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7 pages, LaTeX source in RevTex4 format, with attachment (in the source  
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Brownian motion, virial expansion, the law of large numbers, pair correlation function, foundations of kinetics, 1/f-noise
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\documentclass[twocolumn,showpacs,pra,aps]{revtex4}

\usepackage{dcolumn}
\usepackage{bm}
%\usepackage[english,russian]{babel}

\begin{document}

\title{Molecular Brownian motion and falsity of
the ``\,law of large numbers\,''}

\author{Yuriy E. Kuzovlev}
\email{kuzovlev@kinetic.ac.donetsk.ua} \affiliation{Donetsk Institute
for Physics and Technology (DonPTI NASU), 83114 Donetsk, Ukraine}

%\date{\today}

\begin{abstract}
Real thermal motion of gas particles, free electrons, etc., at long
time intervals (much greater than mean free-flight time) possesses,
contrary to its popular mathematical models, essentially non-Gaussian
statistics, thus implying that chaotic path of a molecular Brownian
particle can not be separated into ``statistically independent''
constituent parts and so the ``law of large numbers'' is not
applicable, even under the Boltzmann-Grad limit. A simple proof of
this statement is suggested basing on only the determinism and
reversibility of microscopic dynamics and besides incidentally
derived virial expansion of the path probability distribution.
\end{abstract}

\pacs{05.20.Dd, 05.40.Fb, 83.10.Mj}

\maketitle

\section{Introduction}
{\bf 1}.\, Thermal chaotic motion of unbound particles of the matter
is mechanism of diffusion as well as many other transport processes.
It is interesting by itself too, since statistics of random walk of a
test or marked ``Brownian'' particle contains rich information about
transport process as a whole including its noise and fluctuations. In
spite of this, dynamical theory of molecular Brownian motion remains
almost undeveloped. Apparently, the common opinion is that anyway it
should confirm the well known beautiful probability-theoretic scheme
based on the celebrated Bernoulli's ``law of large numbers''
\cite{jb} which foretells that at sufficiently large spatio-temporal
scales probability density of displacement, or path, $\,\Delta
\bm{R}\,$, of a Brownian particle (BP) during time $\,t\,$ has
asymptotically universal form of the Gaussian distribution. For
symmetric 3-D random walk that is

$\,V_G(t,\Delta\bm{R})\,=\,(4\pi
Dt)^{-3/2}\,\exp{(-\Delta \bm{R}^2/4Dt)}\,$\,\,.

In fact, it was rigorously proved \cite{sin,gal} that such is the
asymptotic of chaotic walk of hard ball in so non-random environment
as static periodic lattice of elastic scatterers (also hard balls).
It might seem that all the more Brownian motion of gas atoms can not
display something else. Indeed, the Boltzmann-Lorentz equation
\cite{re}, which comes from the Boltzmann equation when adapted to
self-diffusion of gas atoms, produces the Gaussian asymptotic.
Nevertheless, not all is so simple as seems.

{\bf 2}.\, Recall, firstly, that the Boltzmann equation represents
rather rough model resulting from violent truncation of the exact
BBGKY hierarchy of equations and its closure with the help of archaic
Boltzmann's ``molecular chaos'' hypothesis \cite{re,bog,sil}.
However, it was pointed out more than once, e.g. in \cite{re,kac},
that Boltzmann's argumentation \cite{bol} looks enough convincing
only if applied to spatially uniform gas. As to attempts to deduce
``molecular chaos'' from BBGKY equations themselves, they proved to
be unsuccessful \cite{ol}. In opposite, it was shown \cite{i1} (see
also \cite{i2,p1}) that reformulation of BBGKY equations in terms of
inter-particle ``collisions'', instead of continuous interactions,
leads to such new infinite hierarchy of equations which is evidently
incompatible with ``molecular chaos'' if a gas is spatially
non-uniform. The matter is that the very fact of participation of
particles in same collision (or connected group of collisions) {\it
at certain place} (distinguishable due to non-uniformity) is
sufficient reason for mutual statistical dependence of the particles.

Various (higher-order, time-non-local, etc.) corrections to the
Boltzmann equation for comparatively dense gas do not change the
matter, since all they eventually exploit the same ``molecular
chaos'' assumption and therefore in strict sense touch uniform gas
states only.

At the same time, if wishing to consider displacement of BP (test
particle) one has to localize its initial position and thereby
disturb uniformity (translational invariance) of infinitely many
distribution functions (DF) what describe BP under its interactions
with gas. Consequently, after all one inevitably has to deal with
infinite chain of equations. Attempts to construct their approximate
solutions for comparatively rare, or weakly non-ideal, gas (formally,
in the Boltzmann-Grad limit) were made in works \cite{i1} (see also
\cite{i2}) and \cite{p1}. Their findings, of course, qualitatively
differ from the Gaussian asymptotic (but qualitatively confirm the
early phenomenological theory \cite{pjtf,bk12,bk3}).

{\bf 3}.\, Secondly, Gaussian asymptotic of the ball's walk in static
lattice is closely connected with its ergodicity \cite{sin,gal,ar}:
results of time averaging over $\,n\hm\rightarrow\infty $ fragments
of its trajectory (with each fragment consisting of free flight and
collision) almost surely are independent on the trajectory. This is
possible due to the fact that any trajectory is fully determined by
its single fragment (e.g. initial one), and a ratio of number,
$\,d=5n\,$, of quantities describing the trajectory to the number,
$p=5\,$, of its specifying parameters is unrestrictedly large,
$d/p\hm\rightarrow\infty \,$. Infinitely many observations per
parameter are sufficient to exclude specificity of almost any
trajectory.

But a gas is quite another matter. Here any trajectory of Brownian
(test) particle is made of $\,n\approx t/\tau\,$ similar fragments,
with $\,t\,$ being observation time and $\,\tau\,$ mean free flight
time, and is exhaustively characterized by $\,d=6n\,$ quantities.
However, it is specified by not only initial state of the particle
under consideration but also initial states of many other particles.
It is easy to see \cite{p1} that a number $\,m\,$ of such particles
grows with time by far faster than $\,n\,$, so that
$\,n/m\hm\rightarrow 0\,$. Thus now the ratio of number of quantities
what completely describe all details of a trajectory to number,
$\,p=6m\,$, of parameters (initial conditions) what determine these
details, tends to zero, $\,d/p\hm\rightarrow 0\,$, as contrasted with
above case. In my opinion, this big nothing of observations per
parameter is insufficient to get rid of specificity of trajectories
(though standard ``logics'' would give rise to exactly opposite
conclusion about ``stochasticity'' and statistical identity of
trajectories).

By the above reason, hardly results of time averaging (for instance,
such as ``particle's diffusivity'' or ``collision probabilities'')
will turn out to have ``almost surely'' same limits for different
trajectories. There are no grounds for ergodicity, the ``law of large
numbers'' (which needs in beforehand prescribed ``collision
probabilities'' \cite{jb,i2,p3}) and Gaussian asymptotic. Even in
finite gas of $\,N<\infty\,$ particles (in box or torus) necessary
grounds for ergodicity arise not earlier than after time $\,>N\tau\,$
\cite{p1}. This gives illustration of footnote remark in \cite{ar}
that a value of ergodicity in physics looks overestimated since limit
$\,N\hm\rightarrow \infty $ is much more important for physics than
limit $\,t\rightarrow\infty \,$. Besides, in essence, analogous
warning was highlighted already by N.\,Krylov in the forties. See his
book \cite{kr} where he tried to disclose such prejudices that ``some
probabilistic law does exist regardless of a theoretical construct
and full-scale experiments'' or that ``obviously independent
phenomena should possess independent probability distributions''.

{\bf 4}.\, In principle, similar reasonings \cite{i2,bk3,i3,i4}, as
well as equations like the BBGKY hierarchy, are applicable to any
realistic many-particle system, and therefore particular case of gas
has general physical significance. But BBGKY equations yet are so bad
investigated that none supplementary tools would be superfluous. In
the present paper, basing in fact on determinism and reversibility of
Hamiltonian microscopic dynamics only, we will derive a kind of
virial expansion for true probability distribution of the BP's
displacement, $\,V_0(t,\Delta\bm{R})\,$. Then connect coefficients of
this expansion with joint correlation functions of BP and gas and
with usual many-particle DF. To make the consideration purified of
non-principal complications, the limit of Boltzmann-Grad gas is
convenient. Finally, show that all these taken together imply an
interesting differential inequality for $\,V_0(t,\Delta\bm{R})\,$
which definitely forbids Gaussian asymptotic of
$\,V_0(t,\Delta\bm{R})\,$. Instead, it points out essentially
non-Gaussian asymptotic possessing long (though somehow truncated)
power-law tail.

It must be underlined that we will not use the BBGKY or any other
differential equations. Thus we may forget all previous ``world
outlook'' text and start afresh, without detriment to our
results. Anyway, they will appear in agreement with non-Gaussian
distribution of BP's path obtained in \cite{p1} from approximative
solution of the BBGKY equations. From the other hand, the present
paper, along with its short prototype \cite{pro}, logically continues and
improves paper \cite{p3} where gas density perturbation was under
attention in place of individual test particle. Careful comparison of
the one and another will be theme of separate work.

\section{Many-particle correlations and virial expansion of
Brownian path probability distribution} {\bf 1}.\, Let a gas of $N\gg
1\,$ atoms in volume $\,\Omega $ contains a ``Brownian particle''
(BP). Consider, under the thermodynamical limit $N\rightarrow\infty
$, $\Omega\rightarrow\infty $, $N/\Omega \hm =\nu_{\,0}\hm =
\,$\,\,const\,, statistical ensemble of phase trajectories of this
system which responds to canonical equilibrium distribution of its
initial state at $\,t=0\,$. At that we suppose that at $\,t>0\,$ BP
may be subject to constant external force $\,\bm{f}\,$.

The time reversibility of Hamiltonian dynamics and its
phase trajectories implies many relations between statistical
characteristics of their ensemble \cite{jetp1,jetp2,p}. In
particular, if Hamiltonian of the system is invariant of inversion
$\,\bm{p}\rightarrow -\bm{p}\,$, where
$\,\bm{p}=\{\bm{P},\bm{p}_1,...\,,\bm{p}_N\}\,$ are momentums of BP
and atoms, then according to \cite{jetp1} one can write
\begin{equation}
\begin{array}{c}
\langle A(\bm{q}(t))B(\bm{q}(0))\,e^{-\,\mathcal{E}(t)/T}\,\rangle
=\langle B(\bm{q}(t))A(\bm{q}(0))\rangle\,\label{sim0}
\end{array}
\end{equation}
Here angle brackets $\langle ...\rangle\,$ designate the ensemble
average, $\,\bm{q}(0)=\bm{q}=\{\bm{R},\bm{r}_1,...\,,\bm{r}_N\}$ are
space coordinates of BP and atoms, $\,\mathcal{E}(t)=\bm{f}\cdot
[\bm{R}(t)-\bm{R}(0)]\,$ is work made by the external force during
time interval $\,t\,$, $\,A(\bm{q})\,$ and $\,B(\bm{q})\,$ are
``arbitrary functions'', and $\,T\,$ is initial temperature of the
system. Notice that equality (\ref{sim0}) holds also when BP has
internal degrees of freedom. In case of no external force, when
$\,\bm{f}=0\,$ and hence $\,\mathcal{E}(t)=0\,$, it looks so obvious
that even does not need in derivation or references.

