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%%% MACROS
\def\mic{microscopic}
\def\mac{macroscopic}
\newcommand{\lte}{local thermodynamic equilibrium}
\newcommand{\vx}{\vec{x}}
\newcommand{\vv}{\vec{v}}
\newcommand{\Er}[1]{\langle#1\rangle}
\newcommand{\KL}[1]{\left(#1\right)}
\newcommand{\eps}{\varepsilon}
\newcommand{\Ibb}[1]{ {\rm I\ifmmode\mkern -3.6mu\else\kern -.2em\fi#1}}
\newcommand{\Rl}{{\Ibb R}}
\begin{document}
\title{On the Problem of Defining Local Thermodynamic Equilibrium}
\author{H. Roos\\
\quad \\Institut f\"ur Theoretische Physik, Universit\"at
G\"ottingen\\ Bunsenstr. 9, D-3400 G\"ottingen, Germany}
\date{\today}
\maketitle
\section{Introduction}
There are two statements which may come to your mind when reading the
title of this paper: ``Everybody knows what \lte\ means", and: ``Nobody
has ever given a really satisfactory definition of \lte". I can defend
both of them under suitable conditions; and I shall do so in the following
concentrating mainly on the second one. This article will not give
answers, let alone a precise definition of \lte; my aim is to bring the
related problems into sharp focus.
The generic case of a macroscopic system is one which requires
thermodynamics in its description, but which is not in global equilibrium;
any viscous fluid, any elastic medium with internal friction exhibit
generation and dissipation of heat. That is, the concepts of heat --- and
thus thermodynamics --- enter necessarily; but global thermodynamic
equilibrium can be expected only under special circumstances. There is
another line of arguments based on principal grounds: starting from
Galilei group theoretic considerations in defining the concept of
continuous media one finds that one cannot avoid the notion of internal
energy and thus thermodynamics \cite{RSS}. It is to be stressed that we
are speaking of thermo{\sl dynamics}, not merely of thermostatics.
Equilibrium relations are well established; and there are relations for
non-equilibrium as well, e.\ g., Fick's law, Fourier's law, Onsager's
reciprocity relations. These are successful phenomenological relations
akin to the spirit of linear response theory; but they presuppose tacitly
that the medium under consideration can be considered as a thermodynamic
equilibrium system {\sl locally}, i.\ e., quantities used to describe
systems in thermal equilibrium are assumed to exist at every point, that
is, they are now continuous functions of space and time. This is how one
would define local thermodynamic equilibrium phenomenologically; and this
is not a deep insight, hence my first statement: ``Everybody knows what
\lte\ means."
Of course, not every \mac\ system can be considered as a continuous system
of that sort; there are many cases where a microscopic view is necessary
which takes the particle structure into account. Take as an example the
solar photo- and chromosphere above relative cool solar spots governed by
strong magnetic fields where rather turbulent motions of particles
prevail; there are good arguments that this is not a \lte\ situation
\cite{St}.
The question of systems treated \mic ally brings us to the following
problem: under which circumstances can we extract a continuous media
picture from the particle point of view, i.\ e., under which
cirstumstances can we define, and how can we define, the notion of \lte\
from the viewpoint of Statistical Mechanics.
Equilibrium statistical mechanics is by now a well established theory,
also from a mathematically rigorous point of view. The essential
idealization involved is the passage to the thermodynamic limit, i.\ e.,
to a system of infinitely many particles in an infinite volume but with
finite mean density; thereby one is, firstly, freed from problems of
boundary effects, and, secondly, able to describe phase transitions in
a clear-cut way. It is worthwhile to recall the basic axiom of
equilibrium statistical mechanics: {\sl Global equilibrium states are
defined by the (classical or quantum) KMS condition}. For finite systems
these are nothing but the Gibbs states; for infinite systems the axiom is
validated, firstly, by its success in reproducing thermodynamics (or
better: thermostatics), and, secondly, by the theorems of Haag, Kastler
and Trych-Pohlmeyer \cite{HKT} and of Pusz and Woronowicz \cite{PW}.
