Some remarks on the interpretation of degree of nonextensivity
{Ho\.za 69; 00-689 Warsaw, Poland}
{Konopnickiej 15; 25-405 Kielce, Poland}
Recently we have demostrated that the nonextensitivity parameter $q$ occuring in some applications of Tsallis statistics (known also as index of the corresponding L\'evy distribution) is, in $q>1$ case, given entirely by the fluctuations of the parameters of the usual exponential distribution. We show here that this interpretation is valid also for the $q<1$ case. The parameter $q$ is therefore a measure of fluctuations of the parameters of the usual exponential distribution.\\ \noindent PACS numbers: 05.40.Fb 24.60.-k 05.10.Gg\\ {Keywords:} Nonextensive statistics, L\'evy distributions, Thermal models\\ [3ex]
hep-ph0002145.tex
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\begin{document}
\title{\bf Some remarks on the interpretation of degree of nonextensivity}
\author{G.Wilk$^{1}$\thanks{e-mail: wilk@fuw.edu.pl} and Z.W\l
odarczyk$^{2}$ \thanks{e-mail: wlod@pu.kielce.pl}\\[2ex] 
$^1${\it The Andrzej So\l tan Institute for Nuclear Studies}\\
    {\it Ho\.za 69; 00-689 Warsaw, Poland}\\
$^2${\it Institute of Physics, Pedagogical University}\\
    {\it  Konopnickiej 15; 25-405 Kielce, Poland}}  
\date{\today}
\maketitle

\begin{abstract}
Recently we have demostrated that the nonextensitivity parameter $q$
occuring in some applications of Tsallis statistics (known also as
index of the corresponding L\'evy  distribution) is, in $q>1$ case,
given entirely by the fluctuations of the parameters of the usual 
exponential distribution. We show here that this interpretation is
valid also for the $q<1$ case. The parameter $q$ is therefore a
measure of fluctuations of the parameters of the usual exponential
distribution.\\   

\noindent
PACS numbers: 05.40.Fb 24.60.-k  05.10.Gg\\
{\it Keywords:} Nonextensive statistics, L\'evy distributions,
Thermal models\\ 
[3ex]

\end{abstract}

\newpage

Recently we have demostrated that the nonextensitivity parameter $q$
occuring in some applications of Tsallis statistics \cite{T} (known
also as index of the corresponding L\'evy  distribution) is, in $q>1$
case, given entirely by the fluctuations of the parameters of the usual 
exponential distribution \cite{WWq}. It means that:
\begin{itemize}
\item when in some exponential formula describing distribution of a
quantity $\varepsilon$ of physical interest:
\begin{equation}
L_{q=1}(\varepsilon)\, =\, C_{q=1}\, \exp\left[ -
                    \frac{\varepsilon}{\chi}\right] , \label{eq:Lq1}
\end{equation}
one allows the parameter $\chi$ to fluctuate around some mean value
$\chi_0$, and 
\item if these fluctuations are described by simple Gamma
distribution of the form  
\begin{equation}
f(\frac{1}{\chi})\, =\, \frac{1}{\Gamma(\alpha)}\, \mu\, 
 \left(\frac{\mu}{\chi}\right)^{\alpha-1}\, \exp\left( -\,
\frac{\mu}{\chi} \right)  \label{eq:FRES}
\end{equation}
depending on two parameters
\begin{equation}
\alpha \, =\, \frac{1}{q-1}\qquad {\rm and} \qquad \mu = \alpha
               \chi_0 , \label{eq:param}
\end{equation}
\item then, as result, one gets the following power-like distribution
for the quantity $\varepsilon$ of interest:
\begin{equation}
L_q(\varepsilon)\, =\, C_q\, \left[ 1\, -\, (1\, -\, q)\, 
                    \frac{\varepsilon}{\chi_0}\right]^{\frac{1}{1-q}},
                    \label{eq:Lq}
\end{equation}                    
known also as L\'evy distribution with index $q$, where
\begin{equation}
q\, =\, 1\, +\, 
       \frac{\left\langle\left(\frac{1}{\chi}\right)^2\right\rangle\,
       -\, \left\langle\frac{1}{\chi}\right\rangle^2}
       {\left\langle \frac{1}{\chi}\right\rangle^2}, \label{eq:q}
\end{equation}
i.e., where it is entirely given by the relative variance of  the
parameter $1/\chi$ of the initial
distribution (\ref{eq:Lq1}) ($<...>$ denotes the corresponding
averages with respect to distribution $f(\chi)$). 
\end{itemize}
The proof presented in \cite{WWq} was limited to the $q>1$ case and
the physical discussion provided there was also concentrated on such
situation. Because $q$ can be also interpreted as the so called
nonextensivity parameter occuring in some applications of Tsallis 
statistic \cite{T}, it would be interesting to check if such
interpretation can be extended to the $q<1$ case as well. We shall
demonstrate below that this is indeed the case \cite{FOOT1}.\\