Let $\,B(\bm{q})=\Omega\,\delta(\bm{R}-\bm{R}_0)\exp{[-\sum_j
U(\bm{r}_j)/T\,]}$ and $A(\bm{q})=\delta(\bm{R}-\bm{R}^{\prime})\,$.
Then right-hand side of (\ref{sim0}) in the thermodynamical limit
takes form
\[
\begin{array}{c}
\langle B(\bm{q}(t))A(\bm{q})\rangle\, \rightarrow \\
\rightarrow\,\mathcal{F}\{t,\bm{R}_0,\phi|\bm{R}^{\prime}\}\,\equiv\,
V_0(t,\bm{R}_0|\bm{R}^{\prime})\,+\\
+\sum_{n\,=1}^{\infty } (\nu_{\,0}^n/n!)\int^n F_n(t,\bm{R}_0,
\bm{r}_1\,...\,\bm{r}_n|\bm{R}^{\prime})\prod_{j\,=1}^n
\phi(\bm{r}_j)\,\,,
\end{array}
\]
where\, $\,\phi(\bm{r})=\exp{[-\,U(\bm{r})/T\,]}-1\,$,\, symbol
$\,\int^n \,$ means integration over $\,\bm{r}_1...\,\bm{r}_n\,$,
function\, $\,V_0(t,\bm{R}_0|\bm{R}^{\prime})\,$ is conditional
probability density of finding BP at $\,t\geq 0\,$ at point
$\,\bm{R}_0\,$\, under condition that BP had started at $\,t=0\,$
from point $\bm{R}^{\prime}\,$,\, and
$\,F_n(t,\bm{R}_0,\bm{r}_1\,...\,\bm{r}_n|\bm{R}^{\prime})\,$ is
joint conditional probability density of simultaneous finding BP at
point $\,\bm{R}_0\,$ and some atoms at points $\,\bm{r}_j\,$\, under
the same condition.

Notice that the series here converges and the limit functional
$\,\mathcal{F}\{t,\bm{R}_0,\phi|\bm{R}^{\prime}\}\,$ is well defined
at least when\,\, $\,\int |\phi(\bm{r})|\,d\bm{r}\,<\,\infty\,$\,, in
analogy with the generating functional originally considered by
Bogolyubov \cite{bog}. At that, functions $\,F_n\,$ are fully
analogous to usual non-normalized DF of infinite gas \cite{bog,sil}.
In particular, in respect to arguments $\,\bm{r}_j\,$ (coordinates of
atoms) the role of normalization is played by the ``principle of
decay of correlations'':
$F_n(t,\bm{R}_0,...\,\bm{r}_k...\,|\bm{R}^{\prime}) \rightarrow
F_{n-1}(t,\bm{R}_0,...\bm{r}_{k-1},
\bm{r}_{k+1}...\,|\bm{R}^{\prime})\,$ if $\,\bm{r}_k\rightarrow
\infty\,$, and \,$F_1(t,\bm{R}_0,\bm{r}_1|\bm{R}^{\prime})
\rightarrow V_0(t,\bm{R}_0|\bm{R}^{\prime})\,$ if
$\,\bm{r}_1\rightarrow \infty\,$. But, due to the initial
localization of BP, in respect to argument $\,\bm{R}_0\,$ all the DF
are normalized in literal sense. In particular, $\,\int
V_0(t,\bm{R}_0|\bm{R}^{\prime})\,d\bm{R}_0 =1\,$.

Ratios $\,F_n(t,\bm{R}_0,\bm{r}_1\,...\,\bm{r}_n|\bm{R}^{\prime})
/V_0(t,\bm{R}_0|\bm{R}^{\prime})\,$ represent conditional DF of gas,
under condition that both initial and current positions of BP are
known.

Correspondingly,\,
$\,V_0(0,\bm{R}_0|\bm{R}^{\prime})=\delta(\bm{R}_0-\bm{R}^{\prime})\,$\,
and
\[
\begin{array}{c}
F_n(0,\bm{R}_0,\bm{r}_1...\,\bm{r}_n|\bm{R}^{\prime})=
\delta(\bm{R}_0-\bm{R}^{\prime})
\,F_n^{(eq)}(\bm{r}_1...\,\bm{r}_n|\bm{R}_0)\,\,, \\
\mathcal{F}\{0,\bm{R}_0,\phi|\bm{R}^{\prime}\}=
\delta(\bm{R}_0-\bm{R}^{\prime})\,
\mathcal{F}^{(eq)}\{\phi |\bm{R}_0\}\,\,,\\
\mathcal{F}^{(eq)}\{\phi |\bm{R}_0\}\,\equiv \,1\,+ \\
+\sum_{n\,=1}^{\infty }(\nu_{\,0}^n/n!)\int^n
F_n^{(eq)}(\bm{r}_1...\,\bm{r}_n|\bm{R}_0)
\prod_{j\,=1}^n \phi(\bm{r}_j)\,\,,
\end{array}
\]
where $\,F_n^{(eq)}(\bm{r}_1...\,\bm{r}_n|\bm{R}_0)\,$ are
conditional equilibrium DF of gas under fixed position of BP and
$\,\mathcal{F}^{(eq)}\{\phi |\bm{R}_0\}\,$\, is their generating
functional. And, of course, all DF obey translational invariance:
$\,V_0(t,\bm{R}_0|\bm{R}^{\prime})=V_0(t,\bm{R}_0-\bm{R}^{\prime})\,$,
\, $\,F_n(t,\bm{R}_0,
\{\bm{r}\}|\bm{R}^{\prime})=F_n(t,\bm{R}_0-\bm{R}^{\prime},
\{\bm{r}\}-\bm{R}^{\prime}|0)\,$, etc.

{\bf 2}.\, By definition,
$\,F_n(0,\bm{R}_0,\bm{r}_1\,...\,\bm{r}_n|\bm{R}^{\prime})\,$ include
all equilibrium correlations between BP and atoms. But any evolution
of (ensemble of states of) the system is conjugated with disturbance
of detailed balance of collisions and therefore births additional
correlations. The latter reflect joint participation of particles in
excess collisions or ``joint nonparticipation'' in missing ones.
These correlations might be qualified as non-equilibrium ones, but
with those reservation that sometimes they describe evolution of not
so much a system on its own as our knowledge about it (indeed, if
$\,\bm{f}=0\,$ then the system always remains in equilibrium state
and the only thing what is under change is our information about
position of BP).

Traditionally, contributions of these correlations to DF are termed
``correlation functions'' (CF) \cite{re,bog,sil,bal}. Let us
designate them as
$\,V_n(t,\bm{R}_0,\bm{r}_1\,...\,\bm{r}_n|\bm{R}^{\prime})\,$ and
pick out them from $\,F_n(t>0,...)\,$ by definition as follows:
\[
\begin{array}{c}
\mathcal{F}\{t,\bm{R}_0,\phi|\bm{R}^{\prime}\}\,=\,
\mathcal{F}^{(eq)}\{\phi |\bm{R}_0\}\,\,
[\,V_0(t,\bm{R}_0|\bm{R}^{\prime})\,+\\ +\sum_{n\,=1}^{\infty }
(\nu_{\,0}^n/n!)\int^n V_n(t,\bm{R}_0,\bm{r}_1...\,\bm{r}_n|
\bm{R}^{\prime})\prod_{j\,=1}^n \phi(\bm{r}_j)\,]\,
\end{array}
\]
Of course, it is not solely possible definition of the ``correlation
functions'' but it is most suitable for our needs below. Importantly,
it ensures satisfaction of the necessary requirements that
$\,V_n(0\,,...\,)=0\,$ at $\,n\geq 1\,$ and
$\,V_n(t,\bm{R}_0,...\,\bm{r}_k...| \bm{R}^{\prime})\rightarrow
0\,$\, at\, $\,\bm{r}_k\rightarrow \infty\,$.

Particularly,
\[
\begin{array}{c}
F_1(t,\bm{R}_0,\bm{r}_1|\bm{R}^{\prime})\, = \\
=V_0(t,\bm{R}_0|\bm{R}^{\prime})F_1^{(eq)}(\bm{r}_1|
\bm{R}_0)+V_1(t,\bm{R}_0,\bm{r}_1|\bm{R}^{\prime})\,\,\,,
\end{array}
\]
where function $\,V_1(t,\bm{R}_0,\bm{r}_1|\bm{R}^{\prime})\,$
represents pair correlations between positions of BP and atoms. This
equality can be considered as consequence of similar equality for
pair DF and CF defined in the full two-particle phase space:
\begin{equation}
\begin{array}{c}
F_1(t,\bm{R}_0,\bm{r}_1,\bm{P}_0,\bm{p}_1|\bm{R}^{\prime})\,=\,
V_0(t,\bm{R}_0,\bm{P}_0|\bm{R}^{\prime})\times \label{cf1}
\\ \times\, F_1^{(eq)}(\bm{r}_1,\bm{p}_1|\bm{R}_0)+
V_1(t,\bm{R}_0,\bm{r}_1,\bm{P}_0,\bm{p}_1|\bm{R}^{\prime})\,\,\,,
\end{array}
\end{equation}
where $\,F_1^{(eq)}(\bm{r}_1,\bm{p}_1|\bm{R}_0)=
F_1^{(eq)}(\bm{r}_1|\bm{R}_0)\,G^{(eq)}(\bm{p}_1)\,$,\, with
$\,G^{(eq)}(\bm{p})\,$\, being the Maxwellian distribution of atomic
momentum, and
$\,V_1(t,\bm{R}_0,\bm{r}_1,\bm{P}_0,\bm{p}_1|\bm{R}^{\prime})\,$ is
direct analogue of the usual ``pair correlation function''
\cite{sil,bal}. It produces the above defined pair correlation in
configurational space after integration over momentums:
\begin{equation}
\begin{array}{c}
V_1(t,\bm{R}_0,\bm{r}_1|\bm{R}^{\prime})=\int \int
V_1(t,\bm{R}_0,\bm{r}_1,\bm{P}_0,\bm{p}_1|\bm{R}^{\prime})\,
d\bm{P}_0\,d\bm{p}_1 \label{p1}
\end{array}
\end{equation}

{\bf 3}.\, Next, turn to left side of (\ref{sim0}). Multiply and
divide it by $\,\langle B(\bm{q})\rangle\,$\, and use the fact that
any expression $\,\langle \Phi\,B(\bm{q})\rangle /\langle
B(\bm{q})\rangle\,$, with $\,B(\bm{q})\geq 0\,$ and $\,\Phi\,$ being
some functional of system's phase trajectory, in view of determinism
of evolution along any concrete trajectory, can be interpreted as
$\,\Phi$'s\, average over new ensemble of trajectories induced by
new, non-equilibrium, ensemble of initial coordinates and momentums
of the system. Namely, the ensemble determined by probability
distribution $\,\rho (\bm{q},\bm{p})= B(\bm{q})\,
\rho_0(\bm{q},\bm{p})/\langle B(\bm{q})\rangle\,$, where
$\,\rho_0(\bm{q},\bm{p})\,$ denotes original equilibrium
distribution, in whose terms $\,\langle B(\bm{q})\rangle = \int\int
B(\bm{q})\rho_0(\bm{q},\bm{p})\,d\bm{q}d\bm{p}\,$\,. Noticing also,
from the other hand, that $\,\langle B(\bm{q})\rangle
=\mathcal{F}^{(eq)}\{\phi |\bm{R}_0\}\,$, we have
\[
\begin{array}{c}
\langle A(\bm{q}(t))B(\bm{q}(0))\,e^{-\,\mathcal{E}(t)/T}\,\rangle
\rightarrow\\ \rightarrow\,V\{t,\bm{R}^{\,\prime}|\phi ,\bm{R}_0
\}\,\,e^{-\,\bm{f}\cdot [\,\bm{R}^{\,\prime}-\,\bm{R}_0]/T}\,
\mathcal{F}^{(eq)}\{\phi |\bm{R}_0 \}\,\,\,,
\end{array}
\]
where $\,V\{t,\bm{R}^{\prime}|\phi ,\bm{R}_0 \}\,$ is probability
density of finding BP at time $\,t\,$ at point $\,\bm{R}^{\prime}\,$
under conditions that initially it was placed at point $\,\bm{R}_0\,$
while the gas was in such perturbed spatially nonuniform
non-equilibrium state which would be equilibrium in presence of
external potential $\,U(\bm{r})\,$. Spatial distribution of mean
concentration of atoms in such nonuniform state is expressed by
\[
\begin{array}{c}
\nu\{\bm{r}|\phi ,\bm{R}_0 \} = [1+\phi(\bm{r})]\,\delta
\ln\mathcal{F}^{(eq)}\{\phi |\bm{R}_0 \}/\delta\phi(\bm{r})
\end{array}
\]

Hence, in total, we come to formally exact relation
\begin{equation}
\begin{array}{c}
V\{t,\bm{R}^{\prime}|\phi ,\bm{R}_0 \}\,\,e^{-\,\bm{f}
\cdot[\,\bm{R}^{\,\prime}-\,\bm{R}_0]/T}
\,=\,V_0(t,\bm{R}_0|\bm{R}^{\,\prime})\,+\label{r}\\
+\sum_{n\,=1}^{\infty } (\nu_{\,0}^n/n!)\int^n
V_n(t,\bm{R}_0,\bm{r}_1...\,\bm{r}_n|\bm{R}^{\,\prime})
\prod_{j\,=1}^n \phi(\bm{r}_j)\,\,\,
\end{array}
\end{equation}
which connects, from one (left) hand, probability distribution of
BP's path in initially non-equilibrium nonuniform gas and, from the
other (right) hand, probability distribution of BP's path, along with
generating functional of ``non-equilibrium correlations'' between
this previously accumulated path and current BP's environment, in
initially equilibrium uniform gas. In case $\,\bm{f}=0\,$, therefore,
right-hand side of (\ref{r}) represents wholly equilibrium Brownian
motion.