Stationarity and dynamical stability are minimal requirements for a state
to be called an equilibrium state; the first mentioned theorem is a
stability theorem stating in essence that these two conditions --- apart
from technical assumptions --- are necessary and sufficient to imply the
KMS condition. The second one is a passivity theorem which can be
rephrased by saying that it is precisely the KMS states which are
distinguished by the second law of thermodynamics in Kelvin's formulation:
there are no cyclic processes converting heat into mechanical work if the
state of the systems obeys the KMS condition.
Physical intuition tells us that in a closed system of very many particles
``most" initial states evolve towards equilibrium under the microdynamical
time evolution. This can hold only if the states are not measured down to
the last atom, that is, if not all (compatible) \mic\ observables are
measured. In order to convert the intuition into a theorem one has to
specify what equilibrium means, which are the relevant observables,
and, moreover, to define the notion of ``most states". This then could
replace the above axiom, albeit subject to another axiom that the
exceptional states are irrelevant; but no such theorem is known to me.
For non-equilibrium systems the situation is far less clear. In order to
arrive at a formulation of similar mathematical rigor we would like to
perform the thermodynamic limit, too; but now it has to be done
``pointwise", i.\ e., every \mac\ point, which is actually a \mac ally
small volume, has to be considered as an infinite system. This involves a
scaling limit. Problems connected to this difficulty have been tackled,
but up to now no rigorous definition of \lte\ has been given; and this is
the justification of my second statement in the first paragraph. I shall
elaborate on that.
The intuitive idea behind the definition of global equilibrium by the
Gibbs states or, in the microcanonical ensemble, via equipartition of
probability among the microstates, is that inevitably there are small
perturbations not taken into account explicitly in the Hamiltonian,
which, in the course of time, lead to a maximally mixed state, the (mean)
energy being the only conserved quantity (apart from the particle density
and external parameters). These perturbations --- ``a grain of dust in the
system" --- act as thermalizing agents. I want to stress that the
definition of equilibrium is not a natural outcome of the dynamical laws
of mechanics or quantum mechanics; actually it is an additional
assumption, a first ``hypothesis on thermalization", which goes beyond the
dynamical laws. My thesis is that we need another thermalizing hypothesis
singling out states exhibiting thermal equilibrium {\sl locally}.
\section{Phenomenological considerations}
Let us start with a definition:\\
{\sl A system is in \lte\ at a given time if there is a well-defined
temperature $T(x)$ at any \mac\ point $x$.}\\
The term ``well-defined" is meant not only mathematically but physically:
the temperature is defined as the parameter governing thermal equilibrium
between two systems, i.\ e., coupling a heat reservoir at temperature
$T(x)$ to the system at point $x$ would not change the state of the
system. As an additional requirement we demand continuity of $T(x)$ in the
interior of the volume under consideration. Furthermore, in order that the
concept is of any use, \lte\ should prevail for at least a finite time
interval.
It is not hard to invent necessary conditions for \lte\ with a \mic\ point
of view in mind. Consider a continuous system which is to be described
hydrothermodynamically. In order to speak of \lte\ we should be able to
exhibit a ``macroscopically small" volume $V$ at the ``macroscopic point
$x$", i.\ e., small compared to length scales $L$ on which typical
macroscopic quantities change markedly. Furthermore, it is certainly
necessary to have ``many" particles in $V$. Finally, the time needed to
reach local equilibrium should be small compared to time scales on which
macroscopic changes occur.