The essential difference between these two cases is, for the purpose
of present discussion, that whereas for $q>1$ probability distribution
$L_q(\varepsilon)$ is well defined for the whole range of variable
$\varepsilon$, $\varepsilon \in (0,\infty)$, for $q<1$ it is defined
only for $\varepsilon \in [0,\chi_0/(1-q)]$. As it was done in
\cite{WWq} we shall deduce the form of function $f(1/\chi)$, describing
fluctuations in $\chi$, which would lead from the exponential
distribution $L_{q=1}$ to the power-like L\'evy distribution $L_{q<1}$
\begin{equation}
L_{q<1}(\varepsilon;\chi_0)\, =\, C_q\, \left[1\, -\,
       \frac{\varepsilon}{\alpha' \chi_0}\right]^{\alpha'}\, =\,
       C_q\, \int^{\infty}_0\, \exp\left(- \frac{\varepsilon}{\chi}\right)
       \, f\left(\frac{1}{\chi}\right)\, d\left(\frac{1}{\chi}\right)
       \label{eq:QL1}
\end{equation}       
(for simplicity we denote $\alpha' = \frac{1}{1-q}$). From the
representation of the Euler gamma function we have
\begin{equation}
\left[1\, -\, \frac{\varepsilon}{\alpha' \chi_0}\right]^{\alpha'}\, =\,
\left(\frac{\alpha' \chi_0}{\alpha' \chi_0 - \varepsilon}\right)^{-\alpha'}\,
=\, \frac{1}{\Gamma(\alpha')}\, \int^{\infty}_0\, d\eta\, 
    \eta^{\alpha' - 1}\, \exp\left[ - \eta\, \left(1\, +\, 
    \frac{\varepsilon}{\alpha' \chi_0 - \varepsilon}\right)\right] . 
    \label{eq:EGF}
\end{equation}
Changing now variables under the integral in such a way that
$\chi\, =\, \frac{\alpha' \chi_0 - \varepsilon}{\eta}$
one immediately obtains Eq. (\ref{eq:QL1}) with $f(1/\chi)$ given by
the following gamma distribution
\begin{equation}
f\left(\frac{1}{\chi}\right)\, =\, \frac{1}{\Gamma(\alpha')}\,
     \left( \alpha' \chi_0 - \varepsilon \right)\,
     \left(\frac{\alpha' \chi_0 - \varepsilon}{\chi}\right)^{\alpha' - 1}\,
     \exp\left( - \frac{\alpha' \chi_0 - \varepsilon}{\chi}\right) 
     \label{eq:RES}
\end{equation}
with parameters $\alpha'$ and $\mu(\varepsilon) = \alpha' \chi_0 -
\varepsilon$. This time, contrary to the $q>1$ case of \cite{WWq},
fluctuations depend on the value of the variable in question, i.e.,
the mean value and variance are both $\varepsilon$-dependent:
\begin{equation}
\left\langle \frac{1}{\chi}\right\rangle\, =\, \frac{1}{\chi_0 -
\frac{\varepsilon}{\alpha'}}\qquad {\rm and}\qquad \left\langle
\left(\frac{1}{\chi}\right)^2\right\rangle\, -\,
\left\langle\frac{1}{\chi}\right\rangle^2\, =\, \frac{1}{\alpha'}\cdot
\frac{1}{\left(\chi_0 - \frac{\varepsilon}{\alpha'}\right)^2} .
\label{eq:MV}
\end{equation}
However, the relative variance
\begin{equation}
\omega\, =\,  \frac{\left\langle\left(\frac{1}{\chi}\right)^2\right\rangle\,
       -\, \left\langle\frac{1}{\chi}\right\rangle^2}
       {\left\langle \frac{1}{\chi}\right\rangle^2}\, 
       =\, \frac{1}{\alpha'}\, =\, 1\, -\, q \label{eq:RESULT}
\end{equation}
remains $\varepsilon$-independent (exactly like in the case of $q>1$)
and depends only on parameter $q$. It means therefore that the
parameter $q$ in L\'evy distribution $L_q(\varepsilon)$ describes the
relative variance of fluctuations of parameter $\chi$ in
$L_{q=1}(\varepsilon)$ for all values of $q$ (both for $q>1$, where
$\omega = q - 1$, cf. \cite{WWq} and for $q<1$ as given above, where
$\omega = 1-q$).\\ 

In \cite{WWq} we have proposed a general explanation of the meaning
of function $f(\chi)$ describing fluctuations of some variable $\chi$.
The question one is interested in is why, and under what
circumstances, it is the gamma distribution that describes 
fluctuations. To this end we have started with general Langevin type
equation \cite{FP} for the variable $\chi$ 
\begin{equation}
\frac{d\chi}{dt}\, +\, \left[\frac{1}{\tau}\, +\, \xi(t)\right]\,
\chi\, =\, \phi\, =\, {\rm const}\, >\, 0  \label{eq:LE}
\end{equation}
(with damping constant $\tau$ and source term $\phi$). For stochastic
processes defined by the white gaussian noise form of $\xi(t)$ (cf.
\cite{WWq} for details) it can be shown that distribution function
for the variable $\chi$ satisfies the Fokker-Planck equation
($K_{1,2}$ are the corresponding intensity coefficients, cf.
\cite{WWq}) 
\begin{equation}
\frac{df(\chi)}{dt}\, =\, -\, \frac{\partial}{\partial \chi}K_1\,
f(\chi)\, +\, \frac{1}{2}\, \frac{\partial^2}{\partial \chi^2}K_2\,
f(\chi) , \label{eq:FPE}
\end{equation}
i.e., it is indeed given by the Gamma distribution in variable
$1/\chi$ of the form (\ref{eq:FRES}) with $\mu = \alpha\chi_0$.
Notice that it differs from Eq. (\ref{eq:RES}) only in the form of
parameter $\mu$, which in (\ref{eq:RES}) depends also on the physical
quantity of interest $\varepsilon$.\\

