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Rigorous results in statistical mechanics, Dynamical foundations of kinetics, Kinetic theory of fluids, the Boltzmann kinetic equation, Boltzmann-Grad limit
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\documentclass[showpacs,pre,preprint,aps]{revtex4-1}
%\documentclass[twocolumn,showpacs,pra,aps]{revtex4-1}
%\draft
%\usepackage{graphicx}
\usepackage{dcolumn}
\usepackage{amssymb}
%\usepackage{bm}
%\usepackage{footnote}
\begin{document}
\title{Dynamical Virial Relations \\ %
and Invalidity of the Boltzmann Kinetic Equation}
% in spatially inhomogeneous situations}
\author{Yu. E. Kuzovlev}
\email{kuzovlev@fti.dn.ua} \affiliation{Donetsk Institute
for Physics and Technology of NASU, 83114 Donetsk, Ukraine}
\date{17 April 2012}
\begin{abstract}
A sequence of exact relations is found which connect %
one- and many-particle time-dependent distribution %
functions of non-equilibrium low-density gas with their derivatives %
in respect to mean density. It is shown that, %
at least in the context of spatially non-uniform %
gas evolutions, these relations forbid the ``molecular %
chaos propagation'' and imply inapplicability of the %
Boltzmann kinetic equation even under the %
Boltzmann-Grad limit and regardless of degree of the non-uniformity.
Statistical and physical meaning of this result is explained.
\end{abstract}
\pacs{05.20.Dd, 05.20.Jj, 05.40.-a, 51.10.+y}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Two main questions under our interest here will be %
(i) rigorous results about non-equilibrium many-particle systems %
and (ii) theoretical status of %
the celebrated L.\,Boltzmann's kinetic equation %
\cite{bol,bog,re,lp,bal,ub} from viewpoint of such results. %
The Boltzmann equation (BE) can be written as
%
\begin{eqnarray}
\begin{array}{c}
\dot{F}(t,r,v)=-v\nabla F(t,r,v)\, % + \\ %
+\, n\,{\bf C}_2\,F(t,r,v)*F(t,r,w)\,\,, \label{be}
\end{array}
\end{eqnarray}
where\, $\,F(t,r,v)\,$\, is one-particle distribution function %
normalized to volume,\, \,$\,n\,$\, is mean gas density,\, and %
$\,{\bf C}_2\,$ is the Boltzmann's pair collision operator %
(``collision integral'') %
which acts onto $\,F\,$'s velocity argument as
%
\begin{equation}
\begin{array}{c}
{\bf C}_2\,F(v)*F(w)\,= \int d^3w \,|v-w| %
\int d^2b\, \times \\ \times\, %
[F(v^{\,in}(b,v,w)) F(w^{\,in}(b,v,w))-F(v)F(w)]\, %
\label{bo}
\end{array}
\end{equation}
with\, $\,b\,$\, being the impact parameter vector %
(perpendicular to\, $\,v-w\,$) and\, $\,v^{\,in}\,$ and %
$\,w^{\,in}\,$\, input velocities what lead %
to the given output ones. %
Anybody can agree that today's kinetic theory, %
as well as 100 years ago, is unthinkable %
without BE. Nevertheless, nobody have presented %
a conclusive enough derivation of BE from exact equations of %
statistical mechanics, e.g. the BBGKY equations %
(BBGKYE) \cite{bog,re,lp,bal}. %
In general, the BBGKYE can only produce much more %
complicated equation, %
%
\begin{equation}
\begin{array}{c}
\dot{F}=-v\nabla F + %
n\,{\bf C}_2\,F*F + %
n^2\,{\bf C}_3\,F*F*F +\, \\ %
\,+\,\,n^3\,{\bf C}_4\,F*F*F*F\,+\dots\, %
\end{array}
\,\, \label{gbe}
\end{equation}
(see below), %
where operator $\,{\bf C}_s\,$ ($\,s>2\,$) represents %
ring-like connected clusters of $\,s\,$ collisions %
involving $\,s\,$ particles %
(for $\,s=3\,$ described e.g. in \cite{lp}). %
Therefore, BE follows from BBGKYE %
under formal ``low-density limit'' only, when\, %
$\,n\rightarrow 0\,$\, \cite{bog,re,lp,bal}. %
But this is non-physical %
limit, since it enforces the mean free path of %
gas atoms,\, $\,\lambda\sim 1/\pi a^2 n\,$, - %
with\, $\,a\,$\, denoting characteristic radius %
of atom-atom interaction, - to become greater than free %
flights of atoms,\, $\,\sim |v|t\,$, during any given evolution %
time\, $\,t\,$ (so that the resulting BE is in fact addressed %
to collisionless gas).