To make this a bit more precise in the case of a fluid consider a small
volume $V$ flowing along with mean velocity $\Er{v}$. Define $l:=V^{1/3}$
and $L:=\KL{\Er{\nabla v}/ \Er{v}}^{-1}$. $L$ denotes the length
scale on which \mac\ velocity changes occur. Our first requirement then
reads $l\ll L$ or, equivalently,
$$l\,\Er{\nabla v}\ll \Er{v}\;.$$
Let $\rho$ be the mass density, $m $ the particle mass, then the second
requirement is
$$l\gg \KL{m\over \rho}^{1/3}\;.$$
A lower bound for the relaxation time is given by the mean time $t_1$
needed for a particle to traverse the volume $V$ which is flowing along
with velocity $\Er{v}$: if $\Delta v$ is the mean velocity fluctuation,
then $t_1= l/\Delta v$. The time scale for macroscopic changes can be
estimated by the time needed for a particle to traverse the length $L$:
$t_2:= L/\Er{v}\approx \Er{\nabla v}^{-1}$, which has to be large compared
to $t_1$, consequently,
$$l\,\Er{\nabla v}\ll \Delta v\;.$$
This is what I have to say about the definition of \lte\ on a
phenomenological level.
\section{The Boltzmann equation} As a step intermediate between the \mac\
description and the full microscopic picture let us consider the example
of a system described by the Boltzmann equation. This is an intermediate
--- mesoscopic --- description because it already involves an
idealization, e.\ g.\ the Boltzmann-Grad limit \cite{Gr} which leads from
the full (classical) dynamical equations containing the exact particle
interaction to an equation for the one-particle distribution function
$f(\vx,\vv,t)$. The Boltzmann-Grad limit can be interpreted as letting the
particle number $N$ go to infinity and simultaneously shrinking the mass
$m$ and diameter $\sigma$ (resp.\ the range of the interaction) of the
particles such that the mass density and the cross section remain finite.
The resulting system is to be considered as an ideal gas with no
interaction left apart from an (idealized) hard core interaction.
Let us note that only in this case of vanishing interaction is the
description by the one-particle distribution function sufficient:
even for thermodynamics, where we do not need the full information
about the microscopic state, we would in general get wrong results when
calculating the density of the potential energy with the help of
$f(\vx_1,\vv_1,t)f(\vx_2,\vv_2,t)$ instead of the correct
two-particle distribution function which need not factorize.
There is a special class of solutions: Hilbert's ``normal" solutions
\cite{Hi} which are defined via their method of construction by a series
expansion. Their first moments give the mass density $\rho$, the mean
velocity $\vec{u}$ and the density of the kinetic energy
$\eps_{\rm kin}$:
\begin{eqnarray*}
\rho(\vec{x},t)&=& \rho_0\int f(\vec{x},\vec{v},t)\,d^3v\;,\\
u_k(\vec{x},t)&=&
\rho_0\int v_k f(\vec{x},\vec{v},t)\,d^3v\,,\;k=1,2,3\;,\\
\eps_{\rm kin}(\vec{x},t) &=&
\rho_0 \int |\vec{v}|^2 f(\vec{x},\vec{v},t)\,d^3v\;,
\end{eqnarray*}
the constant $\rho_0$ is the global mass density. Conversely, any
normal solution is completely characterized by its first moments at a
fixed initial time. That is, the above quantities provide a
complete description of our system, they are determined by their
initial values alone, and are sufficient to recover $f(\vx,\vv,t)$.
This property is not shared by non-normal solutions.
Let us define the internal energy density
$$\eps_{\rm int}(\vx,t):=
\eps_{\rm kin}(\vx,t) - |\vec{u}(\vx,t)|^2/2\rho(\vx,t)\;;$$
it is then in the spirit of our idealization --- the
Boltzmann gas is an ideal gas --- to put $\eps_{\rm int}(\vx,t)$
proportional to the temperature $T$ at the point $\vx$ at time $t$.
We may replace $\eps_{\rm kin}(\vx,t)$ by $T(\vx,t)$ thus arriving at
a hydrothermal description of our system with a well-defined temperature
at any \mac\ point, i.\ e., exhibiting \lte.