As an illustration of the genesis of Eq. (\ref{eq:LE}) we have
discussed in \cite{WWq} the case of fluctuations of temperature
(i.e., the situation where $\chi=T$) \cite{FOOT2}. Suppose that we
have a thermodynamic system, in a small (mentally separated) part of
which the temperature fluctuates around some mean value $T_0$ (which
can be also understood as an equilibrium temperature) with
$\Delta T \sim T$. The unevitable exchange of heat between this
selected region and the rest of the system is described by Eq.
(\ref{eq:LE}) in which 
\begin{equation}
\phi = \phi_{q<1}\, =\, \frac{1}{\tau}\left(T_0 -
                 \frac{\varepsilon}{\alpha'}\right) 
\qquad {\rm whereas}\qquad \phi = \phi_{q>1} = \frac{T_0}{\tau} .
             \label{eq:FIFI}
\end{equation}
It means that the corresponding process of heat conductivity is, for
$q<1$ case, described by the following equation (here $T'=T_0-\tau
\xi(t)T$) 
\begin{equation}
\frac{\partial T}{\partial t}\, -\, \frac{1}{\tau}
            \, (T'\, -\, T)\,
         +\, \frac{\varepsilon}{\tau \alpha'} =\, 0 , \label{eq:HC}
\end{equation}
which differs from the corresponding equation for $q>1$ case only by
the last term describing the presence the internal heat source. It
has a sense of dissipative transfer of energy from the region where
(due to fluctuation) we have higher $T$. It could be any kind of
convection type flow of energy, for example it could be connected
with emission of particles from that region. The heat release given
by $\varepsilon/(\tau\alpha')$ depends on $\varepsilon$ (but it is
only a part of $\varepsilon$, which is released). In the case of such
energy release (connected with emission of particles) there is
additionale cooling of the whole system. If this process is
sufficiently fast, it could happen that there is no way to reach a
stationary distribution of temperature (because the transfer of the
heat from the outside can be not sufficient for development of the
state of equilibrium). On the other hand (albeit this is not our case
here) for the reverse process we could face the "heat explosion"
situation (which could happen if the velocity of the exotermic
burning reaction grows sufficiently fast; in this case because of
nonexistence of stationary distribution we have fast nonstationary
heating of the substance and acceleration of the respective reaction).\\   

It should ne noticed that in the case of $q<1$ the temperature does
not reach stationary state because, cf. Eq. (\ref{eq:MV}),
$\langle 1/T \rangle\, =\, 1/(T_0 - \varepsilon/\alpha')$,
whereas for $q>1$ we had $<T> = T_0$. As a consequence the
corresponding L\'evy distribution are defined only for $\varepsilon
\in(0, T_0\, \alpha'$) because for $\varepsilon \rightarrow
T_0\alpha'$ the $<T>\rightarrow 0$. Such asymptotic (i.e., for
$t/\tau \rightarrow \infty$) cooling of the system ($T\rightarrow 0$)
can be also deduced form Eq. (\ref{eq:HC}) for $\varepsilon \rightarrow
T_0\alpha'$.\\

To summarize, we have demonstrated that temperature fluctuations lead
to the L\'evy distribution $L_q(\varepsilon)$ with index $q<1$ when
there exists energy source and with $q>1$ in the absence of such
source. In both cases, however, the relative variance of $1/T$ 
fluctuations is described by the parameter $q$ only.\\



\newpage

\begin{thebibliography}{99}
 
 \bibitem{T} C.Tsallis, {\sl J.Stat.Phys.} {\bf 52} (1988) 479;
             for updated bibliography on this subject cf. 
             http://tsallis.cat.cbpf.br/biblio.htm. Recent summary is
             provided in the special issue of {\sl Braz. J. Phys.}
             {\bf 29} (No 1) (1999) (available also at
             http://sbf.if.usp.br/WWW\_pages/Journals/BJP/Vol29/Num1/index.htm). 

 \bibitem{WWq} G.Wilk and Z.W\l odarczyk, {\it Interpretation of the
               nonextensitivity parameter $q$ in some applications of
               Tsallis statistics and L\'evy distributions}, 
               hep-ph/9908459, to be published in {\sl Phys. Rev. Lett.}.

 \bibitem{FOOT1} The relevant references to the up-to-date applications
                 of the Tsallis statistics can be found in \cite{T}
                 whereas \cite{WWq} summarises all recent attempts of
                 its applications in the high energy physics domain.