More physically reasonable idealization, - which allows %
\,$\,\lambda\,$ to be less than typical $\,|v|t\,$, - is %
the ``Bpltzmann-Grad limit'' (BGL) %
when\, $\,n\rightarrow \infty\,$\, %
and simultaneously\, $\,a\rightarrow 0\,$\,, %
in such way that\, $\,\lambda\,$\, %
is kept constant. %
At that,\, $\,a^3n\rightarrow 0\,$\,, %
i.e. gas becomes ``infinitely rare''. %
This gave rise to hopes that the terms with\, %
$\,{\bf C}_3\,$, %
$\,{\bf C}_4\,$,\, etc., in Eq.\ref{gbe} %
must vanish under BGL too. %
%
Moreover, O.\,Lanford and others %
even suggested a proof of this hypothesis for hard-sphere %
gas \cite{lan,vblls}. %
But their interpretation of BBGKYE in this %
singular case hardly is unconditionally acceptable %
(in essence, in my opinion, they considered different equations %
where some of interaction terms were inadvertently rejected %
while another from the very beginning treated %
as pair collision integrals) \cite{hs1}.
In any case, %
contributions with\, $\,{\bf C}_3\,$, $\,{\bf C}_4\,$, %
etc., in Eq.\ref{gbe} appear due to %
violation of the Boltzmann's ``molecular %
chaos hypothesis'', that is due to %
statistical correlations between atoms %
what are in mutually input ({\bf pre-collision}) %
state (aimed to each other). %
Naturally, such correlations take place mainly %
at inter-atom separations comparable %
with \, $\,a\,$\,\,, %
but this does not mean that they vanish under BGL %
(merely they sit just where they are most ``harmful'').
What is physical meaning of the pre-collision %
correlations\,?
Notice that BE (\ref{be})-(\ref{bo}) %
fully corresponds to the ``probability-theoretical %
view'' at physical world:\, it presumes that %
gas evolution consists of elementary random events %
(collisions with various input velocities and impact parameters) %
which occur independently one on another and all possess %
strictly certain (though may be unknown numerically) %
\,{\it a priori}\, (conditional) probabilities. %
Such the view was originated by J.\,Bernoully 300 years ago \cite{jb}. %
%
But in 20-th century N.\,Krylov \cite{kr} showed that %
assumption that any sort of events %
can be furnished with certain %
\,{\it a priori}\, probability, or relative frequency, has %
no support in rigorous statistical mechanics. %
In opposite, mechanics allows different relative frequencies %
on different phase trajectories of a many-particle system. %
%
Then, fluctuations of relative frequencies %
from one phase trajectory (experiment) to another, - %
being averaged over a statistical ensemble of %
trajectories' initial conditions , - %
produce correlations between events and particles %
even in absence of direct %
cause-and-consequence connections between them %
(see item 1 in Sec. ``Footnotes'' below).
Just such statistical correlations %
were found and investigated %
in my works \cite{i1} (see also \cite{i2}) and %
\cite{p1,p0802,p0806,tmf,ig,p1105,mart} %
on spatially inhomogeneous %
evolutions of fluids and random walks of their particles. %
And just such correlations determine %
higher-order terms in Eq.\ref{gbe}, %
impeding reduction of BBGKYE to BE even in spite of BGL. %
%
At that, role of the inhomogeneity %
is to visualize and magnify effects of the uncertainty of %
collision probabilities of fluid's particles. %
%
In particular experiments the uncertainty manifests itself %
in the form of scaless (1/f\,-type) %
low-frequency fluctuations of %
%
both individual particles' mobilities %
and collective transport characteristics of a fluid. %
%
\,\,\,
Here, I suggest new proofs of invalidity of the BE (\ref{be}). %
It means that BE is qualitatively defective model, %
therefore, its quantitatively precise investigation does not %
garantee full physical correctness of the results %
even in application to dilute gas.
