Universal Thermal Radiation Drag on Neutral Objects
Vanik Mkrtchian \dag
V. Adrian Parsegian \ddag
Rudi Podgornik \ddag $\bullet$
Wayne M. Saslow \S
We compute the force on a small neutral polarizable object moving at velocity $\vec v$ relative to a photon gas equilibrated at a temperature $T$ We find a drag force linear in $\vec v$. Its physical basis is identical to that in recent formulations of the dissipative component of the Casimir force. We estimate the strength of this universal Casimir drag force for different dielectric response functions and comment on its relevance in various contexts.
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\title{Universal Thermal Radiation Drag on Neutral Objects}

\author{Vanik Mkrtchian \dag, V. Adrian Parsegian \ddag, Rudi Podgornik
\ddag $\bullet$, Wayne M. Saslow \S }

%\vspace{1cm}

\maketitle

\noindent
\dag Institute for Physical Research, Armenian Academy of Sciences,
378410 Ashtarak-2, Armenia \\
\ddag Laboratory of Structural and Physical Biology, National
Institutes of Health, Bethesda, MD \\
$\bullet$ Faculty of  Mathematics
and Physics, University of
Ljubljana, Jadranska 19, SI-1000 Ljubljana,
Slovenia \\
\S Department of Physics, Texas A\&M University, College
Station, Texas 77840-4242

\begin{abstract}
\small
We compute the force on a small neutral polarizable object
moving at velocity $\vec v$ relative to a photon gas equilibrated at
a temperature $T$ We find a drag force linear in $\vec v$.  Its
physical basis is identical to
that in recent formulations of the dissipative component of the
Casimir force. We estimate the strength of this universal Casimir
drag force for different dielectric response functions and
comment  on its relevance in various contexts.

\end{abstract}

PACS numbers:
72.20.-i, 72.20 Jv, 72.40.+w, 73.50.Pz

%\section{Introduction}

The residual drag force on an AFM tip close to, but not in direct
contact with a substrate {\sl in vacuo} raises an important and
fundamental question on the origin of non-contact friction
\cite{Giess}.   Other experimental techniques also
are sensitive to non-contact friction \cite{Gramila}. Since it became  clear that the Casimir effect, being {\sl par excellence} a non-contact phenomenon, can lead to a dissipative drag (see \cite{Pen} and references therein), it has become a primary focus of theoretical research \cite{Kardar}.  Such  Casimir  dissipative drag  occurs when electromagnetic field fluctuations equilibrate in a specific reference frame, relative to which another system (e.g. a dielectric or  a conducting  body) is in uniform motion \cite{Pen, Vanik, Ramin}. The difference between the frames of reference in the dissipative Casimir effect can be due to relative motion of {\sl different} bodies, as for two conducting plates with relative motion in the  parallel direction, or for a neutral body moving relative to a conducting plate  \cite{AE1, Liebsch, TW, VPpart, Pen, VP1}.  In these cases the radiation equilibrates in one of the plates, and the friction depends upon the proximity of the other one.

We show here that such
friction can also come about when a {\sl single} body moves relative
to a thermal bath of the electromagnetic field excitations, such as those
between the walls of
an oven or in the cosmic microwave  background.  The friction
has no position dependence, {\sl i.e.} it is spatially homogeneous.
The consequence is a
universal dissipative drag acting on all matter in relative motion
with respect to a thermalized photon gas. To estimate  the magnitude of this universal drag we evaluate it as a function of the dominant
frequency of the electromagnetic response of the body for dielectrics
and conductors.