It goes without saying that all the DF and CF depend on the mean gas
density $\,\nu_{\,0}\,$ (and on the force $\,\bm{f}\,$ if any) but
for brevity in (\ref{r}), as well as before it and somewhere below,
corresponding arguments are omitted.

{\bf 4}.\, Further, make a special choice of the function
$\,\phi(\bm{r})\,$. Let $\,\phi(\bm{r})=\phi =\,$const\, inside some
sphere $\,|\bm{r}-\bm{R}_0|< \xi\,$ and (fast but smoothly) vanishes
outside it. Since $\,\phi(\bm{r})\,$ is absolutely integrable, radius
$\,\xi\,$ must be finite but can be as large as we want. For example,
$\,\xi = k\,v_s t_0\,$, where $\,v_s\,$ is speed of sound in our gas,
$\,t_0\,$ is maximal duration of watching BP, and $\,k>1\,$. Then, if
the external force is not too strong and $\,k\,$ is sufficiently
large, we can be sure that at $\,t<t_0\,$ the Brownian motion
mentioned in left part of (\ref{r}) practically takes place in
equilibrium uniform gas with constant mean density $\,\nu =\,$const\,
what corresponds to $\,\phi =\,$const\,. More precisely, $\,\nu\,$
equals to $\,\nu\{\bm{r}|\phi ,\bm{R}_0 \}\,$ taken far from BP, at
$\,|\bm{r}-\bm{R}_0|\gg r_B\,$ with $\,r_B\,$ being radius of pair
interaction between BP and atoms.

Under formulated conditions, clearly, the functional
$\,V\{t,\bm{R}^{\prime}|\phi ,\bm{R}_0 \}\,$ simplifies to mere
function $\,V_0(t,\Delta\bm{R}\,;\,\nu )\,$, where we introduce
$\,\Delta \bm{R}=\bm{R}^{\prime}-\bm{R}_0\,$ and add the density
argument $\,\nu \,$, taking in mind that it is definite function of
$\,\phi \,$ and the seed density, $\,\nu =\nu(\nu_{\,0}\,,\phi )\,$.

On right-hand side of (\ref{r}), under the same conditions, {\it a
fortiori} none correlations might propagate out of the sphere
$\,|\bm{r}-\bm{R}_0|< \xi\,$. Therefore in all integrals
$\,\phi(\bm{r})\,$ can be replaced by the constant.

As the consequence, relation (\ref{r}) transforms into
\begin{equation}
\begin{array}{c}
V_0(t,\Delta\bm{R}\,;\,\nu(\nu_{\,0},\phi
))\,\,e^{\,-\bm{f}\cdot\Delta
\bm{R}/T}\,=\,V_0(t,-\Delta\bm{R}\,;\,\nu_{\,0})
\,+\,\,\,\,\,\,\,\,\,\,\,\,\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\, \sum_{n\,=\,1}^{\infty
} \,\phi^n \,V_n(t,-\Delta\bm{R}\,;\,\nu_{\,0})/n!\, \label{r2}
\end{array}
\end{equation}
with functions $\,V_n\,$, at $\,n>0\,$, defined as
\begin{equation}
\begin{array}{c}
V_n(t,\bm{R}_0-\bm{R}^{\prime}\,;\,\nu_{\,0})\,=\,\nu_{\,0}^n\int^n
V_n(t,\bm{R}_0,\bm{r}_1...\, \bm{r}_n|\bm{R}^{\prime}) \label{clim0}
\end{array}
\end{equation}
The result (\ref{r2}), together with (\ref{clim0}), may be qualified
as ``virial expansion'' of the BP's path probability distribution,
$\,V_0(t,\Delta\bm{R}\,;\,\nu )\,$, in gas with given density.

At $\,n=1\,$, in particular, formulas (\ref{r2}) and (\ref{clim0})
yield
\begin{eqnarray}
\widetilde{\nu }_0\,\,\frac {\partial
V_0(t,\bm{R}^{\prime}-\bm{R}_0\,;\,\nu_{\,0})}{\partial \nu_{\,0}}\,
\,\,e^{\,-\bm{f}\cdot [\bm{R}^{\prime}-\bm{R}_0]/T}\,=
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\label{r21}\\
\,\,\,\,=\,V_1(t,\bm{R}_0-\bm{R}^{\prime}\,;\,\nu_{\,0})\,\equiv
\,\nu_{0}\int V_1(t,\bm{R}_0,\bm{r}_1|\bm{R}^{\prime})\,d\bm{r}_1\,
\,\,,\nonumber
\end{eqnarray}
where, as one can easy verify,
\begin{eqnarray}
\widetilde{\nu }_0\,\equiv \,\left [\frac {\partial \nu
(\nu_{\,0},\phi )}{\partial \phi }\right ]_{\phi =0} =\,\nu_0
+\nu_0^2\,\int [F_2^{(eq)}(\bm{r})-1\,]\,d\bm{r}\,\,\,, \nonumber
\end{eqnarray}
with $\,F_2^{(eq)}(\bm{r})\,$ being usual (unconditional) pair DF of
equilibrium gas with density $\,\nu_{\,0}\,$. Notice that function
$\,V_1(t,\bm{R}_0-\bm{R}^{\prime}\,;\,\nu_{\,0})\,$ can be
interpreted as pair correlation of BP with whole gas but counted per
(elementary volume $\,\nu_0^{-1} \,$ what falls at) one gas atom.

\section{Bounds of pair correlation, rate of change of
the path distribution and failure of Gaussian asymptotic} {\bf 1}.\,
It is time to make some simplifications. Below we will assume that
(i) BP is of molecular size, i.e. $\,r_B\sim r_A\,$, where $\,r_A\,$
is radius of atom-atom interaction, (ii) all interactions are
short-range and repulsive, and (iii) the gas parameter is small,
$\,4\pi r_A^3\nu_0/3\,\ll 1\,$ , i.e. gas is ``dilute'', or ``weakly
non-ideal''. If necessary it can be turned into ``ideal weakly
non-ideal gas'' under the formal Boltzmann-Grad limit, when
$\,\nu_{0}\rightarrow\infty\,$ while $\,r_B\sim r_A \rightarrow 0\,$
in such way that the gas parameter vanishes but mean free paths of
atoms, $\,\lambda =(\pi r_A^2\nu_0 )^{-1}\,$, and BP, $\,\Lambda
=(\pi r_B^2\nu_0 )^{-1}\,$, stay fixed. Besides, below we confine
ourselves by the case of no external force, $\,\bm{f}=0\,$. Then both
sides of (\ref{r2}) and (\ref{r21}) describe equilibrium and hence
{\it spherically symmetric} Brownian motion.

It is not hard to show that under the Boltzmann-Grad limit (BGL), in
(\ref{r2}) and (\ref{r21}),
\[
\nu(\nu_{0},\phi )/\nu_{0} \rightarrow 1+\phi
\,\,\,,\,\,\,\,\,\,\widetilde{\nu }_0/\nu_{0}\rightarrow 1
\]
At that, $\,V_0(t,\Delta\bm{R}\,;\,\nu )\,$ must have finite limit,
since in fact it depends on not bare density $\,\nu\,$ itself but on
a rate of collisions, $\sim \pi r_B^2\nu \,$, or corresponding mean
free paths of BP, $\,\widetilde{\Lambda} =(\pi r_B^2\nu
)^{-1}=\Lambda/(1+\phi)$, and atoms, $\widetilde{\lambda} =(\pi
r_A^2\nu )^{-1}=\lambda/(1+\phi)\,$, which remain constant.

This means that on right-hand side of (\ref{r2}) and (\ref{r21}) all
$\,\,V_n(t,\bm{R}_0-\bm{R}^{\prime}\,;\,\nu_{0})\,$ also have finite
limits. Moreover, for arbitrary function $\,\phi(\bm{r})\,$, whose
spatial scale is fixed in units of $\,\lambda\,$, the BGL yields\,
$\,\nu\{\bm{r}|\phi ,\bm{R}_0 \}/\nu_{0}\rightarrow
1+\phi(\bm{r})\,$, and it follows from (\ref{r}) that any of the
integral quantities $\,\nu_{\,0}^n\int^n V_n(t,\bm{R}_0,\bm{r}_1...\,
\bm{r}_n|\bm{R}^{\prime})\prod_{j\,=1}^n \phi(\bm{r}_j)\,$\, possess
finite limit.

Hence, even in arbitrary ``dilute'', or Boltzmann-Grad, gas {\it
spatial} pair and many-particle correlations keep safe in relative
sense, if counted per atom, although they disappear in literal sense:
$\,V_n(t,\bm{R}_0,\bm{r}_1...\,\bm{r}_n| \bm{R}^{\prime})\rightarrow
0\,$. To make them clear, consider {\it full} phase space
correlations, firstly recollecting those few things what are known
about them from classical books (e.g. \cite{re,bog,sil,bal}),
especially about the pair CF,
$\,V_1(t,\bm{R}_0,\bm{r}_1,\bm{P}_0,\bm{p}_1|\bm{R}^{\prime})\,$.

{\bf 2}.\, According to \cite{re,bog,sil,bal}, in respect to relative
distance between two particles (in our case, BP and an atom),
$\,\bm{r}_1-\bm{R}_0\,$, the pair CF differs from zero inside a
``{\it collision cylinder}'' only which has radius $\,\approx r_B\,$
and is directed in parallel to relative velocity of the particles,
$\,\bm{v}_1-\bm{V}_0\,$. It is known also that characteristic value
of the pair CF in this cylinder is comparable with product of
one-particle DF, that is first right-hand term in (\ref{cf1}). In
other words, in arbitrary ``dilute'' gas the full phase space pair
correlation keeps safe in magnitude, although inside more and more
narrow (hyper-)cylinder only.

Unfortunately, conventional theory what hopes for the ``molecular
chaos'' can not say anything about length of the collision cylinder,
because completely neglects contributions from three-particle
correlations and, all the more, higher-order ones. But taking them
into account means allowing collisions of the pair with ``third
particles'' (the rest of gas). We can surmise that then significant
pair correlations could exist only not too far from the place of
possible contact, $\,|\bm{r}_1-\bm{R}_0|\lesssim r_B\,$. More
concretely, at $\,|r_{\|}|\,\lesssim \,\Lambda^{\prime} \,$, where
$\,r_{\|}=(\bm{v}_1-\bm{V}_0)\cdot
(\bm{r}_1-\bm{R}_0)|/|\bm{v}_1-\bm{V}_0|\,$ designates longitudinal
internal coordinate in the collision cylinder and
$\,\Lambda^{\prime}\,$ is comparable with $\,\Lambda\,$ (for
instance, $\,\Lambda^{\prime} = \lambda \Lambda /(\lambda
+\Lambda)\,$). In other words, effectively the collision cylinder has
finite length $\,\Lambda_C \sim 2\Lambda^{\prime}\,$. Stronger
separated particles hardly are candidates for mutual collision even
if they aim exactly one to another.