Some remarks are in order.\\ 1. The above characterization of normal
solutions as \lte\ states hinges on the proportionality of
$\eps_{\rm int}$ and $T$. But this can be considered a rather arbitrary
definition of $T$ taken over from thermostatics. Remember that the
temperature is originally defined as parameter governing thermal
equilibrium between two systems; there is no such justification like that
here. An ideal gas in global thermal equilibrium exhibits a Maxwellian
velocity distribution; but this is not the case for Hilbert's solutions,
it holds true only in lowest approximation.\\
2. What is the thermalizing hypothesis behind the above definition of
\lte? The Boltzmann equation implies Boltzmann's $H$-Theorem, i.\ e.,
evolution towards global equilibrium for all solutions (and not only for
the normal ones). Thus time reversal invariance is broken. The relevant
assumption for that is hidden in the derivation of the Boltzmann equation:
the assumption of ``molecular chaos", i.\ e., the assumption that the
velocities of colliding particles are statistically independent before the
collision. Of course, this is not the case immediately after the
collision; but it is assumed that statistical independence is restored by
the ``grain of dust" before the next collision occurs. There is a second
part of the thermalizing hypothesis: the assumption that we have to
consider normal solutions, and that $\eps_{\rm int}\propto T$.\\
3. The Boltzmann equation is of course not the most general case of a
classical mesoscopic equation (or set of equations). In general,
one has a hierarchy of equations for the $n$-particle distribution
functions, $n=1,\ldots,\infty$. But in that case, as in the case of
general Boltzmann solutions, the connection to thermohydrodynamics is by
no means obvious. Furthermore, one should start from quantum systems on
the \mic\ level, as is done in a paper of Narnhofer and Sewell
\cite{NS}; cf.\ the comments on that paper in Section \ref{relpap}.
\section{The scaling limit}
Let us now start from a general \mic\ point of view. We have to keep in
mind that defining \lte\ means defining a \mac\ time evolution, too,
starting from the \mic\ interaction. Global thermal equilibrium states are
time invariant ones, and the KMS condition is defined relative to the
\mic\ time evolution. But states exhibiting \lte\ will, in general, not be
stationary on a \mac\ scale although we want them to be locally KMS and
hence time invariant \mic ally.
The following gives an indication of the difficulties we have to face:
defining a \mac\ time evolution may come close to solving the problem of
evolution towards equilibrium of fairly arbitrary initial states. Up to
now only the {\sl return} to equilibrium has been treated satisfactorily,
where the initial states are locally disturbed equilibium states.
Starting from a \mic\ picture requires a twofold idealization:
taking the thermodynamic limit {\sl locally\/} (i.\ e., ``many" particles
in a small volume $dV$), and at the same time shrinking the
(\mac ally) small volume to a point in order to define the local
fields of velocities, pressure, temperature etc. This can actually be
achieved by performing a scaling limit; a possible scheme which, at the
same time, yields a \mac\ time evolution by means of a simultaneaous time
scaling will be described in the following.
Let the \mac\ system be contained in a region $\Gamma\subset\Rl^3$; for
brevity we shall write $x\in \Gamma$ instead of $\vec{x}\in \Gamma$. We
start --- using the language of quantum theory --- with a quasi-local
algebra of observables $\cal A$ generated by a net
$\{{\cal A}(\Lambda)\}_{\Lambda\subset\Rl^3},\; {\cal A}=\overline{\bigcup
{\cal A}(\Lambda)}$, a net $\{H_\Lambda\}$ of corresponding Hamiltonians,
and a net $\{\omega_\Lambda\}$ of states over ${\cal A}(\Lambda)$. We
denote the states over ${\cal A}$ by ${\cal S}({\cal A})$. Our aims are:
1) to ``attach" a copy of ${\cal A}$ to every $x\in \Gamma$
(corresponding to
the local infinite system); 2) to single out suitable nets
$\{\omega_\Lambda\}$ defining ``macroscopic" states, where a \mac\ state
is a set $\{\omega_x\in{\cal S}({\cal A})\}_{x\in \Gamma}$ of states over
${\cal A}$ attached to the points of $\Gamma$, describing the local
properties; 3) to define the dynamics of the macroscopic states based on
the microscopic interaction given by a net of Hamiltonians
$\{H_\Lambda\}$.