Consider the Lorentz
force 
\begin{equation}
	\vec F = \int d^3\vec r \left( \rho \vec E + \vec j
\times \vec B
\right).
\label{eq.1}
\end{equation}
on a dielectric in a field $\vec E(\vec r,
t), ~\vec B(\vec r, t)$. The charge and the
current densities in the dielectric are set by the
polarization, $\vec P(\vec r, t)$, such that $\rho (\vec r, t) = -
\vec\nabla \cdot \vec P$
and $\vec j(\vec r, t) = \partial_{t} \vec
P$, where $\vec P, ~ \vec j$ obviously obey the
continuity equation $\partial_t\rho+\vec\nabla\cdot\vec j=0$.  Assume that the response of the matter to the field is both  linear as well as spatially local.  Thus we write
\begin{equation}
	\vec P (\vec r, t) = \epsilon_{0} \int \chi_e (t -
t') \vec E (\vec
r, t') dt',
\label{eq.1.1}
\end{equation}
in MKS units,
where the dielectric susceptibility $\chi_e$ is dimensionless. In the
frequency domain $\chi_e (\omega)$ is equal to
$(\epsilon (\omega) - 1)/\epsilon_0$ for condensed matter.  For weakly interacting molecules or
atoms $\chi_e= \rho_N \alpha_m$, with $\rho_N$ the atomic or molecular number density of the medium and $\alpha_m$ the polarizability  of the single molecule or atom.

With the polarization proportional to the field, the force is
bilinear in the field.  This bilinearity holds for objects moving at
arbitrary non-relativistic velocities relative to the frame of
reference of the thermal bath.   We require the thermal average of the force acting on a moving body in unbounded space filled with radiation at rest, working in the reference frame where the
particle is instantaneously at rest and the photon gas moves at velocity $\vec v$. The average force is  obtained in terms of the thermal averages of the Fourier components of the
field correlations.  In unbounded space the Fourier components of the
polarization and electric fields
are
\begin{equation}
\vec P(\vec k,
\omega), \vec E(\vec k, \omega)
=\int dt \int d^3\vec r ~\vec P(\vec r, t),
\vec E(\vec r, t)~
e^{-i(\vec k\cdot\vec r-i\omega t)}
\label{eq.2}
\end{equation}
leading to the following form of the thermal
average $\left<
..\right>$ of the Lorentz force
\begin{eqnarray}
\kern-20pt & &\left< \vec F \right> = - i \epsilon_{0} \int d^3\vec
r \int \frac{d^{3} \vec k d \omega}{(2\pi)^{4}} \int \frac{d^{3} \vec k'
d \omega'}{(2\pi)^{4}} e^{-i(\vec k + \vec k') \cdot\vec r + i
(\omega + \omega') t } ~\chi_e(\omega) \times \nonumber\\
	& & \times \left<
\frac{\omega}{\omega'} \left( \vec k' \left( \vec
E (\vec k, \omega) \vec
E (\vec k', \omega') \right) - \left( \vec k'
\cdot \vec E (\vec k,
\omega)\right) \vec E (\vec k', \omega')\right)
\right>,
\nonumber\\
\label{eq.2.1}
\end{eqnarray}
where we took note of the fact that
in empty space the electric field
has no sources.  A slight generalization of the
standard expression (\cite{LL-SM2} Eq.
77.12) for the thermal average of
the correlator of the electric field vectors yields
\begin{equation}
\left<
E_i(\vec k, \omega) E_j(\vec k', \omega') \right> =
(2\pi)^4\delta(\vec
k+\vec k')\delta(\omega+\omega')\left< E_i E_j
\right>_{\vec k, \omega}.
\label{eq.3}
\end{equation}
with
\begin{equation}
<E_i E_j>_{\vec k,
\omega}={2\pi^2\hbar\over {\epsilon_{0} k}}
\left(\frac{\omega^2}{c^2}\delta_{ij}-k_ik_j\right)
\left[ \delta({\omega\over
c}-k)-\delta({\omega\over c}+k)\right]
\left( 1+2n(\omega, \vec k)\right),
\label{eq.4}
\end{equation}
where $k=|\vec k|$ and
\begin{equation}
n(\omega, \vec k)=\frac{1}{e^{\beta\hbar(\omega-\vec
k\cdot\vec v)}-1}
\nonumber
\end{equation}
is the Bose occupation number
for a photon distribution at
temperature $T$, moving with velocity $\vec
v$, and $\beta=(k_B
T)^{-1}$. Eq.~\ref{eq.4} is obtained by generalizing
the  fluctuation-dissipation theorem to a translationally invariant
system, and taking a Gibbs distribution corresponding to a photon gas
moving with velocity $\vec v$, just as done for excitations in superfluidity \cite{LL-SM1}.
\begin{figure}[h]
\begin{center}
\epsfxsize=16cm
%\epsffile{graph1.pdf}
    \epsfig{file=graph1.eps, width=8cm}
\end{center}
\caption{Dependence of the logarithm of the
relaxation time in years on the
logarithm of the primary relaxation frequency ($\omega_0$ in Eq.
\ref{eq.9}) for three
different temperatures, $T=300$~K, $T=1000$~K, $T=3000$~K
(upper, middle and lower curve). The frequency of the minimum is
given by $\hbar \omega_0 =  ~5.9694 ~ k_B T$.}
\label{fig1}
\end{figure}
From Eq. \ref{eq.4} we end up with the following average of the Lorentz force
\begin{equation}
	\left< \vec F \right> = - i
\epsilon_{0} \int d^3\vec r \int
\frac{d^{3} \vec k ~d \omega}{(2\pi)^{4}}
~\chi_e(\omega)~ \vec k
\left< \vec E \cdot \vec E  \right>_{\vec k,
\omega}.
\label{eq.5}
\end{equation}
Substitution of (4) in
(5) with $\vec v=  0$ gives an integral
over $\vec k$ that, by symmetry, is zero.  To obtain a non-zero
result it is convenient to first
eliminate the delta functions by
integrating over $\omega$ and then
expanding in powers of $\vec v$
(as done to obtain the normal fluid
density in the theory of
superfluidity). Integration over $\omega$ then
yields
\begin{eqnarray}
<\vec F> &=& -i~ 2\pi\hbar c ~V~\int{d^3\vec k\over
(2\pi)^3} ~k \vec
k~
\chi_e(ck)\left( 1+2n(ck, \vec k)\right) +
\nonumber\\
& & i~ 2\pi\hbar c ~V~\int{d^3\vec k\over (2\pi)^3} ~k \vec k~
\chi_e(-ck)\left(1+2n(-ck, \vec
k)\right),
\label{force}
\end{eqnarray}
where we have taken the dielectric
to be homogeneous so the integral
over its volume simply gives $V$.