Starting from general properties and %
exact density expansions of %
\, $\,F(t,r,v)\,$\, and many-particle %
distribution functions (DF), %
we will derive a sequence of exact relations %
between them and their derivatives %
in respect to density \,$\,n\,$\,, %
and then, with the help of these relations, demonstrate %
that the pre-collision inter-particle statistical correlations, - %
and hence all higher-order terms in Eq.\ref{gbe}, - %
stay finite and significant even under BGL. %
\section{Many-particle statistical dynamics}
Let $\,N\gg 1\,$ gas atoms are contained, %
by means of an auxiliary external potential, %
in volume\, $\,\Omega = N/n\,$\,. %
Let\,
%
$\,x_i=\{r_i,v_i\}\,$\, denote variables %
of\, $\,i\,$-th,\ atom,
$\,F_i(t)\equiv F(t,x_i)\equiv F_1(t,x_i)\,$\,,\, %
$\,F_{ij}(t)\equiv F_2(t,x_i,x_j)\,$\,,\, %
$\,F_{ijk}(t) \equiv F_3(t,x_i,x_j,x_k)\,$\,,\, %
... \, %
$\,F_{1\dots N}(t) \equiv %
F_N(t,x_1,x_2,\,\dots\, x_N)\,$\,
are one-, two-, %
three-,\, ... \, $\,N\,$-particle DFs,\,
$\,L_i\,$\,,\, %
$\,L_{ij}\,$\,,\, %
$\,L_{ijk}\,$\,,\, ... \, %
$\,L_{1\dots N}\,$\, %
are one-, two-, %
three-,\, ... \, $\,N\,$-particle %
Liouville operators (including the %
auxiliary potential),\, and
$\,S_i(t) = \exp{(L_i \,t)}\,$\,,\, %
$\,S_{ij}(t) = \exp{(L_{ij} \,t)}\,$\,,\, %
... \, %
$\,S_{1\dots N}(t) = \exp{(L_{1\dots N} \,t)}\,$\,
\,are corresponding evolution operators. %
All these objects are fully symmetric functions %
of their arguments (indices).
If all DFs are thought normalized to volume %
and all obey the requirement %
of mutual consistency, %
then
\begin{eqnarray}
\frac 1{\Omega^{s}} \int_{1}\dots %
\int_s F_{1\,\dots\,s}(t)\,\,=\,1 \,\,\,,
\label{nc}\\ \, %
F_{1\,\dots\,s}(t)\, \,=\, %
\frac 1{\Omega} \int_{s+1} %
F_{1\,\dots\,s\, s+1}(t)\,=\, %
%
\frac 1{\Omega^{N-s}} \int_{s+1}\dots %
\int_N F_{1\,\dots\,N}(t)\,\,\,, \nonumber
\end{eqnarray}
%
where\, $\,\int_k \dots \,\equiv \, \int_{\Omega} %
d^3r_k \int d^3v_k \dots \,$\,.\, %
Evolution of all marginal DFs is determined by that %
of the whole system's DF according to %
%
\begin{equation}
\begin{array}{c}
F_{1\,\dots\,N}(t)\, \,=\, %
S_{1\,\dots\,N}(t)\, %
F_{1\,\dots\,N}(0)\,\,\, \label{ev}
\end{array}
\end{equation}
%
Since it conserves full phase volume, %
the equalities (\ref{nc}) will be satisfied at all %
times if it is so at one, ``initial'', %
time moment, e.g. at $\,t=0\,$. %
But some specific form of DFs can realize at one %
moment only. For example, we may choose
%
\begin{equation}
\begin{array}{c}
F_{1\,\dots\,s}(0)\, \,=\, %
\prod_{j=1}^s\, F_j(0)\,\,\,, \label{ch}
\end{array}
\end{equation}
thus assuming ideal ``molecular chaos'' at the initial time moment.
Further, it is convenient to apply a useful formal trick %
and introduce operation \,$\,\circ\,$\, of %
``coherent product'' %
of the evolution operators. %
By definition, for any two %
non-intersecting sets of indices, %
%
\begin{equation}
\begin{array}{c}
S_{i\dots j}\,\circ\,\, S_{k\dots \,l} \,=\, %
S_{i\dots j\,k\dots \,l}\,\,\, \label{cp}
\end{array}
\end{equation}
In essence, the left side here is mere equivalent %
notation for the right-hand side. %
For intersecting sets, of course, one must %
take their union. %
Thus, for instance,\, %
$\,S_{i\dots j}= S_i\circ \dots %
\circ S_{j} \,$\,,\, %
and we can write identities
%
\begin{eqnarray}
S_{1\dots N}\, \,=\, S_{1\dots s} %
\prod_{j\,=\,s+1}^N \circ S_j\,=\, %
S_{1\dots s} %
\prod_{j\,=\,s+1}^N %
\circ [\,1\,+\,(S_j -1)\,]\,=\, %
\label{sid} %
\\ =\, %
%
S_{1\dots s}\,+\, %
\sum_{j\,=\,s+1}^N \, S_{1\dots s} \circ (S_j-1) %
\,+\, \nonumber\\ %
+ \sum_{s+1\,\leq j0\,$\,, %
$\,a\lesssim |r_{12}|< |v_{12}|t\,$\, %
