by the
observer, whose level is the highest in the hierarchy, which
traditionally is presented as a hierarchy of discourses, as in
{physical {chemical { biological }}} -- the specification hierarchy
represents levels as nested subclasses. In a diachronic
interpretation, the specification hierarchy represents stages of
development, as when using the above hierarchy to represent some
stages in the development of the earth. Here development is modeled as
the accumulation of greater specification (as via growth and/or
differentiation). The sequence is then interpreted as having been
generated by a concatenation of historical emergences, from the
outermost to the innermost class. This sequence is held to be
canonical, as when biology is held to be impossible without
chemistry. The specification hierarchy is the core of the philosophy
of nature as received from the Nineteenth Century.

Stanley N. Salthe

 	
Biosemiotics is:

    * biology that interprets living systems as sign systems. 

that is, to use a few more words,

    * the signification, communication and habit formation of living processes

    * semiosis (changing sign relations) in living nature

    * the biological basis of all signs and sign interpretation 

To define biosemiotics as "biology interpreted as sign systems" is to
emphasize not only the close relation between biology as we know it
(as a scientific field of inquiry) and semiotics (the study of signs),
but primarily the profound change of perspective implied when life is
considered not just from the perspectives of molecules and chemistry,
but as signs conveyed and interpreted by other living signs in a
variety of ways, including by means of molecules. In this sense,
biosemiotics takes for granted and respects the complexity of living
processes as revealed by the existing fields of biology - from
molecular biology to brain science and behavioural studies - however,
biosemiotics attempts to bring together separate findings of the
various disciplines of biology (including evolutionary biology) into a
new and more unified perspective on the central phenomena of the
living world, including the generation of function and signification
in living systems, from the ribosome to the ecosystem and from the
beginnings of life to its ultimate meanings. Furthermore, by providing
new concepts, theories and case studies from biology, biosemiotics
attempts to throw new light on some of the unsolved riddles within the
general study of sign processes (semiotics), such as the question
about the origin of signification in the universe. Here, signification
(and sign) is understood in a very general sense, that is, not simply
the transfer of information from one place to another, but the
generation of the very content and meaning of that information in
human as well as non-human sign producers and sign receivers. Sign
processes are thus taken as real: They are governed by regularities
(habits, if not laws) that can be discovered and explained. They are
intrinsic in living nature, but we can access them, not directly, but
indirectly through other sign processes (measurements for instance) -
even though the human representation and understanding of these
processes (in the construction of explanations) builds up as a
separate scientific sign system distinct from the organisms' own sign
processes.


Claus Emmeche (first point due to Kalevi Kull in consultation)

The Cartesian cut is a metaphorical notion alluding to Decartes'
distinction of res cogitans (thinking substance) and res extensa
(extended substance). It plays a crucial role in the long history of
the problem of the relationship between mind and matter and is
constitutive for the natural sciences of today. While the elements of
res cogitans are mental (non-material) entities like ideas, models, or
concepts, the elements of res extensa are material facts, events, or
data. The conventional referents of all natural sciences belong to the
latter regime.
 	
The Heisenberg cut is a concept of physics, hence inconceivable
without the Cartesian cut. It is required for the separation of a
material object from its material environment, e.g., of an observed
system from an observing apparatus. It is possible to formalize the
concept of the Heisenberg cut quantum theoretically. Its main
characteristics is the suppression of nonlocal Einstein-Podolsky-Rosen
(EPR) correlations such that a fundamental holism of quantum systems
is lost and objects and disentangled tools of observation can be
distinguished.

 	Causal Closure :

No outside influences other than those prescribed as forces by
Newton's theory. Only material and efficient causes allowed.

	Openness:

Indeterminacies can arise at any time at any scale. Efficient causes
tend to arise at lower scales and propagate upwards; formal causes
appear at the focal level; and final causes at higher levels to
constrain events downscale.

	Atomism:

Systems are strongly decomposable. Larger units are regarded as
decomposable aggregates of stable least units. That which can be built
up can be taken apart again. Increments of the variables of the theory
can be measured by addition and subtraction.

	Organic:

Propensities (contingent causes) always exist in a context, which
includes other propensities. Propensities in communication grow
progressively more interdependent, so that observation of any element
in isolation (if possible) reveals ever less about how it acts within
the ensemble.

	Reversibility:

The laws specifying motion and change can be calculated in both
temporal directions. There is no inherent arrow of time.

	Historicity:

Genetic events degrade predictability and thereby engender
irreversibility. The effects of genetic events are retained in the
material and kinetic forms of the system. Interaction of propensities
defines a more likely direction or telos for system development.

	Determinism:

Given the initial position of any entity in the system, a set of
forces acting on it, and stable closure conditions, every subsequent
position of each particle or entity in the system is in principle
specific and predictable. Mechanical (efficient) causes are everywhere
ascendant.

	Contingency:

The causes of biotic events involve inherent contingencies. They
resemble propensities (sensu Popper) more than Newtonian forces.