The same equation also can be obtained from energy loss considerations,
following the arguments of Volokitin and Persson \cite{VPpart}. One starts
from the  dissipation rate of the energy in the rest frame of the
fluctuating field, 
\begin{equation}
\frac{dW}{dt} =  \int
d^3\vec r \left( \vec j + \rho \vec v  \right)
\cdot \vec E =
\frac{dW_0}{dt}  - \vec F \cdot \vec v.
\end{equation}
Here we take into
account that in the rest frame of the thermalized
photon gas the total
electric current is given {\sl via} the
linearized Lorentz form $\vec j
\longrightarrow \vec j + \rho \vec
v$. Apart from the heat production $\frac{dW_0}{dt}$ in
the frame of the body,
Eq. \ref{force} is reproduced
immediately.
\begin{figure}[h]
\begin{center}
\epsfxsize=16cm
%\epsffile{graph2.pdf}
    \epsfig{file=graph2.eps, width=8cm}
\end{center}
\caption{Dependence of the relaxation time in
years on temperature. For the dielectric response (upper curve) we plot the minimal
relaxation time obtained for the relaxation frequency that is
proportional to the first Matsubara frequency. We take $\rho_M$ to
be the density of water and the dimensionless measure of atomic polarizability
$\alpha_0 =(N/V)\epsilon\alpha_m \approx 1$. Numerically this
value corresponds to a single particle of molecular polarizability $\epsilon_0 \alpha_m
\approx 1.0 \times 10^{-30}$~m$^3$ and a mass equal to the proton mass $1.67 \times
10^{-27}$~kg. For the metallic response (lower curve) we take the
characteristic time $\tau_0 = \frac{\epsilon_0}{\sigma} \sim 10^{-18}$~s,
well within the range of conductivities of simple metals. }
\label{fig2}
\end{figure}