Corresponding effective volume of the collision cylinder is
$\,\Omega_C\sim \pi r_B^2\Lambda_C =\Lambda_C/\nu_{\,0}\Lambda \sim
\nu_{\,0}^{-1}\,$. Hence, in spite of all limitations, totally the
pair correlation occupies the whole of volume displayed per one atom.

The same conclusions directly follow from relations
(\ref{r2})-(\ref{r21}) as combined with (\ref{p1}). Indeed,
rearranging integrations over $\,\bm{r}_1\,$ and momentums, one can
write
\[
\begin{array}{c}
V_1(t,\bm{R}_0-\bm{R}^{\prime}\,;\,\nu_{\,0})\,=\,\\
=\,\int\int \,[\,\nu_{0}\int
V_1(t,\bm{R}_0,\bm{r}_1,\bm{P}_0,\bm{p}_1|\bm{R}^{\prime})\,d\bm{r}_1\,]\,
d\bm{P}_0\,d\bm{p}_1\,
\end{array}
\]
and then, due to the mentioned knowledge on pair CF,
\[
\begin{array}{c}
\nu_{0}\int
V_1(t,\bm{R}_0,\bm{r}_1,\bm{P}_0,\bm{p}_1|\bm{R}^{\prime})\,d\bm{r}_1\,=
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\nu_{0}\,\Omega_C\,
\overline{V}_1(t,\bm{R}_0-\bm{R}^{\prime},\bm{P}_0,\bm{p}_1)\,\,\,,
\end{array}
\]
where $\,\Omega_C\,$ is volume of those central region of the
collision cylinder where the pair CF significantly differs from zero,
and
$\,\overline{V}_1(t,\bm{R}_0-\bm{R}^{\prime},\bm{P}_0,\bm{p}_1)\,$
represents characteristic or mean value of the pair CF inside this
region. Then the above underlined finiteness, and invariance, of both
sides of (\ref{r21}) under BGL clearly prompts that, firstly,
$\,\Omega_C\,$ is really finite and, secondly, $\,\Omega_C\sim
\nu_0^{-1}\,$ and $\,\Lambda_C\sim \Lambda\,$.

At the same time, integration over momentums in (\ref{p1}), again
with applying our knowledge, yields quite another estimate. For
$\,r_B\lesssim |\bm{r}_1-\bm{R}_0|\lesssim \Lambda_C\,$, simply
\[
\begin{array}{c}
V_1(t,\bm{R}_0,\bm{r}_1|\bm{R}^{\prime})\,\sim \,\\
\sim\,V_1(t,\bm{R}_0-\bm{R}^{\prime}\,;\,\nu_{0})\,
r_B^2/|\bm{r}_1-\bm{R}_0|^2\,\sim \\ \sim\,
\nu_0^{-1}\,V_1(t,\bm{R}_0-\bm{R}^{\prime}\,;\,\nu_{0})/(\Lambda
|\bm{r}_1-\bm{R}_0|^2)\,
\end{array}
\]
It shows that in literal sense the spatial pair correlation
disappears under BGL, because vanishingly small part of differently
oriented collision cylinders only covers a given segment
$\,(\bm{R}_0\,,\bm{r}_1)\,$.

{\bf 3}. Of course, neither $\,\overline{V}_1(t,\Delta
\bm{R},\bm{P}_0,\bm{p}_1)\,$ nor $\,\Omega_C\,$, or $\,\Lambda_C\,$,
above is unambiguously defined if being taken separately. But an
arbitrariness is not strong since $\,\Lambda_C\,$, like generally
free paths in gases, should be insensible to momentums. The more so
as physically the ``third particles'' responsible for $\,\Lambda_C\,$
originate from gas background which stays constant (uniform and
equilibrium) regardless of BP's walk. Nevertheless, we will try more
formal definition of $\,\Omega_C\,$ and $\,\Lambda_C\,$.

Let $\,\Omega =\Omega (\bm{P}_0,\bm{p}_1)\,$ denotes at once a finite
(exactly ar nearly) cylindric region in the $\,\bm{r}_1\,$ space and
volume of this region, presuming that it is centered at point
$\,\bm{R}_0\,$ and elongated in parallel to $\,\bm{v}_1-\bm{V}_0\,$.
Then introduce $\,\Omega_C(\delta)=\Omega_C(\delta
,\bm{P}_0,\bm{p}_1)\,$ as {\it minimum} of all those regions
$\,\Omega \,$ what satisfy
\[
\begin{array}{c}
|\int_{\Omega} V_1\,d\bm{r}_1\,-\,\int
V_1\,d\bm{r}_1|\,<\,\delta\,|\int V_1\,d\bm{r}_1|\,
\end{array}
\]
with $\,0< \delta \ll 1\,$ and
$\,V_1=V_1(t,\bm{R}_0,\bm{r}_1,\bm{P}_0,\bm{p}_1|\bm{R}^{\prime})\,$.
Orally, $\,\Omega_C(\delta)\,$ is minimal region containing at least
$\,100\,(1-\delta )\,$ percents of total pair correlation.

From the other hand, let us turn to identity (\ref{cf1}) and
integrate it over $\,\bm{r}_1 \in \Omega_C(\delta)\,$. It is
important, although trivial, that
$\,F_1(t,\bm{R}_0,\bm{r}_1,\bm{P}_0,\bm{p}_1|\bm{R}^{\prime})\,$ as
well as any distribution function is {\it a fortiori} non-negative.
Therefore the result must be non-negative, and we can write
\[
\begin{array}{c}
\int_{\,\Omega_C(\delta)}
F_1(t,\bm{R}_0,\bm{r}_1,\bm{P}_0,\bm{p}_1|\bm{R}^{\prime})
\,d\bm{r}_1\,=\,\\=\,(\,1-\epsilon )\,\Omega_C(\delta)\,
V_0(t,\bm{R}_0,\bm{P}_0|\bm{R}^{\prime})\,G^{(eq)}(\bm{p}_1)\,
+\\
+\,\int_{\,\Omega_C(\delta)}
V_1(t,\bm{R}_0,\bm{r}_1,\bm{P}_0,\bm{p}_1|\bm{R}^{\prime})
\,d\bm{r}_1\,\geq\,0\,\,
\end{array}
\]
Here\, $\,\epsilon\, \equiv\,
\,\Omega_C^{-1}(\delta)\int_{\,\Omega_C(\delta)}
[1-F_1^{(eq)}(\bm{r}_1|\bm{R}_0)\,]\,d\bm{r}_1\,$. Clearly, for more
or less dilute gas $\,\epsilon\,$ is indifferent to
$\,\Omega_C(\delta)\,$, positive and small,\, $\,\epsilon\, \sim 4\pi
r_B^3\nu_0/3\,\ll 1\,$.

Further, it is easy to make sure that the two above inequalities
together imply a more interesting one:
\begin{equation}
\begin{array}{c}
(\,1-\epsilon )\,\Omega_C(\delta)\,
V_0(t,\bm{R}_0,\bm{P}_0|\bm{R}^{\prime})\,G^{(eq)}(\bm{p}_1)\,+
\,\,\,\,\,\,\,\,\,\,\, \\ \,\,\,\,\,\,\,\,\,\,\,+\,
(1-\delta\,)\,\int
V_1(t,\bm{R}_0,\bm{r}_1,\bm{P}_0,\bm{p}_1|\bm{R}^{\prime})
\,d\bm{r}_1\,\geq\,0\,\, \label{in}
\end{array}
\end{equation}

Next, let us integrate this inequality over momentums. In view of
above remarks on mean free paths, it is natural to expect that
momentum dependence of the volume $\,\Omega_C(\delta)=\Omega_C(\delta
,\bm{P}_0,\bm{p}_1)\,$ is negligible. Otherwise, we can replace
$\,\Omega_C(\delta) \,$ in (\ref{in}) by the $\,\Omega$'s minimax,
i.e. maximum with respect to momentums of the minimums
$\,\Omega_C(\delta ,\bm{P}_0,\bm{p}_1)\,$. Or, alternatively, take
into account that at $\,t\gg \tau\,$, where $\,\tau =\Lambda/v_T\,$
is mean free flight time of BP and $\,v_T\,$ its thermal velocity,
momentum of BP becomes almost uncorrelated with its displacement,
$\,\bm{R}_0-\bm{R}^{\prime}\,$, and obeys almost exactly equilibrium
Maxwellian probability distribution (see \cite{p1} and also
\cite{i1,i2}). Therefore integration of first term in (\ref{in})
merely replaces $\,\Omega_C(\delta) \,$ by equilibrium average of
$\,\Omega_C(\delta ,\bm{P}_0,\bm{p}_1)\,$.

Anyway, integrating (\ref{in}) and combining it with (\ref{p1}) and
first of (\ref{clim0}), we come to inequality
\begin{equation}
\begin{array}{c}
(\,1-\epsilon )\,\Omega_C(\delta)\,
V_0(t,\bm{R}_0-\bm{R}^{\prime}\,;\,\nu_0)\,+ \,\,\,\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\,\,\,\,\,\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\, (1-\delta\,)
\,\nu_0^{-1}\,V_1(t,\bm{R}_0-\bm{R}^{\prime}\,;\,\nu_0)\,\geq\,0
\,\,\,, \label{in1}
\end{array}
\end{equation}
where $\,\Omega_C(\delta) \,$ represents characteristic volume
occupied by ($\,100\,(1-\delta )\,$ percents of total) pair
correlation of two particles (BP and atom) in the space of their
separations. Corresponding effective length of the collisions
cylinder, $\,\Lambda_C = \Omega_C(\delta)/\pi r_B^2\,$, may be
somewhat greater than $\,\Lambda^{\prime }\,$ (see $\,\S\,$2 in this
Sec.). If the correlation exponentially decreases at large
separations (at $\,|r_{\|}|\,>\Lambda^{\prime }\,$) then admittedly
$\,\Lambda_C\,\approx\, \Lambda^{\prime }\,\ln{(1/\delta )}\,$. In
fact, however, there is no need in conjectures or speculations. The
only principal thing is that $\,\Omega_C(\delta)\,$ is finite
quantity of order of $\,\nu_0^{-1}\,$, i.e.
$\,\nu_0\Omega_C(\delta)\sim 1\,$.

{\bf 4}.\, We have approached to curious conclusions.

Inequalities (\ref{in}) and (\ref{in1}) say that pair correlation can
not be ``too negative'', while relation (\ref{r21}) shows that sign
of the summary pair correlation,
$\,V_1(t,\Delta\bm{R}\,;\,\nu_{\,0})\,$, coincides with sign of small
change in the BP's path probability density distribution,
$\,V_0(t,\Delta\bm{R}\,;\,\nu_{\,0})\,$, under infinitesimally small
uniform addition to gas density. Naturally, the latter should squeeze
$\,V_0(t,\Delta\bm{R}\,;\,\nu_{\,0})\,$ and thus decrease it at large
$\,|\Delta\bm{R}|\,$, at least at $\,\Delta\bm{R}^2\gg 4Dt\,$, with
$\,D=D(\nu_0 )\,$ being BP's diffusivity. Hence, just there
$\,V_1(t,\Delta\bm{R}\,;\,\nu_{\,0})\,<\,0\,$ and the restriction
(\ref{in1}) does work.

It is clear also that the faster is decrease of
$\,V_0(t,\Delta\bm{R}\,;\,\nu_{\,0})\,$ at $\,\Delta\bm{R}^2/4Dt
\rightarrow\infty\,$ the stronger is its change there under variation
of gas density. Therefore inequality (\ref{in1}) implies some
limitations of possible shapes of
$\,V_0(t,\Delta\bm{R}\,;\,\nu_{\,0})\,$. To consider them in evident
form, let us combine (\ref{in1}) and (\ref{r21}). This yields
\begin{eqnarray}
c_1\,V_0(t,\Delta\bm{R}\,;\,\nu_{\,0})\,+\,\nu_{\,0}\,\frac {\partial
V_0(t,\Delta\bm{R}\,;\,\nu_{\,0})}{\partial
\nu_{\,0}}\,\geq\,0\,\label{in2}
\end{eqnarray}
Here we introduced\,\, $\,c_1\,\equiv\,  (\,1-\epsilon
\,)\,\Omega_C(\delta)\,\nu_{\,0}^2/(1-\delta\,)\,\widetilde{\nu
}_0\,$\,. According to above consideration, in more or less dilute
gas (or under BGL)\, $\,c_1\,\approx\,
\Omega_C(\delta)\,\nu_{\,0}\,\sim\,1\,$\, independently on the
density.