        Universality:

Laws are applicable everywhere, at all times and over all scales. Time
and space are uniform throughout the universe. There should be no
complications in applying a law from very small scales and short times
(e.g., intranuclear forces) to phenomena at very large scales and long
times (e.g., gravitation.)  

        Granularity:

An event at any one scale can affect a second event at any other scale
only with a magnitude that diminishes as the difference in scale
between the events increases. Ergo, the range of explanation for any
law is circumscribed, and the effects of contingencies cannot
propagate indefinitely without attenuation.

Ecological Causal Closure :

No outside influences other than those prescribed as forces by
Newton's theory. Only material and efficient causes allowed.

	Openness:

Indeterminacies can arise at any time at any scale. Efficient causes
tend to arise at lower scales and propagate upwards; formal causes
appear at the focal level; and final causes at higher levels to
constrain events downscale.

	Atomism:

Systems are strongly decomposable. Larger units are regarded as
decomposable aggregates of stable least units. That which can be built
up can be taken apart again. Increments of the variables of the theory
can be measured by addition and subtraction.

	Organic:

Propensities (contingent causes) always exist in a context, which includes other propensities. Propensities in communication grow progressively more interdependent, so that observation of any element in isolation (if possible) reveals ever less about how it acts within the ensemble.

	Reversibility:

The laws specifying motion and change can be calculated in both
temporal directions. There is no inherent arrow of time.

	Historicity:

Genetic events degrade predictability and thereby engender
irreversibility. The effects of genetic events are retained in the
material and kinetic forms of the system. Interaction of propensities
defines a more likely direction or telos for system development.

	Determinism:

Given the initial position of any entity in the system, a set of
forces acting on it, and stable closure conditions, every subsequent
position of each particle or entity in the system is in principle
specific and predictable. Mechanical (efficient) causes are everywhere
ascendant.

	Contingency:

The causes of biotic events involve inherent contingencies. They
resemble propensities (sensu Popper) more than Newtonian forces.
Universality:

Laws are applicable everywhere, at all times and over all scales. Time
and space are uniform throughout the universe. There should be no
complications in applying a law from very small scales and short times
(e.g., intranuclear forces) to phenomena at very large scales and long
times (e.g., gravitation.)  

      Granularity:

An event at any one scale can affect a second event at any other scale
only with a magnitude that diminishes as the difference in scale
between the events increases. Ergo, the range of explanation for any
law is circumscribed, and the effects of contingencies cannot
propagate indefinitely without attenuation.

 	
Ontic states describe all properties of a physical system
exhaustively. ("Exhaustive" in this context means that an ontic state
is "precisely the way it is," without any reference to epistemic
knowledge or ignorance.) Ontic states are the referents of individual
descriptions, the properties of the system are treated as intrinsic
properties. Their temporal evolution (dynamics) is reversible and
follows universal, deterministic laws. In quantum systems, entangled
(holistic) states are typical examples for ontic states. The concept
of an observation of such an ontic states is ill-defined since in an
ontic description there is no distinction between an observed and an
observing part of a system. In this sense, ontic states are
empirically inaccessible.

 	
Semiosis refers to the generation and usage of signs. What is a sign? 
A sign is the means by which free energy is transformed by
codification into constrained matter or information. Semiosis
transforms energy from states of thermal and kinetic potentiality to
spatiotemporal instantiations within multiple processes of codal
constraints of organized relations. Codification is the formation of
organized connections or relations with other forms of energy
organization. Semiosis, then, is a relational process of codification
by means of which networks of codification develop to transform energy
into spatiotemporal instantiations of matter or information.

Semiosis has been trapped within a semiological or linguistic and
psychological definition which sets up a simple dyadic descriptive
framework of a >this= substituting for a >that=. This nominalist
understanding sees reality as Cartesian inanimates, made up of things
and their associated images, which must be related to each other by
the mechanical intervention of an external agent, who carries out this
act of the representational substitution of one image with another
image. This is a degenerate semiosis which locates information within
an agential consciousness and sets up a framework confined to
isomorphic descriptions which are therefore incapable of developing
emergent or novel properties.

The genuine sign is a triadic process of mediative generative
connections and as such is actually a logical sentence made up of
noun(s), which are the spatiotemporal instantiations, and a predicate
or relational process of mediation. The predicate is the
transformational process of codification that links the noun(s) with
other sign-sentences. Energy encoded in this triadic model is capable
of informing on and about other forms of energy-as-information by
relating its codes and their organizational properties to other forms
of encoded energy-as-information.

A genuine semiosis is a generative process where the signs, activated
within their predicates, seek out and develop pragmatic links with
other semiotic sentences by means of which they interpret, expand and
actually create their identities. The sign as a generative sentence is
a speculative gaze that is focused on past networks, other networks
and the future pragmatics of purely hypothetical and experimental
networks. This semiosic sentence operates within all three cosmic
realms, the physico-chemical, the biological and the socioconceptual
and is the basis for all informational processes of energy.



\end{document}
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