and\, $\,|b|\lesssim a\,$\,, %
where\, %
$\,r_{12}=r_2-r_1\,$\,,\, $\,v_{12}=v_2-v_1\,$\,, %
and \,$\,b = r_{12}- v_{12} %
(v_{12}\cdot r_{12})/|v_{12}|^2\,$\, %
is impact parameter vector %
(already mentioned in Sec.1). %
Thus, under the MC there are no correlations %
between atoms in mutually pre-collision states, - %
defined like the post-collision ones but with\, %
\,$\,v_{12}\cdot r_{12} <0\,$\,, - %
and no correlations between mere close %
though non-interacting atoms %
(for which\, $\,|r_{12}|\,$\, is greater %
than\, $\,a\,$\, but comparable with\, $\,a\,$), %
but the payment for such pleasure is presence of the %
unreservedly far propagating %
post-collision correlations. %
%
If it was really so under BGL, %
then the exact relation (\ref{df}), - %
after its multiplying by\, $\,n\,$\,, - %
would reduce to %
\begin{eqnarray}
n\, \frac {\partial}{\partial n}\,\,\, %
F_1(t)\, =\, n \int_2 %
[\,Z_{12}(t) -1\,]\, F_{1}(t)\, F_{2}(t)\, %
\label{dfa} \,\,
\end{eqnarray}
The multiplication ensures finiteness of both sides here %
under transition to BGL, along with transition from %
\,$\,n\,$\, to physically more meaningful variable %
like\, $\,\kappa =\pi a^2n = 1/\lambda\,$\,, %
so that \,$\,n\, \partial/\partial n\, \Rightarrow %
\,\kappa\, \partial /\partial \kappa\,$\,. %
In combination with Eq.\ref{z12} %
this equality yields- %
\begin{eqnarray}
n\, \frac {\partial}{\partial n}\,\,\, %
F_1(t,r_1,v_1)\, =\, n \int_0^t d\tau %
\int d^3v_2 \int d^2b \,\, %
|v_1-v_2|\,\, %
\times \nonumber\\ \times\, %
[\,F_1(t,r_1 + (v_1^\prime -v_1)\tau, %
v_1^\prime )\, %
F_1(t,r_1 + (v_2^\prime -v_1)\tau, %
v_2^\prime )\,\,-\, %
\nonumber\\ \, %
-\, F_1(t,r_1, v_1)\, %
F_1(t,r_1 + (v_2 -v_1)\tau, v_2)\,]\,=\, %
\nonumber\\ \,=\, %
n \int_0^t d\tau \,\,\exp{(-\tau v_1\nabla_1)} \, %
{\bf C}_2\, F_1(t,r_1 + v_1\tau, v_1)\, %
F_1(t,r_1 + v_2\tau, v_2) \, \,\,\, %
\label{dfc}
\end{eqnarray}
Here\, $\,^\prime\,$\, plays the same role %
as in the BE (\ref{be}). %
And, as in (\ref{be}), transition to BGL
allows to neglect $\,F_1\,$'s %
changes at time and space scales %
related to\, $\,a\,$\,. %
%
Simultaneously, %
according to the previous section, %
the same MC assumption, - that\, %
$\,F_{12}(t)=Z_{12}(t)\,F_1(t)F_2(t)\,$\,, - %
produces the BE (\ref{be}) itself. %
Hence, if this is true assumption %
then Eqs.\ref{dfc} and \ref{be} should be %
compatible one with another. %
%
In fact, however, %
they can not be satisfied simultaneously, %
except purely spatially homogeneous case %
when\, $\,\nabla_1 F_1(t)= 0\,$\, (see Footnote 2) ! %
%
To become convinced of this fact in detail, %
one may e.g. consider linearized evolution of small %
local or periodic perturbations of equilibrium state %
(\,$\,F^{eq}(v_1)\,$\, from above). %
\,\,\,
Consequently, %
contribution of the rejected terms of Eq.\ref{f2e} %
into integral in DVR (\ref{df}) is on order of %
its value (see Eq.\ref{dfa}) under the MC assumption, %
and the latter, %
as applied to spatially inhomogeneous evolutions, %
is incompatible with the exact relation (\ref{df}) %
even in spite of the BGL. %
In other words, %
the ``molecular chaos propagation'' %
fails in spatially inhomogeneous case, %
so that the BE proves to be invalid %
even under BGL \,(see Footnotes 3 and 4). %
\,\,\,
I would like to underline that %
all the aforesaid equally embrace %
the hard-sphere interaction. %
To finish the paper and exclude hopes to ``save'' BE, %
we will supplement just presented proof of %
its invalidity with short notes on the pre-collision correlations. %
%
Additional comments on related many-particle correlations %
and their influence onto\, $\,F_1(t)\,$'s\, %
evolution are placed to Appendix B.