 \bibitem{FP} N.G. van Kampen, {\it Stochastic Processes in Physics
              and Chemistry}, Elsevier Science Pub. B.V.,
              North-Holland, Amsterdam 1987 (Chapter VIII).

 \bibitem{FOOT2} This is interesting and hot subject in high energy
                 physics, as can be judged from: L.Stodolsky, {\sl 
                 Phys. Rev. Lett.} {\bf 75} (1995) 1044; E.V.Shuryak, 
                 {\sl Phys. Lett.} {\bf B423} (1998) 9 and 
                 S.Mr\'owczy\'nski, {\sl Phys. Lett.} {\bf B430} 
                 (1998) 9.

\end{thebibliography}

\end{document}

                              
 \bibitem{RAF} D.B.Walton and J.Rafelski, {\sl Phys. Rev. Lett.} {\bf
               84} (2000) 31.




There is an enormous variety of physical phenomena described most
economically (by introducing only one new parameter $q$) and 
adequately by the so called nonextensive statistics introduced
some time ago by Tsallis \cite{T}. They include all situations
characterized by long-range interactions, long-range microscopic
memories and space-time (and phase-space as well) (multi) fractal
structure of the process (cf. \cite{T} for details). The high energy
physics applications of nonextensive statistics are quite recent, but
already numerous and still growing, cf. Refs.
\cite{WW,BCM,FLUQ,UWW,RHIP,RAF,FOOT1}. All examples mentioned above
have one thing in common: the central formula employed is the
following power-like distribution:  
\begin{equation}
G_q(x)\, =\, C_q\, \left[\, 1\, -
                   \, (1\, -\, q)\, \frac{x}{\lambda}\,
                   \right]^{\frac{1}{1 - q}}, \label{eq:T}
\end{equation}                   
which is just a one parameter generalization of the Boltzmann-Gibbs
exponential formula to which it converges for $q\rightarrow 1$:
\begin{equation}
G_{q=1}\, =\, g(x)\, =\, c\cdot \exp\left[\, -\, \frac{x}{\lambda}\,
\right]                \label{eq:BG}
\end{equation}
When $G_q(x)$ is used as probability distribution (L\'evy
distribution) of the variable $x\in(0, \infty)$ (which will be the
case we are interested in here), the parameter $q$ is limited to $1\leq q
< 2$. For $q<1$, the distribution $G_q(x)$ is defined only for $x\in [0,
\lambda/(1-q)]$. For $q>1$ the upper limit comes from the
normalization condition (to unity) for $G_q(x)$ and from the
requirement of the positivity of the resulting normalisation constant
$C_q$. However, if one demands in addition that the mean value of
$G_q(x)$ is well defined, i.e., that $\langle x\rangle = \lambda
/(3-2q) < \infty$ for $x\in (0,\infty)$, then $q$ is further limited
to $1\leq q< 1.5$ only. In spite of numerous applications of the
L\'evy distribution $G_q(x)$, the interpretation of the parameter $q$
is still an open issue. In this work we shall demonstrate, on the
basis of our previous application of the L\'evy distribution to
cosmic rays \cite{WW}, that this L\'evy distribution $G_q(x)$
(\ref{eq:T}) emerges in a natural way from the fluctuations of the
parameter $1/\lambda$ of the original exponential distribution
(\ref{eq:BG}) and that the parameters of its distribution
$f(1/\lambda)$ define parameter $q$ in unique way.\\

Let us first briefly summarise the result of \cite{WW}. Analysing
experimental distributions $dN(x)/dx$ of depths $x$ of interactions
of hadrons from cosmic ray cascades in the emulsion chambers, we
have shown that the so called {\it long flying component}
(manifesting itself in aparently unnexpected nonexponential
behaviour of  $dN(x)/dx$) is just a manifestation of the L\'evy
distribution $G_q(x)$ with $q=1.3$. This result must be confronted
with our earlier analysis of the same phenomenon \cite{WWCR}. We have
demonstrated there that distributions $dN(x)/dx$ can be also 
described by the fluctuation of the corresponding cross-section
$\sigma\, =\, A\, m_N\, \frac{1}{\lambda}$ (where $A$ denotes mass
number of the target, $m_N$ is the mass of the nucleon and $\lambda$
is the corresponding mean free path). The fluctuation of this
cross-section (i.e., in effect, fluctuations of the quantity
$1/\lambda$) with relative variance  
\begin{equation}
\omega \, =\, \frac{\langle \sigma^2 \rangle\, -\, \langle
\sigma\rangle ^2}{\langle \sigma \rangle ^2} \, \ge \, 0.2 \label{eq:OMEGA}
\end{equation}
allow us to describe the non-exponentiality of the experimental data
as well as the distribution $G_{q=1.3}(x)$ mentioned above.
We therefore argue that these two numerical examples show that
fluctuations of the parameter $1/\lambda$ in the $g(x;\lambda)$
result in the L\'evy distributions $G_q(x;\lambda)$.\\