{\bf 5}.\, Now assume that at  $\,t\gg \tau \,$, when
\[
\begin{array}{c}
\langle \Delta\bm{R}^2 \rangle \,=\, \int \Delta\bm{R}^2\,
V_0(t,\Delta\bm{R}\,;\,\nu_{\,0})\,d\Delta\bm{R}\,\approx \,6Dt\,\gg
3\Lambda^2\,\,\,,
\end{array}
\]
the BP's path distribution tends to the Gaussian one (see
Introduction), $\,V_0(t,\Delta \bm{R}\,;\,\nu_{\,0})\rightarrow
V_G(t,\Delta\bm{R})\,$\,. Then inequality (\ref{in2}) turns into
requirement
\[
\left [\,c_1\,+\,\left (\frac {\Delta\bm{R}^2}{4Dt}-\frac 32 \right
)\frac {\partial\, \ln D}{\partial\, \ln \nu_{\,0}}\,\right
]V_G(t,\Delta\bm{R})\,\geq\,0\,
\]
Since diffusivity certainly is a decreasing function of gas density,
and $\,\partial \ln D/\partial \ln \nu_{\,0}<0\,$, this requirement
inevitably gets broken at sufficiently large values of
$\,\Delta\bm{R}^2/4Dt\,$, while we get contradiction.

Consequently, the theory forbids the Gaussian asymptotic! This
statement, or rather its new proof since it itself is not quite novel
(see Introduction and resume below), is principal result of the
present paper.

{\bf 6}.\, But what kind of asymptotic is allowed instead of the
Gaussian one?

Let $\,V_0(t,\Delta \bm{R}\,;\,\nu_{\,0})\,$ at $\,t\gg \tau \,$ is
characterized, similarly to $\,V_G(t,\Delta \bm{R})\,$, by a single
parameter, that is diffusivity:
\[
V_0(t,\Delta\bm{R}\,;\,\nu_{\,0}) \rightarrow
(4Dt)^{-3/2}\,\Psi(\Delta \bm{R}^2/4Dt)\,\,\,,
\]
where $\int \Psi(\bm{a}^2)\,d\bm{a}\,=1\,$\, (the normalization
condition)\, and $\,\int \bm{a}^2\Psi(\bm{a}^2)\,d\bm{a}\,=3/2$ (that
is $\,\langle \Delta\bm{R}^2 \rangle =6Dt\,$). If take in mind dilute
or Boltzmann-Grad gas, where $\,D\sim \Lambda v_T\,$ and thus
$\,D\propto \nu_{\,0}^{-1}\,$, then inequality (\ref{in2}) reduces to
\[
\begin{array}{c}
(c_1+3/2)\,\Psi(x) +x\,d\Psi(x)/dx \,\geq \,0\,
\end{array}
\]
It requires that function $\,\Psi(x)\,$ must have power-law long
tail: \, $\,\Psi(x)\propto\, x^{-\,\alpha }\,$\, at
$\,x\rightarrow\infty\,$, where $\,\alpha\leq c_1+3/2\,$\,.

But such behavior leads to unboundedness of higher-order statistical
moments of BP's path. Recall, however, that we have one more
parameter, the thermal velocity $\,v_T\,$ (or, equivalently, speed of
sound, $\,v_s\sim v_T\,$) and therefore possibility at $\,t\gg
\tau\,$ to write
\begin{equation}
V_0(t,\Delta \bm{R}\,;\,\nu_{\,0})\rightarrow \frac
{1}{(4Dt)^{3/2}}\, \Psi\left (\frac
{\Delta\bm{R}^2}{4Dt}\right)\Theta \left (\frac {|\Delta \bm{R}|}{v_T
t}\right )\,\,\,\,\,\,\,\,\,\label{as}
\end{equation}
with a cut off function $\,\Theta(x)\,$
which satisfies $\,\Theta(0)=1\,$ and fast enough decreases at
infinity. From the point of view of inequality (\ref{in2}) this is
also allowed variant. And at $\,c_1=2\,$, i.e. $\,\alpha =7/2\,$, and
$\,\Psi(x)= \Gamma(\alpha )\,\pi^{-3/2}\,(1+x)^{-\,\alpha }\,$ it
qualitatively and and in part quantitatively reproduces the
asymptotic of distribution of BP's path found in \cite{p1} in the
frame of the ``collisional approximation'' of BBGKY equations
\cite{i1,i2} (there BP was identical to atoms).

\section{Discussion and resume}
Why it is so that the Gaussian asymptotic fails (and usual intuition
does along with it)? In principle, the answer was provided many years
ago, firstly (as far as I know), in general, by Krylov in book
\cite{kr} (see citation from it in Introduction) and later, in
context of Brownian motion of micro-particles and transport processes
(particularly, electrons and charge transport), in \cite{bk12,bk3}
and then in \cite{i1,i2}.

Briefly, if a system actually is constantly forgetting history of
``Brownian particle'' (BP), that is a number of its previous
collisions, etc., then such system is physically unable to check a
``number of collisions (of this particle) per unit time''. Therefore
one can not describe BP's walk in terms of {\it a priori}
``probabilities of collisions per unit time'' or other beforehand
assigned probabilities of ``elementary'' steps or events. In other
words, the walk can not be divided into ``statistically independent''
pieces, and thus products of ``the art of conjecturing'' like the
``law of large numbers'' become out of job. The resulting
non-ergodicity of Brownian trajectories may be characterized as
``1/f\,-fluctuations'' in ``probabilities of collisions'' and thus in
BP's diffusivity and mobility \cite{i1,i2,p1,pjtf,bk12,bk3,dev}.

In present work, the aforesaid is concretized by inequality
(\ref{in2}). It shows that hypothetical Gaussian asymptotic has too
large relative speed of decrease at infinity: $\,\partial\, \ln V_G
/\partial\, \ln \Delta\bm{R}^2\,=-\Delta\bm{R}^2/4Dt\,\rightarrow
-\infty\,$. So much large that the quantity (see $\,\S\,$3 in
Sec.III)
\[
\overline{F}_1(t,\bm{R}_0-\bm{R}^{\prime},\bm{P}_0,\bm{p}_1)
=\int_{\Omega_C}
F_1(t,\bm{R}_0,\bm{r}_1,\bm{P}_0,\bm{p}_1|\bm{R}^{\prime}) \,\frac
{d\bm{r}_1}{\Omega_C}
\]
(with $\,\Omega_C\,$ shortening $\,\Omega_C(\delta
,\bm{P}_0,\bm{p}_1)\,$) and even quantity
$\,\overline{W}_1(t,\Delta\bm{R})= \int\overline{F}_1(t,\Delta\bm{R},
\bm{P}_0,\bm{p}_1)\,d\bm{P}_0\,d\bm{p}_1\,$ would become negative.
However, in reality this is impossible even under maximally allowable
statistical unbalance of mutually antithetic BP's collisions, since
$\,\overline{F}_1(t,\Delta \bm{R},\bm{P}_0,\bm{p}_1)\,$, by its
probabilistic sense, must be non-negative.

Notice that $\,\overline{F}_1(t,\Delta \bm{R},\bm{P}_0,\bm{p}_1)\,$
and $\,\overline{W}_1(t,\Delta\bm{R})\,$ are direct analogues of
function $\,A_2\,$ from \cite{i1} (or $\,F_2\,$ from \cite{i2}) and
$\,W_2\,$ from \cite{p1} and represent ensemble averages of number
density of those two-particle configurations what match up BP's
collisions with atoms.

But where in present consideration one can see ``fluctuations in
probabilities of collisions''\,? Notice that ratio
$\,\overline{W}_1(t,\Delta\bm{R})/V_0(t,\Delta\bm{R})\,$ is a measure
of conditional probability of BP's collisions. Or, in other words, it
represents {\it a posteriori} probabilities what emerge from factual
BP's path. According to $\,\S \S\,$3-4 in Sec.III,
\[
\begin{array}{c}
\overline{W}_1(t,\Delta\bm{R})\approx V_0(t,\Delta\bm{R})+
c_1^{-1}\,V_1(t,\Delta\bm{R})\,
\end{array}
\]
(argument $\,\nu_0\,$ is omitted). At far tail of the BP's path
distribution $\,V_0(t,\Delta\bm{R})\,$, at
$\,x=\Delta\bm{R}^2/4Dt\rightarrow \infty\,$, it goes to zero much
faster than $\,V_0(t,\Delta\bm{R})\,$, so that
$\,\overline{W}_1(t,\Delta\bm{R})/V_0(t,\Delta\bm{R})\propto\,
x^{-1}\rightarrow 0\,$. If $\,x\ll 1\,$ then, in opposite,
$\,\overline{W}_1 >V_0\,$. So substantial changeability of the {\it a
posteriori} probabilities (in contrast with their constancy imposed
by the ''molecular chaos'' in conventional theory) just shows that
asymptotic like (\ref{as}) reflects what can be named long-scale
``fluctuations in probabilities of collisions''.

Importantly, our consideration has outlined a way to formalizing
connections between usual many-particle distribution and correlation
functions, on one hand, and specific distribution functions (first
introduced in \cite{i1}) for two- and many-particle collisional
configurations, on the other hand. Their deeper investigation would
mean better understanding real statistics of molecular Brownian
motion and related noise and might open new applications of the
virial expansions (\ref{r}) and (\ref{r2}).

I am grateful to Dr. I.\,Krasnyuk for many useful conversations.

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(editors). Non-equilibrium phenomena. I. The Boltzmann equation.
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\bibitem{i1}
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(1988).

\bibitem{i2}
Yu.\,E.\,Kuzovlev, {\it Kinetical theory beyond conventional
approximations and 1/f-noise},\, Math. Physics Archive (University of
Texas),\, 99-64.

\bibitem{p1}
Yu.\,E. Kuzovlev,\, {\it On statistics and 1/f noise of Brownian
motion in Boltzmann-Grad gas and finite gas on torus},\,
arXiv:\,\,cond-mat/0609515,\, 0612325.

\bibitem{pjtf}
Yu.\,E. Kuzovlev and G.\,N. Bochkov,\, {\it On the origin and
statistical parameters of equilibrium 1/f-noise},\,
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transl. to English in \,Sov.\,Tech.\,Phys.\,Lett. (1982)].

\bibitem{bk12}
G.\,N.\,Bochkov and Yu.\,E.\,Kuzovlev,\,{\it On the origin and
statistical characteristics of 1/f-noise},\, Radiophysics and Quantum
Electronics,\, No.3 (1983);\, {\it On the probabilistic
characteristics of 1/f-noise},\, {\it ibid.}, No.9 (1984).

\bibitem{bk3}
G.\,N.\,Bochkov and Yu.\,E.\,Kuzovlev,\, {\it New aspects in studies
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subjects under current consideration.