The region $\Gamma$ will be considered finite in order to leave open the
possibility of introducing \mac\ boundary conditions.
Let us introduce a scaling parameter $\kappa$ which will tend to infinity,
and scale $\Gamma$ by a factor of $\kappa$ in order to ``gain a
microscopic point of view": $\Lambda_\kappa:=\kappa \Gamma$; we assume
that the origin of the macroscopic coordinate system is in $\Gamma$. The
``microscopic" coordinates will be denoted by $\xi= \kappa x$. Now let us
consider an arbitrary strictly local observable $A\in{\cal
A}_{\Lambda_0}$, $\Lambda_0$ fixed. For sufficiently large $\kappa$, and
$x$ in the interior of $\Gamma$, $\Lambda_0$ as well as the translated
region $\Lambda_0+\kappa x$ are contained in $\Lambda_\kappa$; the set of
space translated observables $\{\sigma_{\kappa x}(A)\in{\cal
A}(\Lambda_\kappa); \kappa\in\Rl^+\}$ is considered as representing the
copy of $A$ attached to $x\in\Gamma$.
We now require that the sequence
$\omega_\kappa\equiv\omega_{\Lambda_\kappa}$ be chosen such that
\begin{equation}\label{e.macstate}
\lim_{\kappa\rightarrow\infty}\omega_\kappa(\sigma_{\kappa x}(A))
=: \omega_x(A)
\end{equation}
exists for all strictly local $A$ and all $x\in \Gamma$. By continuity,
$\omega_x$ can be extended to a state over ${\cal A}$. The set
$\{\omega_x\}_{x\in \Gamma}$ is said to be a macroscopic state; and
$\omega_y$ is the local state corresponding to the \mac\ point
$y\in\Gamma$.
It is not hard to construct fairly arbitrary \mac\ states. The
construction is roughly as follows. Start from a given set $\{\omega_x\}$
of local states depending continuously on $x$. Divide $\Lambda_\kappa$
into disjoint subsets $\Lambda_{\kappa l}$ with centers
$\xi_{\kappa l}=\kappa x_{\kappa l}$ making sure that the $x_{\kappa l}$
form a dense set of $\Gamma$ as $\kappa\to\infty$ while the subsets
$\Lambda_{\kappa l}$ increase to $\Rl^3$. Define $\omega_\kappa$ by gluing
together the states $\omega_{x_{\kappa l}}$ belonging to the centers of
the subsets of $\Lambda_\kappa$. Finally, take the limit
$\kappa\to\infty$.
We omit the subtleties of the construction; but we claim that the
resulting \mac\ state has the set $\{\omega_x\}$ as local states. In
this way we may construct a \mac\ state of continuously varying local
inverse temperature $\beta(x)$ by choosing the states $\omega_x$ as
$\beta(x)$-KMS states.
Of course, in order to do so we need a well-defined \mic\ time evolution.
For that end let us assume that the automorphisms $\alpha_t^\Lambda$ of
${\cal A}_\Lambda$, given by
$$\alpha_t^\Lambda(A)=e^{itH_\Lambda}Ae^{-itH_\Lambda},$$ define a
limiting automorphism $\alpha_t$ of ${\cal A}$ or at least a map
$\alpha_t^*$ of a suitable subset of ${\cal S}({\cal A})$ describing the
time evolution of the microscopic system in the Heisenberg or in the
Schr\"odinger picture. The concept of a KMS state is defined in the
Schr\"odinger picture as well \cite{Ro}, but let us stick to the
Heisenberg picture for simplicity.