Actually the above quoted example from cosmic ray physics is not the
only one known at present in the field of high energy collisions. It
turns out \cite{BCM,UWW} that distributions of transverse momenta
$dN(p_T)/dp_T$ are best described by a slightly non-exponential
distribution $G_q(p_T)$ of the L\'evy type with $q=1.01\div1.2$
depending on situation considered. The usual exponential distribution
$dN(p_T)/dp_T = g(p_T) \sim \exp( - \sqrt{m^2+p_T^2}/kT)$ contains as
a main parameter the inverse temperature $\beta = 1/kT$ and the above
mentioned numerical results leading to $G_{q=1.01\div1.2}(p_T)$ can
again be understood as a result of a fluctuation of inverse
temperature $\beta$ in the usual exponential formula $g(p_T)$. This
point is of special interest because of recent discussions on the
dynamical possibility of temperature fluctuations in some collisions,
cf. Ref. \cite{LS,FLUCT,L}. Later on we shall use it to illustrate
our results concerning $q$.\\   

To recapitulate: we claim that (for $q>1$) the parameter $q$ is
nothing but a measure of fluctuations present in L\'evy distributions
$G_q(x)$ describing particular processes under consideration. To make
our statement more quantitative, let us analyse the influence of
fluctuations of parameter $1/\lambda$ which are present in the
exponential formula $g(x) \sim \exp(-x/\lambda)$ on the final result.
Our aim will be a deduction of the form of the function
$f(1/\lambda)$ which leads from an exponential distribution $g(x)$ to
power-like L\'evy distribution $G_q(x)$ and which describes
fluctuation about the mean value $1/\lambda_0$, i.e., such that 
\begin{equation}
G_q(x;\lambda_0)\, =\, C_q\, 
\left( 1\, +\, \frac{x}{\lambda_0}\, \frac{1}{\alpha}\right)^{-a}\, =\,
C_q\, \int^{\infty}_0\, \exp\left( - \frac{x}{\lambda}\right)\,
f\left(\frac{1}{\lambda}\right)\, d\left(\frac{1}{\lambda}\right)
\label{eq:DEF} 
\end{equation}
where for simplicity we have introduced the abbreviation $\alpha =
\frac{1}{q-1}$. 
From the representation of the Euler gamma function we have
\cite{FOOT2}
\begin{equation}
\left( 1\, +\, \frac{x}{\lambda_0}\, \frac{1}{\alpha}\right)^{-a}\, =\,
\frac{1}{\Gamma(\alpha)}\, \int^{\infty}_0\, d\xi\, 
      {\xi}^{\alpha - 1}\, \exp\left[ - \xi\, 
      \left(1\, +\, \frac{x}{\lambda_0}\,
       \frac{1}{\alpha}\right)\right] .
 \label{eq:GF}
\end{equation}
Changing variables under the integral in such a way that
$\frac{\xi}{\lambda_0}\frac{1}{\alpha}=\frac{1}{\lambda}$ one
immediately obtains eq. (\ref{eq:DEF}) with $f(1/\lambda)$ given by
the following gamma distribution 
\begin{equation}
f\left(\frac{1}{\lambda}\right)\, =\,
f_{\alpha}\left(\frac{1}{\lambda},\frac{1}{\lambda_0}\right)\, =\,
\frac{1}{\Gamma(\alpha)}\,  (\alpha\lambda_0)\,
\left(\frac{\alpha\lambda_0}{\lambda}\right)^{\alpha-1}\, \exp\left(
- \frac{\alpha\lambda_0}{\lambda}\right) \label{eq:F}
\end{equation}
with mean value
\begin{equation}
\left\langle \frac{1}{\lambda}\right\rangle \, =\,
 \frac{1}{\lambda_0} \label{eq:MEAN}
\end{equation}
and variation
\begin{equation}
\left\langle \left(\frac{1}{\lambda}\right)^2\right\rangle\, -\, 
\left\langle\frac{1}{\lambda}\right\rangle^2\, =\, 
\frac{1}{\alpha\, \lambda_0^2} . \label{eq:VAR}
\end{equation}
Notice that, with increasing $\alpha$ variance (\ref{eq:VAR})
decreases and asymptotically (for $ \rightarrow \infty$, i.e,
for $q\rightarrow 1$) the gamma distribution (\ref{eq:F}) becomes
a delta function $\delta (\lambda - \lambda_0)$. The relative
variance (cf. eq.(\ref{eq:OMEGA})) for this distribution is given by 
We see therefore that, indeed, the parameter $q$ in the L\'evy
distribution $G_q(x)$ describes the relative variance of the
parameter $1/\lambda$ present in the exponential distribution
$g(x;\lambda)$.\\ 

Some remarks on the numerical results quoted before \cite{WW,WWCR}
are in order here. Notice that the value of $q=1.3$ for cosmic ray
distribution $dN(x)/dx$ obtained in \cite{WW} leads to the relative
variance of the cross section $\omega = 0.3$ whereas in \cite{WWCR}
we have reported value $\omega' =0.2$. This discrepancy has its
origin in the fact that in numerical calculations in \cite{WWCR} we
have used a symmetric Gaussian distribution to decribe fluctuations
of the cross section, whereas the relation (\ref{eq:PROOF}) has been
obtained for fluctuations described by gamma distribution. In the
gaussian approximation we expect that  
\begin{equation}
\frac{q - 1}{q^2}\, <\, \omega'\, <\, q - 1 , \label{eq:INEQ}
\end{equation}
where lower and upper limits are obtained by normalizing the variance
of the $f(1/\lambda)$ distribution to the modial (equal to
$(2-q)/\lambda_0$) and mean (equal to $1/\lambda_0$) values,
respectively. Therefore for $q=1.3$ one should expect that $ 0.18 <
\omega' < 0.3$, which is exactly the case.\\