%7
\section{Excess and pre-collision correlations}
Let us return to Eq.\ref{f2e} and %
consider functions
% two functions by %
\begin{equation}
\begin{array}{c}
\Delta F_{12} (t)\,\equiv\, %
Z_{12}(t)\, \Delta F_{12}^\prime (t)\,\equiv\,
%
F_{12}(t) \,-\, Z_{12}(t)\, %
F_1(t)\,F_2(t)\,\,= \, %
\nonumber\\ \,=\, %
%
\sum_{k\,=\,1}^\infty\, n^k\, %
F_{12}^{(k)}(t)\,=\, %
\\ \,=\, %
n \int_3 \,[\, Z_{123}(t)\,- %
\, Z_{12}(t)\,Z_{13}(t)\,-\, %
Z_{12}(t)\,Z_{23}(t)\, + %
\nonumber\\ \,+ %
\,Z_{12}(t)\,]\, %
F_1(t)\,F_2(t)\, F_3(t)\, \, %
+\, \dots\, \,\, \label{ec}
\end{array}
\end{equation}
They characterize those part of %
statistical correlations between two atoms %
what is due to not their interaction between themselves %
but common pre-history of their interactions %
(collisions) with the rest of gas. %
At that,\, $\,k\,$-th term of the sum %
represents connected chains (clusters) of %
at least \,$\,k+1\,$\, %
collisions (actual or virtual ones) %
conjointly involving \,$\,k+2\,$\, atoms. %
%
Statistical meaning of such ``excess'' %
(or ``historical'' \cite{tmf}) correlations %
was exhaustively explained in \cite{i1,i2} %
(see also Introduction and notes in \cite{hs1,p1,ig,hs}).
It is clear that, first, if\, %
$\,\Delta F_{12} (t)\neq 0\,$\, %
somewhere in two-particle phase space, %
then\, $\,\Delta F_{12} (t)\neq 0\,$\, %
almost everywhere, since anyway there are %
many various clusters of collisions resulting in given %
current states of atoms 1 and 2. %
Second,\, $\,\Delta F_{12} (t)\neq 0\,$\, %
for post-collision states, since\, $\,\Delta F_{12} (t)\,$\, %
must compensate and stop nonphysical %
unrestricted propagation %
of the above mentioned post-collision correlations %
prescribed by the MC approximation\, %
$\,F_{12} (t)= F_{12}^{(0)}\,$\,. %
Hence, undoubtedly\, $\,\Delta F_{12} (t)\neq 0\,$\, %
almost everywhere, including the pre-collision states. %
This reasoning shows that appearance %
of pre-collision correlations along with %
other (``non-collision'') excess correlations %
(at states for which\, %
\, $\,\Delta F_{12} (t)\approx \Delta %
F_{12}^\prime (t)\,$\, %
in Eq.\ref{ec}) is quite inevitable from %
physical point of view. %
Third, on the other hand, %
separation of atoms 1 and 2 %
must decrease a number of the clusters %
determining\, $\,C_{12}(t)\,$\, %
nearly proportionally to visual space angle %
\,$\,\o\,$\, %
of one of the atoms from viewpoint of another, %
\,$\,\o \sim \pi a^2/4\pi |r_{12}|^2\,$\,. %
%
The matter is that, since\, $\,F_{12}^{(k)}(t)\,$\, %
involves \,$\,k+1\,$\, (or more) collisions %
but contains only \,$\,k\,$\, integrations, %
one (or more) of \,$\,k\,$\, integration %
velocities is restricted in respect %
to its direction by a space angle %
\,$\,\sim \o\,$\,. %
%
By these reasons, we can propose %
the following qualitative fit of the correlation function\,: %
%
\begin{eqnarray}
C_{12} (t)\,=\, \frac {a^2} %
{a^2 + 4|r_2-r_1|^2}\,\, %
C_{12}^\prime (t)\,\,\,, \label{ce}
\end{eqnarray}
where function\, $\,C_{12}^\prime (t)\,$\, %
keeps non-zero under BGL %
and smmothly depends on\, $\,|r_{12}|\,$\, %
at\, $\,|r_{12}|\gtrsim a\,$\,. %
Then DVR (\ref{df}) implies (under BGL) that
%
\begin{eqnarray}
\frac {a^2}4 \int d\o \int_0^\infty %
d|r_{12}| \int d^3v_2 \,\,\, %
C_{12}^\prime (t)\,\,=\, %
%
\pi \, a^2\,\, %
\frac {\partial F_1(t)}{\partial \kappa}\, %
\,\,, \label{dff}
\end{eqnarray}
where\, $\,\o = r_{12}/|r_{12}|\,$\,. %
We see that at any fixed distance %
\,$\,|r_{12}|\,$\, the excess correlations, - %
particularly, the pre-collision ones, - %
turn to zero under BGL. %
In this sense, the MC really takes place. %
But contribution of these correlations %
to the ``triple collision integral'' %
(\ref{c3}), to\, $\,{\bf C}_4\,$\,,\,...\,, %
and thus to the whole Eq.\ref{gbe} %
is determined by region \,$\,|r_{12}|\sim a\,$\, %
where all excess (pre-collision) %
correlations stay finite under BGL (Footnote 5). %
Hence, in essence MC fails.