We now write, in a standard way, $\chi_e(x) =
\chi_e'(x) + i \chi_e''(x)$, noting that the real part is an even and the
imaginary part is an odd function of the argument. The expression for the
force can now be obtained in the following form
\begin{eqnarray}
\kern-30pt <\vec F> = -i~ \frac{4\pi\hbar c
V}{(2\pi)^3}\int{d^3\vec
k} ~k \vec k\!\!\!&&\!\!\!\left[ \chi_e'(ck)
\left(n(ck, \vec k) -
n(-ck, \vec k)\right) +  \right. \nonumber\\
& &
\left. + ~i~\chi_e''(ck) \left( 1 + n(ck, \vec k) + n(-ck, \vec
k)\right)\right].
\end{eqnarray}
Expanding the Bose function to the lowest order in velocity, we have
$ n(ck, \vec k) - n(-ck, \vec k) = \coth{\ul12
\beta \hbar c k} +
{\cal O}(v^2)$ and $ 1 + n(ck, \vec k) + n(-ck, \vec k)
= \ul12 {\rm
csch}^2{\ul12 \beta \hbar c k}~\left(\beta \hbar (\vec k
\cdot \vec
v)\right) + {\cal O}(v^3)$. Placed in Eq. \ref{force}, the $\vec k$-space
integral over the term in $\chi_e'(ck)$ is zero,  by symmetry.  The
remaining term then yields
\begin{equation}
<\vec F> = (8\pi \beta
\hbar^2 c) ~V~\int{d^3\vec k\over (2\pi)^3}
~\frac{k~\vec k (\vec k\cdot\vec v)~ \chi_e''(ck)}{ {\rm sinh}^2{\ul12
\beta \hbar c  k} },
\label{eq.6}
\end{equation}
Performing the angular integral over $d^3 \vec
k$ gives a factor of $4\pi/3$. Reverting again to $\omega$ by
substituting $k = \omega/c$  gives
\begin{equation}
<\vec F> = V~\vec
v~\left({\beta \hbar^{2} \over 3\pi c^5}\right)
~\int_0^\infty d\omega
~\frac{\omega^5 ~ \chi_e''(\omega)}{{\rm
sinh}^{2}(\ul12 \beta \hbar
\omega)}.
\label{eq.7}
\end{equation}
This is the fundamental result of our paper. It states that the EM
field fluctuations exert a drag, proportional in the lowest order to
the velocity, on a particle that moves with respect to the frame of
reference in which the EM field fluctuations are thermalized.  Setting this force to $M\vec v/\tau$, where $M$ is the total mass of  the object and $1/\tau$ is the drag time, and using
$\rho_M=M/V$,  yields the result that
\begin{equation}
{1\over\tau} =
\left({\beta \hbar^{2}\over 3\pi \rho_M
c^5}\right)\int_0^\infty d\omega
~\frac{\omega^5
\chi_e''(\omega)}{\sinh^{2}(\ul12 \beta \hbar \omega)}.
\label{eq.8}
\end{equation}
Regard this fundamental result in
three different  contexts: molecules, dielectric and conducting condensed
matter.  First assume that the dielectric response of the medium
can be characterized by a single sharp absorption line at $\omega_0$.
Because $1\over\tau$ is proportional to $\chi_e''(\omega)$, obviously each of
the absorption lines for a molecule or a dielectric will contribute
additively to the integral in Eq. \ref{eq.8}.
We set $\chi_e''(\omega) =
\alpha_0\delta(\omega/\omega_0-1)$, where
$\alpha_0$ is a constant
describing the strength of the response.
This assumption gives
\begin{equation}
\tau = \left({3\pi n c^5\over \beta
\hbar^{2}}\right)~\frac{\sinh^{2}(\frac{\beta \hbar
\omega_{0}}{2})}{\alpha_{0} \omega_{0}^{6}} = \left({3\pi \rho_M c^5
\hbar^4 \over 2^6 \alpha_{0} (k_B T)^5}\right)~\frac{\sinh^{2}(x)}{
x^6},
\label{eq.9}
\end{equation}
after introducing $x = \ul12 \beta \hbar \omega_0$. In
this form the relaxation time depends strongly on the absorption frequency
$\omega_0$.  It has a minimum at a temperature dependent
frequency (see Fig. \ref{fig1}) that
coincides with the minimum of the function $f(x) =
\frac{\sinh^{2}(x)}{ x^6}$, at $x_m = 2.98$, where  $f(x_m) =
0.137$. Because $f(x)$ is the square of the function
$\frac{\sinh(x)}{ x^3}$, it has a broad minimum with a quartic,
rather than a quadratic, dependence upon the deviation from $x_m$.  Taking
this minimum at $x_m$ into account, the smallest possible relaxation
time can thus be obtained from the above equation in the form
\begin{equation}
\tau = \tau_0 \left(
{T_0 \over T}\right)^5,
\label{findie}
\end{equation}
where $\tau_0 T_0^5
= \left({3\pi f(x_m) \over
2^6}\right)~\frac{\rho_M c^5
\hbar^4}{\alpha_{0} k_B^5}$. At this  minimum the  absorption
frequency $\hbar \omega_0 = ~2 ~x_m ~ k_B T$ is proportional to the first
Matsubara frequency. The temperature dependence of this minimal possible
relaxation time is  given in Fig. \ref{fig2}.