\end{thebibliography}

\end{document}

\newpage

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\section*{\,\,\,\,\,\,\,\,\,ПРАВДА О БРОУНОВСКОМ ДВИЖЕНИИ\,\,\,\,
В ГАЗАХ И ВООБЩЕ}
\[
\begin{array}{c}
\text{Ю.\,Е.\,Кузовлев} \\
\text{Донецкий физико-технический институт НАНУ} \\
\text{83114 Донецк, Украина} \\
\text{(e-mail:\, kuzovlev@kinetic.ac.donetsk.ua \,)}
\end{array}
\]

\,\,\,

{\bf Аннотация}. \emph{Реальное тепловое движение частиц газа,
свободных электронов и др.\, на временах много больше времени
свободного пробега обладает, вопреки его популярным математическим
моделям, существенно негауссовской статистикой. Предложено
доказательство этого утверждения для броуновского движения в
слабонеидеальном газе, опирающееся только на детерминизм и
обратимость микродинамики и попутно полученное вириальное разложение
плотности вероятности перемещения броуновской частицы.}

\,\,\,

{\bf 1}. Беспорядочное броуновское движение частиц материи является
механизмом диффузии и многих других процессов переноса. Оно интересно
и само по себе, и в виду того, что статистика блуждания пробной или
меченой ``броуновской'' частицы (БЧ) несет полную информацию о
процессе в целом, включая его шумы и флуктуации. Однако динамическая
теория броуновского движения практически не разработана. Считается,
видимо, что она лишь подтвердила бы ту красивую математическую схему,
которая создана теорией вероятностей, базируется на знаменитом
``законе больших чисел'' \cite{rjb} и предрекает, что в достаточно
грубом масштабе пространства-времени плотность вероятности
перемещения БЧ $\,\Delta {\bf R}\,$ за время $\,t\,$ имеет
универсальный вид гауссова распределения. Для симметричного
трехмерного блуждания это -

\[
V_G(t,\Delta{\bf R})\,=\,(4\pi Dt)^{-3/2}\,\exp{(-\Delta {\bf
R}^2/4Dt)}\,\,
\]

Такова, как строго доказано \cite{rsin,rgal}, асимптотика блуждания
частицы (твердого шарика) в столь неслучайной обстановке, как
периодическая решетка неподвижных рассеивателей - твердых шаров.
Казалось бы, от броуновского перемещения атомов газа тем более нельзя
ожидать ничего иного. И действительно, уравнение Больцмана-Лоренца
для его распределения \cite{rre}, происходящее от уравнения
Больцмана, дает гауссову асимптотику. Тем не менее не все так просто,
как кажется.

Во-первых, уравнение Больцмана-Лоренца - это весьма грубая модель,
которая получается, как и уравнение Больцмана, обрыванием цепочки
точных уравнений ББГКИ и замыканием ее с помощью архаичной
больцмановской гипотезы о ``молекулярном хаосе''
\cite{rre,rbog,rsil}. Но, как не раз отмечалось \cite{rre,rkac},
аргументация Больцмана \cite{rbol} годится лишь для однородного газа.
Попытки же вывести ``молекулярный хаос'' из самих уравнений ББГКИ
безуспешны \cite{rol}. Напротив, показано \cite{ri1} (см. также
\cite{ri2,rp1}), что переформулировка уравнений ББГКИ в терминах
``столкновений'' частиц (взамен их непрерывного взаимодействия)
приводит к уравнениям, которые очевидным образом несовместимы с этой
гипотезой, если газ пространственно неоднороден. Иными словами, сам
факт соучастия частиц в столкновении (или связном конгломерате
столкновений) в конкретном месте пространства уже влечет
статистическую зависимость между ними. Но для анализа перемещения БЧ
(пробного атома газа) необходимо локализовать ее начальное положение
и тем самым нарушить однородность (трансляционную инвариантность)
всех функциий распределения (ФР), относящихся к ней и ее
взаимодействию с газом. Поэтому, согласно сказанному, придется-таки
рассматривать бесконечную цепочку уравнений. Попытки приближенного
решения этой задачи предприняты в \cite{ri1} (или см. \cite{ri2}) и в
\cite{rp1}. Полученное в них, разумеется, качественно отличается от
гауссовой асимптотики (зато качественно подтверждает предшествующую
феноменологическую теорию \cite{rpjtf,rbk12,rbk3}).

Во-вторых, гауссова асимптотика блуждания в неподвижной решетке
обязана его эргодичности \cite{rsin,rgal,rar}: результаты усреднения
по $\,n\hm\rightarrow\infty $ фрагментам его траектории (каждый
состоит из свободного пробега и соударения) почти наверное не зависят
от нее. Это возможно благодаря тому, что всякая траектория полностью
определена одним своим (например, начальным) фрагментом, и отношение
количества $\,d=5n\,$ чисел, описывающих траекторию, к количеству
$p=5\,$ задающих ее параметров неограниченно растет,
$d/p\hm\rightarrow\infty \,$. Иное дело - (слабонеидеальный) газ.
Здесь траектория броуновской (пробной) частицы складывается из
аналогичных фрагментов в количестве $\,n\approx t/\tau\,$, где
$\,t\,$ - время наблюдения, $\,\tau\,$ - среднее время свободного
пробега, и исчерпывающе характеризуется $\,d=6n\,$ числами. Однако
определяется она не только своими начальными условиями, но и
начальными состояниями многих других частиц. Их количество $\,m\,$,
как легко убедиться \cite{rp1}, растет несравненно быстрее $\,n\,$, -
так, что $\,n/m\hm\rightarrow 0\,$. Соответственно, теперь отношение
количества чисел, описывающих траекторию, к количеству ее параметров
$\,p=6m\,$ стремится к нулю, $\,d/p\hm\rightarrow 0\,$. Вряд ли
поэтому результаты усреднения по времени, - к примеру, ``коэффициент
диффузии'' или ``вероятности столкновений'', - окажутся одинаковы для
всех траекторий. Здесь нет оснований для эргодичности, ``закона
больших чисел'' (который требует \cite{rjb,ri2,rp3} подставить в себя
наперед известные ``вероятности столкновений'') и гауссовой
асимптотики. Если же газ состоит из $\,N<\infty\,$ частиц (в ящике
или на торе), то основания появятся не раньше, чем спустя время $\,>
N\tau\,$ \cite{rp1}. В данном контексте симптоматично замечание из
\cite{rar}, что ценность эргодической теории для физики преувеличена,
т.к. предел $\,N\hm\rightarrow \infty $ для  физики важнее предела
$\,t\rightarrow\infty \,$. Хотя, собственно, об этом предупреждал еще
Н.Крылов \cite{rkr}.

В принципе, подобные рассуждения \cite{ri2,rbk3,ri3,ri4} применимы к
любой реальной системе большого числа частиц, равно как и уравнения
ББГКИ или их аналоги, так что случай газа имеет общефизическое
значение. Но эти уравнения так плохо изучены, что дополнительные
средства не будут лишними. Ниже мы выведем своего рода вириальное
разложение для истинного вероятностного распределения перемещения БЧ,
$\,V_0(t,\Delta{\bf R})\,$, основываясь лишь на детерминизме и
обратимости микроскопической динамики. Свяжем коэффициенты этого
разложения с совместными корреляционными функциями БЧ и газа, а затем
c обычными ФР. Наконец, покажем, что все это вместе взятое дает
дифференциальное неравенство для $\,V_0(t,\Delta{\bf R})\,$, которое
запрещает гауссову асимптотику (но допускает распределение,
полученное в \cite{rp1} приближенным решением уравнений ББГКИ).

{\bf 2}. Пусть в газ $N\gg 1$ атомов в объеме $\,\Omega $ помещена
``броуновская частица''\, (БЧ). Рассмотрим, в термодинамическом
пределе $N\rightarrow\infty $, $\Omega\rightarrow\infty $, $N/\Omega
\hm =\nu_{\,0}\hm = \,$\,\,const\,, статистический ансамбль фазовых
траекторий этой системы, отвечающий каноническому равновесному
распределению их начальных условий при $t=0$, полагая, что при $t>0$
на БЧ действует постоянная сторонняя сила ${\bf f}\,$. Обратимость
фазовых траекторий во времени привносит множество соотношений между
характеристиками ансамбля \cite{rjetp1,rjetp2,rp}. В частности, можно
согласно \cite{rjetp1} написать
\begin{equation}
\begin{array}{c}
\langle A({\bf q}(t))B({\bf q}(0))\,e^{-\,\mathcal{E}(t)/T}\,\rangle
=\langle B({\bf q}(t))A({\bf q}(0))\rangle\,\label{rsim0}
\end{array}
\end{equation}
Здесь скобки $\langle ...\rangle\,$ означают усреднение по ансамблю,
$\,{\bf q}(0)={\bf q}=\{{\bf R},{\bf r}_1,...\,,{\bf r}_N\}$ -
координаты БЧ и атомов, $\mathcal{E}(t)={\bf f}\cdot [{\bf R}(t)-{\bf
R}(0)]$ - работа сторонней силы за время $t\,$, $A({\bf q})$ и
$B({\bf q})$ - ``произвольные функции'', а $\,T$ - начальная
температура системы. Данное равенство справедливо и при наличии у БЧ
внутренних степеней свободы.

Возьмем $\,B({\bf q})=\Omega\,\delta({\bf R}-{\bf R}_0)\exp{[-\sum_j
U({\bf r}_j)/T\,]}$ и $A({\bf q})=\delta({\bf R}-{\bf
R}^{\prime})\,$. Правая сторона (\ref{rsim0}) примет вид
\[
\begin{array}{c}
\langle B({\bf q}(t))A({\bf q})\rangle =\mathcal{F}\{t,{\bf
R}_0,\phi|{\bf R}^{\prime}\}\,\equiv\,
V_0(t,{\bf R}_0-{\bf R}^{\prime})\,+\\
+\sum_{n\,=1}^{\infty } (\nu_{\,0}^n/n!)\int^n F_n(t,{\bf R}_0,{\bf
r}_1\,...\,{\bf r}_n|{\bf R}^{\prime})\prod_{j\,=1}^n \phi({\bf
r}_j)\,\,,
\end{array}
\]
где мы ввели $\,\phi({\bf r})=\exp{[-\,U({\bf r})/T\,]}-1\,$, символ
$\int^n $ обозначает интегрирование по ${\bf r}_1...\,{\bf r}_n\,$,
функция $V_0(t,{\bf R}_0-{\bf R}^{\prime})$ - плотность вероятности
обнаружить БЧ при $t\geq 0\,$ в точке ${\bf R}_0$, а $\,F_n(t,{\bf
R}_0,{\bf r}_1\,...\,{\bf r}_n|{\bf R}^{\prime})$ - плотность
вероятности того же события и одновременно обнаружения атомов в
точках ${\bf r}_j\,$, при условии, что БЧ стартовала из точки ${\bf
R}^{\prime}\,$. Ясно, что\,\, $\,V_0(0,{\bf r})=\delta({\bf r})\,$,
$\,\int V_0(t,{\bf r})\,d{\bf r} =1\,$, и
\[
\begin{array}{c}
F_n(0,{\bf R}_0,{\bf r}_1...\,{\bf r}_n|{\bf R}^{\prime})=\delta({\bf
R}_0-{\bf R}^{\prime})\,F_n^{(eq)}({\bf
r}_1...\,{\bf r}_n|{\bf R}_0)\,\,, \\
\mathcal{F}\{0,{\bf R}_0,\phi|{\bf R}^{\prime}\}=\delta({\bf
R}_0-{\bf R}^{\prime})\,\mathcal{F}^{(eq)}\{\phi |{\bf R}_0\}\,\,,\\
\mathcal{F}^{(eq)}\{\phi |{\bf R}_0\}\,\equiv \,1\,+ \\
+\sum_{n\,=1}^{\infty }(\nu_{\,0}^n/n!)\int^n F_n^{(eq)}({\bf
r}_1...\,{\bf r}_n|{\bf R}_0)\prod_{j\,=1}^n \phi({\bf r}_j)\,\,,
\end{array}
\]
где $F_n^{(eq)}({\bf r}_1...\,{\bf r}_n|{\bf R}_0)$ - равновесные ФР
и $\mathcal{F}^{(eq)}\{\phi |{\bf R}_0\}$ - их производящий
функционал. По отношению к атомам все ФР нормированы как обычно
\cite{rbog,rsil}, то есть\, $\,F_n(...\,{\bf r}_k...\,|{\bf
R}^{\prime})\rightarrow F_{n-1}(...\,{\bf r}_{k-1},{\bf
r}_{k+1}...\,|{\bf R}^{\prime})\,$, если ${\bf r}_k\rightarrow
\infty\,$, и $F_1^{(eq)}(\bm{r}_1|\bm{R}^{\prime}) \rightarrow 1\,$
при $\bm{r}_1\rightarrow \infty$. Заодно этим требованием выражается
``принцип ослабления корреляций''.