\,\,\,
And last remarks. %
(i)\, Both the left-hand integral in Eq.\ref{dff} %
and right-hand derivative there define some %
(one and the same) characteristic length. %
Of course, it must be nothing but the\, %
$\,\lambda\,$\, (since\, $\,C_{12}(t)\,$\, %
can not be indifferent to \,$\,\lambda\,$\,). %
At the same time, obviously, any of constituent parts of\, %
$\,C_{12}(t)\,$\,, namely,\, %
$\,[Z_{12}(t)-1]F_1(t)F_2(t)\,$\, and\, %
$\,F_{12}^{(k)}(t)\,$\, ($\,k=1,2,\dots\,$), %
extends up to\, $\,|r_{12}|\sim v_0 t\,$\,, %
with\, $\,v_0\,$\, being characteristic velocity %
of gas atoms. %
Hence, indeed all these parts are required in order %
to introduce\, $\,\lambda\,$\, %
in place of\, $\,v_0t\,$\,.
\,\,\,
(ii)\, The reasonings and conclusions %
of this section (as well as that at end of Sec.5 and in Appendix B) %
in no way rely on some measure of non-uniformity of gas state. %
Therefore, all they are equally valid, - %
and thus BE is equally invalid, - %
for arbitrary weakly non-uniform gas (Footnotes 4 and 6). %
\section{Conclusion}
Considering spatially inhomogeneous %
evolution of low-density gas, we introduced a %
non-standard representation of its time-dependent %
distribution functions (DF) in the form of their %
density expansion, \, and then %
exploited it to derive original %
exact ``dynamical virial relations'' (DVR) %
connecting DFs with their density derivatives. %
\,Then we applied DVR to analysis of behavior %
of two-particle correlation function %
under the Boltzmann-Grad limit (BGL), %
in order to examine validity of the %
``folklore'' opinion %
that evolution of one-particle DF under BGL %
exactly obeys the Boltzmann kinetic equation (BE) %
while many-particle DFs undergo the %
``molecular chaos propagation''. %
We showed that the corresponding %
approximate approaches to gas %
kinetics \cite{bog,re,lp,bal}, which seem well %
grounded in the ``low-density limit'', %
at the same time appear non-grounded under BGL, %
since contradict the DVR and thus fail when applied to %
spatially non-uniform situations (independently on degree %
of the non-uniformity). %
This fact does not mean, of course, %
that idea of the Boltzmann collision operator %
(integral) is defective. This is excellent concept %
if one applies it to a separate collision. %
But it by itself is unable to comprise %
those inter-atom statistical correlations %
what arise from uniqueness (``non-ergodicity'' \cite{tmf}) %
of histories of collisions in particular experiments %
(see Introduction and comments in %
\cite{i1,i2,p1,p0802,tmf,ig,p1105,eiphg,p0705}). %
One of ways to take into account all these %
correlations is the approach to correct solution of %
the BBGKY equations (the ``collisional approximation'') %
suggested in \cite{i1} (see also \cite{i2,p1}). %
In principle, this approach allows to consider %
a wide variety of phenomena and problems %
(``from molecular Brownian motion to shock waves'' %
\cite{p0705}). %
In \cite{i1,p1} this approach was used to investigate %
statistical characteristics of molecular random walks %
in fluids (in particular, the related 1/f noise). %
The results then were confirmed from viewpoint of %
corresponding exact ``virial relations'' %
\cite{p0802,tmf,p1105}. %
Thus, some ancient prejudices, %
pointed out in \cite{kr}, like %
``molecular chaos'', were overcome %
with substantial physical profit. %
Therefore, it seems reasonable to try to %
apply the mentioned approach (strengthened with the DVR) %
to macroscopic gas (fluid) kinetics and hydrodynamics too. %
All the more so as their difference %
from Boltzmannian kinetics and classical %
hydrodynamics may be practically important %
even for weakly non-equilibrium %
(non-uniform) processes (see Footnote 7). %
%app
\,\,\,
\appendix
\section{Correlation (cumulant) %
distribution functions and the DVR}
Let us introduce irreducible many-particle correlation, %
or cumulant, functions (CF) by
\begin{equation}
\begin{array}{c}
F_{12} \,= \,F_{1}F_{2} + C_{12}\,\,, %