The crucial point in this scheme is the definition of the time
evolution of the {\sl macroscopic\/} states. If a disturbance of the
\mic\ system spreads with a finite velocity one cannot expect any
change of $\omega_x$ in time induced by the properties of the system at
neighbouring points unless time is scaled, too. For the sake of
generality let us introduce a time scaling factor
$\gamma=\gamma(\kappa),\;\lim_{\kappa\rightarrow\infty}\gamma(\kappa)
=\infty$ and write $\tau=\gamma t$, where $\tau$ is now the \mic\ time. In
the spirit of equation~(\ref{e.macstate}) we would like to define
$\omega_{t,x}$ by
\begin{equation}\label{e.timescal}
\omega_{t,x}(A) :=\lim_{\kappa\rightarrow\infty}\omega_\kappa\circ
\alpha_{\gamma t}^{\Lambda_\kappa}(\sigma_{\kappa x}(A))\;,
\end{equation}
provided the limit exists. Its existence is to be considered as an
additional requirement on a net $\{\omega_\kappa\}$ defining a macroscopic
state.
There is a case for taking $\gamma(\kappa)=\kappa^2$ (van Hove limit)
\cite{vH,O}; this choice corresponds to the fact that in Brownian motion
the mean distance travelled in time $t$ is proportional to $\sqrt{t}$. On
the other hand, if there is a finite propagation velocity of a \mic\
perturbation (which is the case for lattice systems under fairly general
assumptions \cite[Thm.\ 6.2.11]{BR}) one has to choose
$\gamma(\kappa)\propto\kappa$ in order to arrive at finite \mac\
propagation velocities. This choice can be justified in still another
way:\\ {\sl The requirement of Galilei invariance on the \mac\ as well as
on the \mic\ scale forces $\gamma=\kappa$}.\\
This follows rather easily by consideration of the transformation
properties of \mac\ quantities $f(x,t)$ given by limits of \mic\
quantities which might depend on micro-states $\omega_\kappa$ and one or
more observables $A_\kappa$:
$$f(x,t)=
\lim_\kappa F_\kappa(\omega_\kappa,A_\kappa;\kappa x,\gamma(\kappa)t)$$
(the dependence of $\omega_\kappa$ and $A_\kappa$ on $\kappa$ is assumed
to be only through the size of the micro-system considered, they should be
the ``same" for all $\kappa$, $\{\omega_\kappa\}$ forming a macro-state,
and $A_\kappa\to A$); applying Galilei boosts to the macroscopic variables
$(x,t)$ on the one hand, and to the microscopic variables
$(\xi=\kappa x,\tau=\gamma(\kappa)t)$ on the other hand, one finds that
necessarily $\gamma(\kappa)\propto\kappa$. Of course, one may doubt that
there is a cogent reason for assuming microscopic invariance: all frames
of reference which can be realized in the laboratory are connected by
{\sl macroscopic} Galilei transformations.
The hard task is not to invent a scheme as the one sketched above, but to
prove the existence of the \mac\ time evolution. I know of only two
extreme cases where it has been done: the first one is the trivial case of
a free Fermi gas \cite{FR}, the second one is an inhomogeneous mean field
model \cite{DRW}. But let us assume for a moment that this task has been
performed. Then the scaling limit allows the definition of local thermal
quantities, and of \lte: for the microscopic system attached to a point
$x$ we have the \mic\ time evolution given by $\alpha_t$, and we can ask
whether $\omega_x$ is a $\beta(x)$-KMS state, i.\ e., whether the
temperature at point $x$ is given by $1/k\beta(x)$, try to compute its
entropy density, etc.; any such \mac\ state $\{\omega_x\}$ is then to be
called a \lte\ state.
But this is too na\"{\i}ve a view!
{\bf Thesis:} {\sl Thermodynamics necessitates two ``hypotheses on
thermalization": a first one to describe global equilibrium, i.\ e., to
single out a set of states which are to be interpreted as equilibrium
states; and a second one in order to enforce\/ {\rm local}
equilibrium.}
Note that this goes beyond the \mic\ picture.
To see that another hypothesis is needed for \lte\ one may take a look at
those two models for which the \mac\ time evolution is known to exist.