Let us now proceed to the above mentioned analysis of transverse
momentum distributions in heavy ion collisions, $dN(p_T)/dp_T$
\cite{FLUQ}. It is interesting to notice that the relatively small
value $q \simeq 1.015$ of the nonextensive parameter obtained there,
if interpreted in the same spirit as above, indicates that rather
large relative fluctuations of temperature, of the order of $\Delta
T/T \simeq 0.12$, exist in nuclear collisions. It could mean
therefore that we are dealing here with some fluctuations existing in
small parts of the system in respect to the whole system (according
to interpretation of \cite{L}) rather than with fluctuations of the
event-by-event type in which, for large multiplicity $N$,
fluctuations $\Delta T/T = 0.06/ \sqrt{N}$ should be negligibly small
\cite{LS}.\\ 

We shall now propose a general explanation of the meaning of the
function $f(\chi)$ describing fluctuations of some variable $\chi$.
In paticular, we shall be interested in question why, and under what
circumstances, it is the gamma distribution that describes
fluctuations. To this end let us start with the well known equation
for the variable $\chi$, which in the Langevin formulation has the
following form \cite{FP}  
Let us concentrate for our purposes on the stochastic process which is
defined by the {\it white gaussian noise} $\xi(t)$ with ensemble mean 
\begin{equation}
\langle \xi(t) \rangle\, =\, 0 \label{eq:EM}
\end{equation}
and correlator $\langle \xi(t)\, \xi(t + \Delta t) \rangle$, which
for sufficiently fast changes is equal to 
\begin{equation}
\langle \xi(t)\, \xi(t + \Delta t) \rangle\, =\, 2\, D\,
\delta(\Delta t) .\label{eq:COR}
\end{equation}
Constants $\tau$ and $D$ define, respectively, the mean time for
changes and their variance by means of the following conditions:
\begin{equation}
\langle \chi(t)\rangle\, =\, \chi_0\, \exp\left( - \frac{t}{\tau} \right)
\quad {\rm and} \quad
\langle \chi^2(t=\infty)\rangle\, =\, \frac{1}{2}\, D\, \tau .
\label{eq:COND}
\end{equation}
Thermodynamical equilibrium is assumed here (i.e., $t >> \tau$, in
which case the influence of the initial condition $\chi_0$ vanishes
and the mean squared of $\chi$ has value corresponding to the state of
equilibrium). Making use of the Fokker-Plank equation \cite{FOOT3}
we get for the distribution function the following  expression
\begin{equation}
f(\chi)\, =\, \frac{c}{K_2(\chi)}\, \exp\left[\, 2\,
\int^{\chi}_0 d\chi'\, \frac{K_1(\chi')}{K_2(\chi')}\, \right]
\label{eq:EF} 
\end{equation}
where the constant $c$ is defined by the normalisation condition for
$f(\chi)$: $\int^{\infty}_0 d\chi f(\chi) = 1$. $K_1$ and $K_2$
are the intensity coefficients which for the process defined by eq.
(\ref{eq:LE}) are equal to (cf., for example, \cite{ADT}):
\begin{eqnarray}
K_1(\chi)\, &=&\, \phi\, -\, 2\, \frac{\chi}{\tau}\, +\, D\, \chi
,\nonumber\\ 
K_2(\chi)\, &=&\, 2\, D\, \chi^2 . \label{eq:KK}
\end{eqnarray}
It means therefore that as result we have the following distribution
function 
Returning to the $q$-notation (cf. eq. (\ref{eq:DEF})) we have
therefore 
\begin{equation}
q\, =\, 1\, +\, \tau\, D , \label{eq:DEFQ}
\end{equation}
i.e., the parameter of nonextensitivity is given by the parameter $D$
describing the {\it white noise} and by the damping constant $\tau$.
This means then that the relative variance $\omega(1/\chi)$ of distribution
(\ref{eq:FRES}) is (as in eq. (\ref{eq:PROOF})) given by $\tau D$.\\