\,\,\,\,\,\, %
F_{123} \,= \,F_{1}F_{2}F_3 + %
\\ +\, C_{12}F_3 + %
C_{23}F_1 + C_{13}F_2 + C_{123}\,\,\,, \label{cfs}
\end{array}
\end{equation}
and so on. %
Higher-order CFs can be defined with the help of %
generating functionals: %
\begin{eqnarray}
% \begin{array}{c}
1\,+\sum_{s=1}^\infty F_s(t)\,\frac {\psi^s}{s!} \,=\, %
\nonumber \\ %
\,=\, \exp\, [\,F_1(0)\,\psi \,]\,\, %
\{\,1\,+\sum_{s=1}^\infty G_s(t)\, %
\frac {\psi^s}{s!}\}\, =\, \nonumber \\ %
%
\,=\, \exp\, [\,F_1(t)\,\psi \,+ %
\sum_{s=2}^\infty C_s(t)\, \frac {\psi^s}{s!}\,] %
\,\,,\,\, \,\,\, \,\,\, %
\label{gfs}
% \end{array}
\end{eqnarray}
where, of course,\, %
%
\begin{eqnarray}
F_s(t)\,\psi^s \,=\, %
\int_1\dots \int_s F_{1\,\dots\, s}(t) %
\,\psi(x_1)\dots \psi(x_s)\,\,\, \nonumber
\end{eqnarray}
and so on. %
In these shortened notations, %
the DVR (\ref{dg}) altogether can be %
accumulated into single generating DVR %
\begin{eqnarray}
\frac {\partial }{\partial n}\, %
\,\sum_{s=1}^\infty G_s(t)\, %
\frac {\psi^s}{s!}\,\,\, =\, %
\nonumber\\ \,=\, %
\int dx \frac {\delta }{\delta \psi(x)}\, %
\sum_{s=1}^\infty G_s(t)\, %
\frac {\psi^s}{s!}\,\,\, \,\,\,\label{dgs}
\end{eqnarray}
(to be supplemented with equality %
(\ref{g0}) and thus also (\ref{gz})). %
%
Obviously, this generating DVR is equivalent to %
\begin{eqnarray}
\frac {\partial C_{1\,\dots\,s}(t)}{\partial n}\,\, %
=\, \int_{s+1} C_{1\,\dots\,s\,s+1}(t)\, \,\, %
\, \,\, (s\geq 2) \,\,, \,\,\,\,\,\, %
\label{dc}\\
%
\frac {\partial F_{1}(t)}{\partial n}\,\, %
=\, \int_{2} C_{12}(t)\, \,\, \,\, %
\label{dc1}
\end{eqnarray}
Generalization of these DVR to dense gases (fluids), - %
when the ``initial molecular chaos'' (\ref{ch}) is too %
bad choice for initial DFs (since they must %
include effects of atom-atom interaction %
and thus depend on the mean %
density), - will be considered elsewhere.
\section{Many-particle correlations %
and falsity of the BE's ``derivations''}
According to the BBGKY equations (\ref{bbgky}) %
or the DVR (\ref{dg}) and/or (\ref{dc1}), %
the pair pre-collision correlations %
arise in company with various many-particle ones. %
Their importance for correct approach %
to gas (fluid) kinetics was demonstrated %
already in \cite{i1} (see also \cite{i2,p1,tmf}). %
%
For one more demonstration, %
let us criticize the ``derivation'' %
of BE (\ref{be}) suggested in \cite{bal}. %
Assume, as there, that the irreducible (``pure'') %
three-particle correlations (see Appendix 1), - %
described by CF\, $\,C_{123}(t)\,$\,, - %
can be neglected at\, $\,a^3n\ll 1\,$\, %
and hence under BGL. %
Then the second of Eqs.\ref{bbgky}, %
when written in terms of CFs, reduces to %
%
\begin{equation}
\begin{array}{c}
\dot{C}_{12}\,=\, L_{12}\, C_{12}\, +\, %
L_{12}^\prime\, F_1\,F_2\, =\, %
L_{12}^0\, C_{12}\, +\, %
L_{12}^\prime\, F_{12}\,\, \label{bale}
\end{array}
\end{equation}
Solving Eq.\ref{bale} (with zero initial condition) %
and inserting the result,\, %
$\,C_{12}\,$\,, %
into the first of Eqs.\ref{bbgky}, %
one can come (at\, $\,a^3n\ll 1\,$\,) % %
to the BE \cite{bal}. %
If this is true derivation of BE, %
then it should be compatible with the DVR, %
at least, with Eq.\ref{df} %
(i.e. Eq.\ref{dc1}). %
In fact, however, this is not the case. %
Indeed, integration of Eq.\ref{bale} over\, %
$\,x_2\,$\,, after multiplying it %
by\, $\,n\,$\,, yields
%
\begin{eqnarray}
\frac {\partial }{\partial t}\, %
\,n \int_2 C_{12}\,=\, %
L_1^0\, n\int_2 C_{12}\, +\, %
n\int_2 L_{12}^\prime\, F_{12}\, =\, %
\nonumber \\ \,=\, %
L_1^0\, n\int_2 C_{12}\, +\, %
\left[\,\frac {\partial }{\partial t}\,-\, %
L_1^0\,\right]\, F_1\,\,\,, %
% n\,{\bf C}_2\,F_1*F_2
\nonumber
\end{eqnarray}
where we used also the first of %
the BBGKY equations (\ref{bbgky}). %
Combining this equality with the DVR (\ref{df}), %
we come to equality %
%
\begin{equation}
\begin{array}{c}
[\,\partial /\partial t\,-\,L_1^0\,]\, %
[\,n\, \partial F_1/\partial n\,\, -\,F_1\,] %
\,=\, 0\,\,\,, \nonumber
\end{array}
\end{equation}
which certainly is wrong.