A different formula for
$\tau$ is obtained for metals, which have
constant conductivity at frequencies below the collision time
of their charge-carriers.
In this case $\chi_e(\omega) \approx -\frac{\sigma}{i
\epsilon_0 \omega}$.
Inserting this into Eq. \ref{eq.8}, the inverse
relaxation time for drag
now takes the form
\begin{equation}
{1\over\tau} =
\left({\beta \hbar^{2} \sigma \over 3\pi \rho_M c^5
\epsilon_0} \right)
\int_0^\infty d\omega
~\frac{\omega^4}{\sinh^{2}(\ul12 \beta \hbar
\omega)},
\end{equation}
so that 
\begin{equation}
\tau = \tau_0 \left(  {T_0 \over
T}\right)^4,
\label{finmet}
\end{equation}
with $\tau_0 =
\frac{\epsilon_0}{\sigma}$ and $T_0^4 = \frac{45 c^5
\hbar^3 \rho_M}{16
\pi^2 k_B^4}$. For most common metals the value of
$\tau_0$ is between $10^{-19} - 10^{-17}$~s.  Taking its geometrical average
we obtain the temperature dependence of the relaxation time as
given in Fig. \ref{fig2}.

The times given in the figures are relatively
long,  corresponding to the general weakness of the Casimir interaction and
the even lower strength of the dissipative Casimir effect.
Two circumstances under which such long times might be observable
are ovens and the cosmos.

For ovens, on trapping molecules with
unusually large polarizabilities it might  be possible to observe a
resonance in an ion or atom trap with a quality factor ${\cal Q}$
that is determined by the
dissipative Casimir interaction.  Small metal particles in a high
temperature oven should also be susceptible to the effects of
the Casimir drag.

For the cosmos, it is
believed that hydrogen atoms  condensed from protons and electrons at about
3000~K, and that the  coupling of radiation and matter due to Compton
scattering becomes  ineffective below this condensation temperature
\cite{Peebles}.
However, as is clear from Fig.~2, it should not be  difficult for
molecules to remain coupled to the cosmic microwave background
when the temperature was between about 300~K (and perhaps a bit less)
and the 3000~K condensation temperature.  This
coupling could have an influence on the structure and anisotropies
observed in recent experiments on the cosmic microwave background.
It could also influence the behavior of molecules formed from
the residue of novas and supernovas, and then subject to drag from a
still-hot cosmic microwave (i.e., electromagnetic) background.

The universal thermal drag calculated in this contribution is yet
another important facet of the Casimir effect, which in this context
appears to play a role not only in the static interactions between
dielectrically inhomogeneous bodies but also in the dissipative
dynamics of particles in homogeneous space.

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***What follows is an
aside: 

Kevin Cahill notes that the Pioneer spacecraft is subject to a
small
acceleration of unknown source that is directed radially toward the

Sun, with
$a~8.7\times10^{-7}$~m/s, as given in a report by M.M.Nieto of
Los 
Alamos. Note
that the temperature of the microwave background at the
earth is 
about 2.3~K,
and the dipole anisotropy of the microwave
background is about 
3.3~mK, which
translates to a drift velocity of the
earth relative to the microwave 
background
of
$v_d/c=(3.3/2.3)\times10^{-3}$, or $v_d=4.4\times10^5$~m/s. This 
is about
ten
times the velocity of Mercury around the Sun.