As illustration of the genesis of eq. (\ref{eq:LE}) used to derive
eq. (\ref{eq:DEFQ}), we turn once more to the fluctuations of
temperature \cite{LS,FLUCT,L} discussed before (i.e., to the
situation when $\chi = T$). Let $\xi(t)$ describes stochastic
changes of temperature in time. If the mean temperature of the system
$\langle T\rangle = T_0$ then, as result of fluctuations in some
small selected region, the actual temperature $T'$ equals 
\begin{equation}
T'\, =\, T_0\, -\, b\, \xi(t)\, T  ,\label{eq:TTT}
\end{equation}
where the constant $b$ is defined by the actual definition of the
stochastic process under consideration, i.e., by $\xi(t)$, which is
assumed to satisfy conditions given by eqs. (\ref{eq:EM}) and
(\ref{eq:COR}). The inevitable exchange of heat between this selected
region and the rest of the system leads to the equilibration of the
temperature. The corresponding process of heat conductance is
described by the following equation \cite{LLH} 
where $c_p,~\rho$ and $a$ are, respectively, the specific heat,
density and the coefficient of the external conductance. Using $T'$
as defined in (\ref{eq:TTT}) we finally get the linear differential
equation (\ref{eq:LE}) for the temperature $T$ with coefficients:
$\tau = \frac{c_p\rho}{a}$, $\phi = \frac{a}{c_p\rho}T_0 = T_0/\tau$
and $b=\tau$:
\begin{equation}
\frac{\partial T}{\partial t}\, +\, \left[\, \frac{a}{c_p\, \rho}\,
+\, \frac{a}{c_p\, \rho}\, b\, \xi(t)\, \right]\, T\, =
\, \frac{a}{c_p\, \rho}\, T_0 . \label{eq:RESF}
\end{equation}
This result demonstrates clearly that one can think of a deep
physical interpretation of the parameter $q$ of the corresponding
L\'evy distribution describing the distributions of the transverse
momenta mentioned before. In this respect our work differs from works
in which $G_q(x)$ is shown to be connected with $G_{q=1}(x) = g(x)$
by the so called Hilhorst integral formula (the trace of which
is our eq. (\ref{eq:GF})) \cite{FOOT2,C} but without discussing the
physical context of the problem. Our original motivation was to
understand the apparent success of Tsallis statistics (i.e., the
situations in which $q>1$) in the realm of high energy collisions.\\

To summarise: if fluctuations of the variable $\chi$ can be described
in terms of the Langevin formulation, their distribution function
$f(1/\chi)$ satisfies the Fokker-Plank equation and is therefore
given by the Gamma distribution in the variable $1/\chi$. Such
fluctuations of the parameter $1/\chi$ in the exponential formula of
physical interest, $g(x/\chi)$, lead immediately to a L\'evy
distribution $G_{q>1}(x/\chi)$ with $q$ parameter given by the
relative variance of the fluctuations described by $f(1/\chi)$. 
It should be stressed that in this way we address the interpretation
of only very limited cases of applications of Tsallis statistics.
They belong to the category in which the power laws physically appear
as a consequence of some continuous spectra within appropriate
integrals. It does not touche, however, a really {\it hard} case of
applicability of Tsallis statistics, namely when {\it zero} Lyapunov
exponents are involved \cite{FOOT4}. Nevertheless, this allows us to
interpret some nuclear collisions data in terms of fluctuations of
the inverse temperature, providing thus an important hint to the
origin of some systematics in the data, understanding of which is
crucial in the search for the new state of matter: the Quark Gluon
Plasma \cite{FLUQ,FLUCT}.\\ 

Acknowledgement: We are grateful to Prof. St. Mr\'owczy\'nski for 
fruitful discussions and comments.

 \bibitem{WW} G.Wilk and Z.W\l odarczyk, {\sl Nucl. Phys.} {\bf B} 
              ({\sl Proc. Suppl.}) {\bf A75} (1999) 191.

 \bibitem{BCM} I.Bediaga, E.M.F.Curado and J.M.de Miranda, {\it A
               nonextensive thermodynamical equilibrium approach in
               $e^+e^- \rightarrow hadrons$}, hep-ph/9905255.
              
 \bibitem{FLUQ} W.M.Alberico, A.Lavagno and P.Quarati, {\it Non-extensive
                statistics, fluctuations and correlations in high energy
                nuclear collisions}, nucl-th/9902070; to be published in
                {\sl Eur. Phys. J.} {\bf C}.

 \bibitem{UWW} O.V.Utyuzh, G.Wilk and Z.W\l odarczyk, {\it The effect
               of nonextensive statistics on fluctuations investigated
               in event-by-event analysis of data}, hep-ph/9906500.

 \bibitem{RHIP} O.V.Utyuzh, G.Wilk and Z.W\l odarczyk, {\it The fractal
                properties of the source and BEC}, presented at the
                $12^{th}$ Indian Summer School {\it Relativistic Heavy
                Ion Physics}, Prague, Czech Rep., $30$ August - 3 Sept.
                1999, to be published in {\sl Czech J. Phys.} (1999),
                hep-ph/9910355.
                
 \bibitem{FOOT1} Actually, the recently proposed use of quantum
                 groups in studying Bose-Einstein correlations
                 (cf. D.V.Anchishkin, A.M.Gavrilik and N.Z.Iogorov, 
                 {\it Two-particle correlations from the $q$-boson 
                 viewpoint}, CERN-TH/99-177 preprint and nucl-th/9906034)
                 belong also to that cathegory because (as demonstrated
                 in C.Tsallis, {\sl Phys. Lett.} {\bf A195} (1994) 329)
                 there is close correspondence between deformation 
                 parameter of quantum groups and nonextensitivity 
                 parameter of Tsallis statistics. The same can be said 
                 on the works on intermittency using the so called 
                 L\'evy stable distributions (cf., for example, 
                 Ph. Brax and R.Peschanski, {\sl Nucl. Phys.} {\bf B253} 
                 (1991) 225 or S.Hegyi, {\sl Phys. Lett.} {\bf B387} 
                 (1996) 642).
                 