Hence, the above BF's ``derivation'', based %
on approximation (\ref{bale}), is erroneous. %
And, in order to get a correct description of %
\,$\,F_1\,$'s evolution, one should %
seriously think about role of %
many-particle correlations.
%fn
\section*{Footnotes}
%\,\,\,
{\bf 1}.\,
Similar statements were formulated, independently on the Krylov's %
work, 30 years ago and confirmed at the level of semi-phenomenological %
statistical theory by the author of this %
communication together with Prof. G.\,Bochkov %
in \cite{pjtf,pr157} and \cite{bk12,bk3}), where the idea of %
realistic ``non-ergodic'' random walks %
(of charge carriers), - which have no certain %
diffusivity and mobility and do not obey %
Boltzmannian kinetics %
(``quasi-Gaussian random walks \cite{i2}), - %
was suggested and mathematically designed %
(about my recent considerations on this subject %
see \cite{p1008,eiphg}). For understanding %
our principal idea of 1/f-noise, my preprint %
\cite{i2} may be useful and, besides, my %
recent presentation % placed at\, %
\,\url{http://yuk-137.narod.ru/What_is_1_by_F_noise.pdf}
%\begin{verbatim}
%yuk-137.narod.ru/What_is_1_by_F_noise.pdf
%\end{verbatim}
\,\,\,
{\bf 2}.\,
At that, Eq.\ref{dfc} turns to %
\,$\,n\,\partial F/\partial n\,=\,%
t\,n{\bf C}_2\, F*F\,$\,, while Eq.\ref{be} %
can be written as %
\,$\,t\,\partial F/\partial t\,=\,%
t\,n{\bf C}_2\, F*F\,$\,, %
\,therefore, both they are satisfied by %
a function depending on time and density %
via their product\, $\,\xi=nt\,$\, %
and obeying equation\, %
\,$\,\partial F/\partial \xi\,=\,%
{\bf C}_2\, F*F\,$\,.
\,\,\,
{\bf 3}.\,
Notice that the %
modified formulation \cite{lp} of the ``molecular
chaos'' hypothesis, like %
\,$\,F_{12}(t)=Z_{12}(t_0)\,F_1(t)F_2(t)\,$\,, - %
with a fixed\, $\,t_0\,$\, (at\, $\,t>t_0\,$\,), - %
is worse than the above formulation, since %
appears incompatible with virial relation (\ref{df}) %
even in homogeneous case.
\,\,\,
{\bf 4}.\,
It is useful to add that %
the incompatibility of (both hypothetical) equations %
(\ref{dfa}) and (\ref{be}) %
is sufficient but not necessary condition %
for invalidity of one (or both) of them. %
Therefore, both they can be wrong %
for homogeneous gas too. %
This suspicion is confirmed by argumentation %
presented in Sec.7 %
(another matter that under our conditions %
for existence of the thermodynamic limit, - %
see Sec.3, - homogeneous gas means %
merely equilibrium gas). %
\,\,\,
{\bf 5}.\,
Seemingly, possibility of such rather subtle circumstance %
was not foreseen by the authors of \cite{lan,vblls}, %
and this oversight caused subtle errors in their %
formal construction (see notes in \cite{hs1,i2,p0806,hs}). %
\,\,\,
{\bf 6}.\,
This circumstance highlights that the Boltzmann equation %
is from mathematical tale but not from real life:\, %
it describes ensemble of ``dice tosses'' %
(``coin tossings'') instead of ensemble of dynamical %
trajectories (which in fact appears to be irreducible %
to ``dice tosses'' \cite{p0802,tmf}).
\,\,\,
{\bf 7}.\,
This statement comes from %
\,(i) essential difference of our theory of %
self-diffusion of gas atms \cite{i1} from the %
classical one, and %
\,(ii) observation %
that in case of (infinitesimally) weak non-equilibrium %
equations of our approach (``collisional approximation'') %
\cite{i1} are quite similar to the equations describing %
the self-diffusion (the only difference is that %
the place of the Boltzmann-Lorentz operators %
is occupied by the linearized Boltzmann operators). %
%
%} %
%bib
%\newpage
\,\,\,
%----------------------------------------
%\section*{References}
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