Here are some numbers.
Relative velocities become important here.  A 
google
search gave
$10^4$~m/s as an approximate value for the velocity of 
Pioneer,
about
three or four times slower  than the fastest of the planets,
Mercury.

The temperature of the microwave background at the earth is about

2.3~K, and the
dipole anisotropy of the microwave background is about
3.3~mK, which 
translates
to a drift velocity of the earth relative to the
microwave background 
of
$v_d/c=(3.3/2.3)\times10^{-3}$, or
$v_d=4.4\times10^5$~m/s.  This is 
about ten
times the velocity of Mercury
around the Sun.

Let's assume that Pioneer is moving radially.  Unless by
accident 
Pioneer is
moving opposite to the microwave background, the
radiation pressure 
of the
microwave background would not cause it to move
radially toward the 
Sun.

So I kind of doubt that this mechanism is
operative.  By the way, 
radiation
pressure should give a drag force going
as $PA(v_d/c)$, where $P \sim 
u$ (radiation
pressure and energy density)
$\sim (k_B T)^4/(\hbar c)^3$.  Hence the 
drag rate
should be about
$(1/\tau) \sim (A/M) (k_B T)^4/(\hbar^3 c^4)$.  
Kevin:  If you
accept this
line of reasoning, could you try to calculate a number 
from that?  I
think
that $v_d/\tau$ would give the relevant acceleration.  Kevin 
now seems to

agree that this mechanism is unlikely to explain the unusual drag 
that
has been 
observed. ****





The spirit of the calculation is that the
rate of absorption of 
momentum is 
calculated from 
\begin{equation}

	\vec F=\int d^3\vec r (\rho\vec E+\vec j\times\vec B),

\label{eq.1}
\end{equation}
where the volume integral is over the
polariable material, $\rho$ and 
$\vec j$ 
are the local charge and
currrent densities, and $\vec E$ and $\vec 
B$ are the 
localvalues of the
external fields.  

where the polarization $\vec P$ satisfies
$\rho=-\vec\nabla\cdot\vec 
P$
and $\vec j=\partial_t \vec P$, so that

$\partial_t\rho+\vec\nabla\cdot\vec j=0$. 




Our result is that, for a
particle of mass $m$, polarizability 
$\alpha(\omega)$,
and with $n$ the
Bose factor, the relaxation time $\tau$ is given
by:

${1\over\tau}={2\hbar\over3\pi mc^5}\int_0^\infty d\omega

Im{\alpha(\omega)}
\omega^5({-dn\over d\omega}).$

Thus we note that

\begin{equation}
E_i(\vec r_1, t_1) E_j(\vec r_2,
t_2)
=\int_{-\infty}^\infty{d\omega\over 2\pi}\int{d^3\vec k\over

(2\pi)^3} 
E_i^{(1)}({\vec k, \omega}) e^{i[\vec k\cdot\vec r_1-i\omega
t_1]} 
\int_{-\infty}^\infty{d\omega\over 2\pi}\int{d^3\vec k\over
(2\pi)^3} 
E_j^{(2)}({\vec k', \omega'}) e^{-i[\vec k'\cdot\vec
r_1-i\omega' 
t_2]} 
\label{eq.2}
\end{equation}
For free space, the
thermal average of this quantity depends only on 
the
relative positions
and times, so that it must reduce to the form 
\begin{equation}
<E_i(\vec
r_1, t_1) E_j(\vec r_2, t_2)>
=\int_{-\infty}^\infty{d\omega\over
2\pi}\int{d^3\vec k\over 
(2\pi)^3} 
<E_i E_j>_{\vec k, \omega}e^{i[\vec
k\cdot(\vec r_1-\vec 
r_2)-i\omega(t_1-t_2)]}.