Начальные ФР $\,F_n(0,{\bf R}_0,{\bf r}_1\,...\,{\bf r}_n|{\bf
R}^{\prime})\,$ включают все равновесные корреляции между БЧ и
атомами. Но любая эволюция (ансамбля состояний) системы сопряжена с
нарушением детального баланса столкновений и потому рождает
дополнительные корреляции, обусловленные соучастием частиц в
избыточных (или их ``совместным неучастием'' в недостающих)
столкновениях. Их можно назвать еще неравновесными корреляциями, с
оговоркой, что иногда они описывают эволюцию не столько системы самой
по себе, сколько нашей информации о ней. Соответствующие добавки к ФР
обычно именуются ``корреляционные функции''
\cite{rre,rbog,rsil,rbal}. Обозначим их через $\,V_n(t,{\bf R}_0,{\bf
r}_1\,...\,{\bf r}_n|{\bf R}^{\prime})\,$ и вычленим из
$\,F_n(t>0,...)\,$, определив так:
\[
\begin{array}{c}
\mathcal{F}\{t,{\bf R}_0,\phi|{\bf
R}^{\prime}\}=\mathcal{F}^{(eq)}\{\phi |{\bf R}_0\}\, [\,V_0(t,{\bf
R}_0-{\bf R}^{\prime})\,+\\ +\sum_{n\,=1}^{\infty }
(\nu_{\,0}^n/n!)\int^n V_n(t,{\bf R}_0,{\bf r}_1...\,{\bf r}_n|{\bf
R}^{\prime})\prod_{j\,=1}^n \phi({\bf r}_j)]\,
\end{array}
\]
Отсюда видно, что $\,V_n(0\,,...\,)=0\,$, и
$\,V_n(\,t\,,...)\rightarrow 0\,$, если хотя бы одна из точек $\,{\bf
r}_k\rightarrow \infty\,$. В частности,
\[
\begin{array}{c}
F_1(t,{\bf R}_0,{\bf r}_1|{\bf R}^{\prime})\, = \\
=V_0(t,{\bf R}_0-{\bf R}^{\prime})F_1^{(eq)}({\bf r}_1|{\bf
R}_0)+V_1(t,{\bf R}_0,{\bf r}_1|{\bf R}^{\prime})\,\,\,,
\end{array}
\]
где функция $\,V_1(t,{\bf R}_0,{\bf r}_1|{\bf R}^{\prime})\,$
происходит от обычной ``парной корреляционной функции''
\cite{rsil,rbal} в полном двухчастичном фазовом пространстве:
\begin{equation}
\begin{array}{c}
V_1(t,{\bf R}_0,{\bf r}_1|{\bf R}^{\prime})=\int \int V_1(t,{\bf
R}_0,{\bf r}_1,{\bf P}_0,{\bf p}_1|{\bf R}^{\prime})\, d{\bf
P}_0\,d{\bf p}_1 \label{rp1}
\end{array}
\end{equation}
Последнюю можно соотнести с полными ФР точно так же, как
$\,V_1(t,{\bf R}_0,{\bf r}_1|{\bf R}^{\prime})\,$ с координатными:
\begin{equation}
\begin{array}{c}
F_1(t,{\bf R}_0,{\bf r}_1,{\bf P}_0,{\bf p}_1|{\bf R}^{\prime})\,=\,
V_0(t,{\bf R}_0-{\bf R}^{\prime},{\bf P}_0)\times \label{rcf1}
\\ \times\, F_1^{(eq)}({\bf r}_1,{\bf p}_1|{\bf R}_0)+V_1(t,{\bf R}_0,{\bf
r}_1,{\bf P}_0,{\bf p}_1|{\bf R}^{\prime})
\end{array}
\end{equation}

Обратимся теперь к левой стороне (\ref{rsim0}). Умножим и поделим ее
на $\langle B({\bf q})\rangle$\, и воспользуемся тем, что выражение
$\langle \Phi\,B({\bf q})\rangle /\langle B({\bf q})\rangle\,$, в
котором $\Phi$ - некоторый функционал фазовой траектории, можно,
благодаря детерминизму движения по конкретной траектории, понимать
как усреднение $\Phi$ по новому ансамблю, порожденному новым,
неравновесным, распределением начальных кординат и импульсов системы,
а именно, $\,\rho ({\bf q},{\bf p})\propto B({\bf q})\,\rho_0({\bf
q},{\bf p})\,$, где $\rho_0({\bf q},{\bf p})$ - истинное равновесное
распределение. Заметив еще, что $\langle B({\bf q})\rangle
=\mathcal{F}^{(eq)}\{\phi |{\bf R}_0\}\,$, имеем
\[
\begin{array}{c}
\langle A({\bf q}(t))B({\bf q}(0))\,e^{-\,\mathcal{E}(t)/T}\,\rangle
=\\ =\,V\{t,{\bf R}^{\prime}|\phi ,{\bf R}_0 \}\,\,e^{-\,{\bf f}\cdot
[\,{\bf R}^{\prime}-\,{\bf R}_0]/T}\,\mathcal{F}^{(eq)}\{\phi |{\bf
R}_0 \}\,\,\,,
\end{array}
\]
где $\,V\{t,{\bf R}^{\prime}|\phi ,{\bf R}_0 \}\,$ - плотность
вероятности найти БЧ в момент $t$ в точке ${\bf R}^{\prime}$ при
условиях, что вначале она была в точке ${\bf R}_0$, а газ находился в
таком возмущенном неоднородном состоянии, какое было бы равновесным в
присутствии потенциала $\,U({\bf r})\,$. Средняя концентрация атомов
в этом состоянии равна
\[
\begin{array}{c}
\nu\{{\bf r}|\phi ,{\bf R}_0 \}\hm = [1+\phi({\bf r})]\,\delta
\ln\mathcal{F}^{(eq)}\{\phi |{\bf R}_0 \}/\delta\phi({\bf r})
\end{array}
\]

В сумме приходим к точному соотношению
\begin{equation}
\begin{array}{c}
V\{t,{\bf R}^{\prime}|{\phi ,\bf R}_0 \}\,\,e^{-\,{\bf f}\cdot[\,{\bf
R}^{\prime}-\,{\bf R}_0]/T}
\,=\,V_0(t,{\bf R}_0-{\bf R}^{\prime})\,+\label{rr}\\
+\sum_{n\,=1}^{\infty } (\nu_{\,0}^n/n!)\int^n V_n(t,{\bf R}_0,{\bf
r}_1...\,{\bf r}_n|{\bf R}^{\prime})\prod_{j\,=1}^n \phi({\bf
r}_j)\,\,,
\end{array}
\end{equation}
связывающему вероятностное распределение пути БЧ в изначально
неравновесном неоднородном газе и аналогичное распределение, вкупе с
производящим функционалом ``неравновесных корреляций'' между прошлым
путем БЧ и ее текущим окружением, для исходно равновесного
однородного газа. Если же $\,{\bf f}=0\,$, то правая сторона
относится к целиком равновесному броуновскому движению. Отметим, что
похожее соотношение для плотности газа вместо распределения БЧ
рассматривалось нами в \cite{rp3}.

{\bf 3}. Какова польза данного соотношения, легче понять в формальном
``пределе Больцмана-Грэда'' \cite{rol}, когда плотность газа
$\nu_0\,$ растет, а радиусы взаимодействий (коротких и отталкивающих)
БЧ с атомами, $r_b\,$, и межатомного, $r_a\,$, уменьшаются таким
образом, что ``газовые параметры'' $\,r_a^3\nu_0 \,$ и $\,r_b^3\nu_0
\,$ идут к нулю, но длины свободного пробега атомов $\,\lambda\hm
=(\pi r_a^2\nu_0 )^{-1}\,$ и БЧ $\,\Lambda =(\pi r_b^2\nu_0 )^{-1}\,$
фиксированы. Получается ``идеальный слабонеидеальный газ'', в котором
$\,F_n^{(eq)}\rightarrow 1\,$ в том смысле, что, например,
$\,\nu_0\int [\,F_1^{(eq)}({\bf r}_1|{\bf R}_0)-1] \,d{\bf
r}_1\,\hm\sim\, \nu_0 r_b^3\,\rightarrow 0\,$\,. Поэтому
\[
\begin{array}{c}
\nu\{{\bf r}|\phi ,{\bf R}_0 \}\,\rightarrow \,
\nu_{\,0}\,[\,1+\phi({\bf r})\,]\,=\,\nu_{\,0}\,\exp{[-U({\bf
r})/T\,]}
\end{array}
\]

Данное упрощение делает совсем очевидным, что эффект возмущений
плотности слева в (\ref{rr}) в пределе Больцмана-Грэда зависит от
$\,\phi({\bf r})\,$, но не от $\,\nu_0\,$. Тогда и правая часть
(\ref{rr}) зависит только от $\,\phi({\bf r})\,$, т.е. при заданной
функции $\,\phi({\bf r})\,$ (и фиксированных $\,\Lambda\,$ и
$\,\lambda\,$) существуют конечные пределы
\[
\begin{array}{c}
\lim\,\,\nu_{\,0}^n\int^n V_n(t,{\bf R}_0,{\bf r}_1...\,{\bf
r}_n|{\bf R}^{\prime})\prod_{j\,=1}^n \phi({\bf r}_j)\,\neq
0,\,\infty
\end{array}
\]
Выбирая же $\,\phi({\bf r})=\phi =\,$const\, в достаточно большом
шаре (например, $\,|{\bf r}-{\bf R}_0|\hm <v_s t_0\,$, где $\,v_s\,$
- скорость звука, $\,t_0>t\,$), за который корреляции заведомо не
выходят, заключаем, что (при фиксированных $\,\Lambda\,$,
$\,\lambda\,$ и $\,t\,$) существуют пределы
\begin{equation}
\begin{array}{c}
\lim\,\,\nu_{\,0}^n\int^n V_n(t,{\bf R}_0,{\bf r}_1...\,{\bf
r}_n|{\bf R}^{\prime})\,=\,V_n(t,{\bf R}_0-{\bf R}^{\prime})\,
\label{rclim}
\end{array}
\end{equation}
Теперь и слева в (\ref{rr}) фактически, с точки зрения БЧ, газ
однороден на старте, да и во время наблюдения за БЧ. Но его плотность
в $\,(1+\phi)\,$ раз больше, чем справа. Соответственно (\ref{rr})
превращается в
\begin{equation}
\begin{array}{c}
V_0(t,-\Delta{\bf R}\,;1+\phi )\,\,e^{\,{\bf f}\cdot\Delta {\bf R}/T}
\,=\,\,\,\,\,\,\,\,\,\,\,\,\\
\,\,\,\,\,\,\,\,\,\,\,\,=\,\sum_{n\,=\,0}^{\infty } \,\phi^n
\,V_n(t,\Delta{\bf R}\,;1)/n!\, \label{rr2}
\end{array}
\end{equation}
Здесь мы ввели $\,\Delta {\bf R}={\bf R}_0-{\bf R}^{\prime}\,$ и
третий аргумент - (безразмерную) плотность газа, измеренную в
единицах $\,\nu_0\,$, так что $\,V_n(...\,;1)=V_n(...\,)\,$.

Как явствует из (\ref{rclim})-(\ref{rr2}), в слабонеидеальном газе
пространственные неравновесные корреляции не исчезают в относительном
смысле - в расчете на \,(объем $\,\nu_0^{-1} \,$, приходящийся на
один)\, атом,\, хотя они исчезающе малы в буквальном смысле:
$\,V_n(t,{\bf R}_0,{\bf r}_1...\,{\bf r}_n|{\bf
R}^{\prime})\rightarrow 0\,$.