 \bibitem{WWCR} G.Wilk and Z.W\l odarczyk, {\sl Phys. Rev.} {\bf D50}
                (1994) 2318.
 
 \bibitem{FLUCT} T.C.P.Chui, D.R.Swanson, M.J.Adriaans,
                 J.A.Nissen and J.A.Lipa, {\sl Phys. Rev. Lett.} {\bf
                 69} (1992) 3005; C.Kittel, {\sl Physics Today} {\bf 5} 
                 (1988) 93; B.B.Mandelbrot, {\sl Physics Today} {\bf
                 1} (1989) 71. Cf also: 

 \bibitem{L} L.D.Landau, I.M.Lifschitz, {\it Course of Theoretical 
             Physics: Statistical Physics}, Pergmon Press, New
             York 1958.

 \bibitem{FOOT2} This use of what is essentially the
                 Mellin transformation has been discussed in
                 different contexts of Tsallis statistics in a
                 number of places, cf., for example, D.Prato, {\sl
                 Phys. Lett.} {\bf A203} (1995) 165; P.A.Alemany,
                 {\sl Phys. Lett.} {\bf A235} (1997) 452 or C.Tsallis
                 et al., {\sl Phys. Rev.} {\bf E56} (1997) R4922.  

 
 \bibitem{FOOT3} Our discussion (and results that follow) resembles 
                 to some extent approaches where Tsallis-type microscopic
                 distributions were derived as exact solution of the
                 standard Fokker-Planck equation, see for example, 
                 L.Borland, {\sl Phys. Lett.} {\bf A245} (1998) 67 and 
                 {\sl Phys. Rev.} {\bf E57} (1998) 6634. 
                 
 \bibitem{ADT} C.A.Ahmatov, Y.E.Diakov and A.Tchirkin, {\it Introduction
               to Statistical Radiophysics and Optics}, Nauka,
               Moscow, 1981 (in Russian).

 \bibitem{LLH} L.D.Landau and I.M.Lifschitz, {\it Course of Theoretical
               Physics: Hydrodynamics}, Pergamon Press, New York 1958 
               or {\it Course of Theoretical Physics: Mechanics of
               Continous Media}, Pergamon Press, Oxford 1981.
 
 \bibitem{C} C.f., for example, S.Curilef, {\sl Z.Phys.} {\bf B100} 
             (1996) 433 and references therein.

 \bibitem{FOOT4} See for example M.L.Lyra and C.Tsallis, {\sl Phys.
                 Rev. Lett.} {\bf 80} (1998) 53, C.Anteneodo and
                 C.Tsallis, {\sl Phys. Rev. Lett.} {\bf 80} (1998)
                 5313, and references therein. 
 
, which for $q>1$ has the form
\begin{equation}
f(\chi)\, =\,
\frac{\mu}{\Gamma(\alpha')}
       \left(\frac{\mu}{\alpha'}\right)^{\alpha' -1}\, 
       \exp\left( - \frac{\mu}{\alpha'}\right) \label{eq:GD}
\end{equation}

hep-ph0002145.extracted_bib
\begin{thebibliography}{99}
 
 \bibitem{T} C.Tsallis, {\sl J.Stat.Phys.} {\bf 52} (1988) 479;
             for updated bibliography on this subject cf. 
             http://tsallis.cat.cbpf.br/biblio.htm. Recent summary is
             provided in the special issue of {\sl Braz. J. Phys.}
             {\bf 29} (No 1) (1999) (available also at
             http://sbf.if.usp.br/WWW\_pages/Journals/BJP/Vol29/Num1/index.htm). 

 \bibitem{WWq} G.Wilk and Z.W\l odarczyk, {\it Interpretation of the
               nonextensitivity parameter $q$ in some applications of
               Tsallis statistics and L\'evy distributions}, 
               hep-ph/9908459, to be published in {\sl Phys. Rev. Lett.}.

 \bibitem{FOOT1} The relevant references to the up-to-date applications
                 of the Tsallis statistics can be found in \cite{T}
                 whereas \cite{WWq} summarises all recent attempts of
                 its applications in the high energy physics domain.

 \bibitem{FP} N.G. van Kampen, {\it Stochastic Processes in Physics
              and Chemistry}, Elsevier Science Pub. B.V.,
              North-Holland, Amsterdam 1987 (Chapter VIII).

 \bibitem{FOOT2} This is interesting and hot subject in high energy
                 physics, as can be judged from: L.Stodolsky, {\sl 
                 Phys. Rev. Lett.} {\bf 75} (1995) 1044; E.V.Shuryak, 
                 {\sl Phys. Lett.} {\bf B423} (1998) 9 and 
                 S.Mr\'owczy\'nski, {\sl Phys. Lett.} {\bf B430} 
                 (1998) 9.

\end{thebibliography}