\label{eq.3}
\end{equation}
Consistency between (2) and (3) requires that




For $T=2.3$~K, $\omega=1.8\times10^{12}$~s$^{-1}$, $m$ about 20 times

the proton mass, and $\alpha_0=1.0\times10^{-24}$~cm$^3$, (9)
gives
$\tau=1.3\times10^{21}$~s$^{-1}$. The lifetime of the universe is

thought to be about $4\times10^{17}$~s. The present mechanism hence 
is
not effective for such a temperature and material properties.  
However,
for larger polarizabilities, and at previous epochs where 
the universe was
much hotter, the present mechanism provides a means 
by which neutral,
polarizable matter can equilibrate with respect to 
the background
radiation.

Note that charged particles are subject to drag from Compton

scattering, with $1/\tau_C\sim(\hbar\omega_T^4/m_e c^4)r_0^2$, where 
$r_0
= 2.82\times10^{-15}$~m is the classical radius of the electron,

and
$\omega_T=k_B T/\hbar$ is a thermal frequency.  Drag would act

dominantly on
electrons. In practice, in the early universe the electrons
would 
transfer their momentum to the protons, so the mass used should be

the proton mass. The quantity $R=\alpha(\omega_T/c)/r_0^2$ is a 
relative
measure of the effectiveness of the present mechanism to 
that of Compton
scattering, assuming that the masses are the same in 
each case.  For
$T=2.3$~K, this yields $R\approx 750$, so that the 
present mechanism can
be much more effective than Compton radiation, 
provided that there is a
resonance at thermal frequencies.  





Returning to (1), we fourier
analyze $\vec P(\vec r)$ and $\vec 
E(\vec r)$ in
space and time, over the
volume $V$ of the object using the relation 
$\vec P(\vec r,
\omega)=n_p\alpha_{tot}(\omega)\vec E(\vec r, 
\omega)$.  Here
$n_p$ is the
particle number density $N/V$ of the object, and 
$\alpha(\omega)$
(of
dimension (length)$^3$) is the frequency-dependent polarizability 
of
the
object, per particle.

Using (3), evaluation of the thermal average of
(1) yields   
\begin{equation}
	<F_j>=-iN\int_{-\infty}^\infty{d\omega\over
2\pi}\int{d^3\vec k\over 
(2\pi)^3} k_j 
<E_l E_l>_{\vec k, \omega}
\alpha(\omega).    
\label{eq.5}
\end{equation}

This result can be cast
also into a somewhat more transparent form by 
assuming that the dielectric
body is composed of particles of mass 
$m_{0}$ with a volume $V_0$ per
particle. Introducing the thermal de 
Broglie wavelength of a particle of
mass $m_{0}$ as $\Lambda = ({2\pi 
\beta \hbar^{2} \over m_{0}})^{1/2}$ we
can write Eq. \ref{eq.9} ws
\begin{equation}
{1\over\tau} =
\left(\frac{\Lambda^{2} \omega_{0}^{6}}{6 
\pi^{2}}\right)~V_{0}~\alpha_{0}
~sech^{2}(\frac{\beta \hbar 
\omega_{0}}{2}).

\label{eq.10}
\end{equation}

A better estimate would be obtained if we
use a complete 
phenomenological form for the dielectric function of the
material. 
For definiteness sake let us assume that we are dealing with a

hydrocarbon, such as tetradecane, in vacuum. In this case a good

approximative form of its dielectric response would
be
\begin{equation}
\alpha''(\omega) = \sum_{i=1}^{3}
\frac{\left(\epsilon(0)-1\right) 
\gamma_{i} \omega_{i}^{2}~\omega}{\left(
\omega_{i}^{2} - 
\omega^{2}\right)^{2} + \gamma_{i}^{2}
\omega^{2}},
\end{equation}
corresponding to three consecutive resonances
in the UV region. The 
integral in Eq. \ref{eq.8} now has to be evaluated
numerically.

\bibitem{1} Although this
is a standard result taken from the 
derivative of the
field energy (and
assuming the term in $d(E^2)/dt$ has zero 
time-average), it
can also be
derived as the time-average of $\int d\vec r(\rho\vec 
E+\vec
j\times\vec
B)$, on using Faraday's Law and assuming that $d(\vec 
P\times\vec
B)/dt$
has zero time-average.  Here we take 
$\rho=-\vec\nabla\cdot\vec P$
and
$\vec j=\partial_t\vec P$. 
  



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