Интерпретируем это утверждение на основе того немногого, что уже
известно о корреляционных функциях \cite{rbog,rsil,ri1,ri2,rp1,rbal},
в первую очередь, о $\,V_1(t,{\bf R}_0,{\bf r}_1,{\bf P}_0,{\bf
p}_1|{\bf R}^{\prime})\,$. По отношению к $\,{\bf r}_1-{\bf R}_0\,$
эта функция отлична от нуля в ``цилиндре столкновений'' с радиусом
$\,r_b\,\,$, параллельном относительной скорости пары частиц (БЧ и
атома), причем здесь сравнима с произведением одночастичных ФР, т.е.
с первым слагаемым справа в (\ref{rcf1}). Но приближенная теория,
которой принято довольствоваться \cite{rsil,rbal}, ничего не говорит
о длине цилиндра. Зато формула (\ref{rclim}) подсказывает, что в
точной теории эффективная длина цилиндра столкновений конечна и имеет
порядок $\,\Lambda\,$. Действительно, пусть она равна
$\,c_1\Lambda\,$, тогда объем цилиндра равен \,$\,\pi
r_b^2\,c_1\Lambda =\,c_1\nu_0^{-1} \,$, и
\begin{equation}
\begin{array}{c}
\int V_1(t,{\bf R}_0,{\bf r}_1,{\bf P}_0,{\bf p}_1|{\bf
R}^{\prime})\,d{\bf r}_1\, = \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=
\,c_1\,\nu_0^{-1}\,\overline{V}_1(t,\Delta {\bf R},{\bf P}_0,{\bf
p}_1)\,\,\,,\label{rvm}
\end{array}
\end{equation}
где\, $\,\overline{V}_1(t,\Delta {\bf R},{\bf P}_0,{\bf p}_1)\,$\,
представляет собой среднюю величину
$\,V_1(t,{\bf R}_0,{\bf r}_1,{\bf P}_0,{\bf p}_1|{\bf R}^{\prime})\,$
внутри цилиндра. Комбинируя это
выражение с (\ref{rp1}) и (\ref{rclim}) при $n=1$, убеждаемся в
конечности предела:
\begin{equation}
\begin{array}{c}
V_1(t,\Delta {\bf R})\,=\,c_1\,\int \int \overline{V}_1(t,\Delta {\bf
R},{\bf P}_0,{\bf p}_1)\,d{\bf P}_0\, d{\bf p}_1 \label{rmc}
\end{array}
\end{equation}
В то же время в буквальном смысле из (\ref{rp1}) и (\ref{rmc})
находим \,\,\,$\,V_1(t,{\bf R}_0,{\bf r}_1|{\bf R}^{\prime})\hm\sim
\, V_1(t,\Delta {\bf R})\,r_b^2/|{\bf r}_1-{\bf R}_0|^2\hm\sim \,
\nu_0^{-1}\,V_1(t,\Delta {\bf R})/(\Lambda|{\bf r}_1-{\bf R}_0|^2)\hm
\,\rightarrow \,0\,$\,\,,\, поскольку при интегрировании
по импульсам
лишь очень малая часть разнонаправленных
цилиндров столкновений
покрывает (фиксированную) точку $\,{\bf r}_1\,$.

{\bf 4}. Мы подошли к любопытным выводам.

Усредним, варьируя координату $\,{\bf r}_1\,$, тождество (\ref{rcf1})
по эффективному цилиндру столкновений:
\[
\begin{array}{c}
\overline{F}_1(t,\Delta {\bf R},{\bf P}_0,{\bf p}_1)\,=\,\\
=\,V_0(t,\Delta {\bf R},{\bf P}_0)\,G^{(eq)}({\bf
p}_1)\,+\,\overline{V}_1(t,\Delta {\bf R},{\bf P}_0,{\bf p}_1)\,\,
\end{array}
\]
Здесь $\,G^{(eq)}({\bf p}_1)\,$ - максвелловское распределение, а
результат $\,\overline{F}_1(t,\Delta {\bf R},{\bf P}_0,{\bf p}_1)\,$
можно трактовать как среднюю по статистическому ансамблю плотность
числа двухчастичных конфигураций, отвечающих столкновениям
\cite{ri1}. Проводя затем интегрирование по импульсам и привлекая
(\ref{rmc}), получим
\begin{equation}
\begin{array}{c}
W_1(t,\Delta {\bf R})\,\equiv\, \int\int \overline{F}_1(t,\Delta {\bf
R},{\bf P}_0,{\bf p}_1)\,d{\bf P}_0\, d{\bf p}_1\,=\\
=\, V_0(t,\Delta {\bf R})\,+\,c_1^{-1}\,V_1(t,\Delta {\bf
R})\,\,\label{rcf2}
\end{array}
\end{equation}

Далее для простоты ограничимся случаем $\,{\bf f}=0\,$. Из
соотношения (\ref{rr2}), где обе части сейчас относятся к
равновесному (и, значит, сферически симметричному) броуновскому
движению, выразим\, $\,V_1(t,\Delta{\bf R})\hm =V_1(t,\Delta{\bf
R}\,;1)\,$ через производную левой части по относительной плотности
газа: $\,V_1(t,\Delta {\bf R})\,\hm =\, [\,\partial V_0(t,\Delta{\bf
R}\,;1+\phi )/\partial \phi\,]_{\,\phi\,= \,0}\,$\,\,.\, Объединяя
это с (\ref{rcf2}) и замечая, что функции $\,\overline{F}_1\,$ и
$\,W_1\,$ по своему происхождению от ФР $\,F_1\,$ неотрицательны,
приходим, в силу $\,W_1\geq 0\,$, к неравенству
\begin{equation}
c_1 V_0(t,\Delta {\bf R}\,;1)+\left [\frac {\partial V_0(t,\Delta{\bf
R}\,;1+\phi )}{\partial \phi }\,\right ]_{\,\phi\,= \,0}\,\geq 0
\label{rr3}
\end{equation}

Теперь предположим, что при $\,t\gg \tau \,$, где $\,\tau =\Lambda
/v_0\,$ - среднее время свободного пробега, а $\,v_0\,$ - характерная
тепловая скорость БЧ, вероятностное распределение пути БЧ
приближается к гауссову (см. пункт 1): $\,V_0(t,\Delta {\bf
R})\rightarrow V_G(t,\Delta{\bf R})\,$\,. Тогда, поскольку
коэффициент диффузии БЧ $\,D\,$ в газе Больцмана-Грэда устроен как
$\,D=\Lambda v_0=v_0^2\tau \,$ и вместе с $\,\Lambda \,$ обратно
пропорционален плотности газа, левая часть (\ref{rr2}) должна
получаться из  $\,V_G(t,\Delta{\bf R})\,$ заменой $\,\Lambda
\rightarrow \Lambda /(1+\phi )\,$, $\,D \rightarrow D/(1+\phi )\,$,
т.е.
\[
V_0(t,\Delta {\bf R}\,;1+\phi )\,\rightarrow \,\left (\frac
{1+\phi}{4\pi Dt}\right )^{3/2}\exp{\left [-(1+\phi)\,\frac {\Delta
{\bf R}^2}{4Dt}\right ]}\,\,
\]
Подставляя это выражение в неравенство (\ref{rr3}), приходим к \,\,
$\,V_G(t,\Delta{\bf R})\,[c_1+3/2\hm -\Delta {\bf R}^2/4Dt\,]\,\geq
0\,$\,\,, т.е. к явному противоречию.

Следовательно, теория не разрешает гауссову асимптотику! Это
утверждение, - а вернее, новое его доказательство, т.к. само оно не
ново (см. пункт 1), - принципиальный результат настоящей статьи.

Конечно, его нельзя назвать строгим уже по той причине, что мы
обошлись без конкретного определения операции усреднения - черты
сверху (хотя эвристически это плюс нашего рассмотрения). А ведь от
нее, в виду (\ref{rvm}), зависит значение коэффицента $\,c_1\,$.
Однако это означает просто, что в (\ref{rr3}) следует подставить
наименьшее из возможных значений. Оправдываемо и наше допущение, что
$\,c_1\,$ - это действительно коэффициент, а не функция $\,t,\Delta
{\bf R},{\bf P}_0\,$ и $\,{\bf p}_1\,$ в (\ref{rvm}) или $\,t\,$ и
$\,\Delta {\bf R}\,$ в (\ref{rmc}). Физически эффективная длина
цилиндра столкновений $\,c_1\Lambda\,$ для пары частиц ограничена
столкновениями с ``третьей частицей'', выбивающими пару из числа
кандидатов в столкновения. Поскольку же ``третьи частицы'' берутся в
основном из однородного фона, постоянство $\,c_1\,$ вполне
естественно.

{\bf 5}. В заключение еще несколько замечаний.

Во-первых, формулировка ``предела Больцмана-Грэда'' выше
подразумевала, что по размерам БЧ сравнима с атомами газа. Поэтому
выводы пункта 4 не относятся к ``макроскопической'' БЧ (для нее, по
крайней мере, в пределе бесконечно большой массы, гауссова
асимптотика остается вне подозрений).

Во-вторых, а что разрешено неравенством (\ref{rr3})? Пусть $\,V_0\,$
характеризуется при $\,t\gg \tau \,$, подобно $\,V_G\,$, единственным
параметром - коэффициентом диффузии:\, $\,\,V_0(t,\Delta {\bf R})\hm
\rightarrow (2Dt)^{-3/2}\,\Psi(\Delta {\bf R}^2/2Dt)\,$\,\,, где
$\,\int \Psi({\bf a}^2)\,d{\bf a}\,=1\,$ (условие нормировки). Тогда
неравенство (\ref{rr3}) сводится к
\[
\begin{array}{c}
(c_1+3/2)\Psi(x)\hm +x\,d\Psi(x)/dx\hm \,\geq 0\,\,\,,
\end{array}
\]
требуя, чтобы функция $\,\Psi(x)\,$ имела степенной длинный хвост:\,
$\,\Psi(x)\propto\, x^{-\,\alpha }\,$\, при $\,x\rightarrow\infty\,$,
где $\,\alpha \leq c_1+3/2\,$\,. Но такое поведение означало бы
бесконечность старших статистических моментов перемещения БЧ.
Вспомним, однако, у нас есть еще один параметр - $\,v_0\,$, так что
можно написать\, $\,\,V_0(t,\Delta {\bf R})\hm \rightarrow
(2Dt)^{-3/2}\,\Psi(\Delta {\bf R}^2/2Dt)\,\Theta (|\Delta {\bf
R}|/v_0t)\,$\,, где $\,\Theta(0)=1\,$ и $\,\Theta(x)\,$ достаточно
быстро убывает на бесконечности. С точки зрения (\ref{rr3}), такой
вариант тоже разрешается. И он воспроизводит, если взять $\,c_1=2\,$,
асимптотику распределения, найденную в \cite{rp1} в рамках
``столкновительного приближения'' для уравнений ББГКИ \cite{ri1,ri2}.
При этом в роли БЧ выступал атом газа. Данная асимптотика отражает
неэргодичность броуновских траекторий, которую можно охарактеризовать
как ``фликкерные'' флуктуации коэффициента диффузии (и подвижности)
БЧ \cite{ri1,rbk3,rrdev}.

Подчеркнем, что формализмы, использованные здесь и в \cite{rp1}, не
имеют ничего общего между собой (за исключением объекта их
приложения). Поэтому замечательную близость их результатов вполне
естественно расценить как свидетельство адекватности и того, и
другого.

Наконец, в-третьих, выше на примере парных ФР мы фактически наметили
связь между обычными корреляционными функциями и специальными ФР (для
парных и многочастичных столкновительных конфигураций), которые были
впервые введены в \cite{ri1}. Так, здешняя функция
$\,\overline{F}_1\,$ - это эквивалент функции $\,A_2\,$ из
\cite{ri1}, а функция $\,W_1\,$ из формулы (\ref{rcf2}) - эквивалент
$\,W_2\,$ из \cite{rp1}. Распространение этой связи на более высокие
порядки объединит два формализма и, несомненно, укажет новые
применения ``вириальных разложений'' (\ref{rr}) и (\ref{rr2}).

Признателен И.\,Краснюку за многочисленные полезные обсуждения.

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Г.\,Н. Бочков, Ю.\,E. Кузовлев, ЖЭТФ {\bf 72}, 238 (1977).

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Г.Н. Бочков, Ю.E. Кузовлев, ЖЭТФ {\bf 76}, 1071 (1979).

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\bibitem{rrdev}
Данная статья посвящена 25-летию первой работы, \cite{rpjtf}, на ту
же тему.

\end{thebibliography}
\end{document}
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