The mean field model is definitely not trivial, but it suffers from the
same defect as the first one: starting from an initial \mac\ state with
well-defined local temperature $\beta(x)$, which can be constructed, one
ends up --- after arbitrarily short \mac\ times --- with a state which is
no longer locally KMS. This is to be expected in the first model because
there is no interaction; it is to be expected of the second model, too:
here we have to take $\gamma=1$, i.\ e., time is not scaled at all
because ``mean field interaction" implies infinitely long range
interaction, the velocity of spreading of a disturbance is infinite; and
this means an immediate destruction of local equilibrium due to the
effects of different temperatures at different points.
Similar results are to be expected in more realistic models (for
which the above scheme is very hard to implement), hence the requirement
of a ``thermalizing agent", cf.\ the molecular chaos hypothesis in the
case of the Boltzmann equation.
For the general case one might think of introducing an averaging over
\mic\ times together with the limiting procedure; this is suggested by a
paper of Ojima \cite{O} on entropy production. A second possibility would
be to adopt the ideas of information thermodynamics of finite volume
systems \cite{R} to the infinite case. But at the moment these are only
vague ideas which will not be followed here.
\section{Related papers} \label{relpap} There are many papers related to
our topic; but this is not the place to review them all. I want to mention
only a few ones dealing with nonequilibrium thermodynamics, in order to
point out where or how a definition of \lte\ has been circumvented. I
hasten to stress that this is a highly subjective random selection.
The first paper I want to cite is the work of Davies \cite{D} treating a
model for heat conduction. Here heat baths are coupled to every point of
the system, their temperatures are to be adjusted according to the
interaction. But this means introducing \lte\ by hand.
As a representative of papers deriving classical macroscopic, or
mesoscopic, equations --- possibly showing irreversibility --- from
microscopic ones let me cite a paper by Narnhofer and Sewell \cite{NS} on
``Vlasov hydrodynamics of a quantum mechanical model". They derive the
Vlasov hierarchy from the \mic\ quantum dynamics by means of a
``hydrolocal" limit. But in that stage the concept of \lte\ does not
enter. The set of Vlasov equations describe classical correlation
functions on a mesoscopic level comparable to the Boltzmann equation, but
in greater generality. Another effort is needed to derive macroscopic
hydrothermal equations from these correlation functions. One important
result of this paper is a theorem stating that the Vlasov dynamics
respects ``molecular chaos": once the correlation functions factorize as
time $t=0$, they do so at all times. So it seems that this hypothesis is
an important part of a thermalizing hypothesis in this case.
Sewell wrote a series of papers \cite{Se1,Se2,Se3} determining
restrictions imposed on \mac\ phenomenological laws by microphysics. A lot
is achieved by that method; but due to the assumption of the very
existence of the \mac\ laws the question of the definition of \lte\ could
be avoided. The last paper contains a nice model, a heavy quantum particle
coupled to an infinite chain of small particles, which, in an appropriate
limit, yields a classical equation of motion with a frictional force for
the massive particle; i.\ e., the particle tends to the (trivial)
equilibrium state of being at rest. But, of course, {\sl local}
thermodynamic equilibrium is trivial here.
Verbeure and collaborators have succeeded in deriving quantum central
limit
theorems \cite{GVV1,GVV2,GVV3} which can be applied in the study of
fluctuations. This has been done in a paper by Verbeure and Zagrebnov
\cite{VZ}. They showed the existence of a \mac\ limiting algebra of
fluctuations. But this is not a \mac\ dynamics in the sense sketched
above. First of all, it refers to an equilibrium situation, and
furthermore, the resulting \mac\ algebra of fluctuations is dependent on
the state underlying the construction.
\section{Conclusion}
Our conclusion can be kept very short: whatever has been done, a second
thermalizing hypothesis is still missing.
\vspace{7ex}
{\bf Acknowledgement} I would like to thank R. N. Sen for inviting
me to Beer Sheva, and, moreover, for many fruitful discussions. Actually
he deserved to appear as co-author of this contribution.
\newpage
\frenchspacing
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