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%%%%%%%%
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\skew5\widetilde{\buildrel {\scriptscriptstyle\supset} 
\over \iota}\delta o \varsigma$}
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\supset\prime} \over \varepsilon \rho\gamma o 
\nu$}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\def\NN{{\cal N}}\def\EE{{\cal E}}\def\CC{{\cal C}}
\let\a=\alpha\let\b=\beta\let\g=\gamma\let\D=\Delta\let\dpr=\partial
\let\d=\delta\let\t=\tau\let\m=\mu\let\ig=\int\def\ie{{\it i.e.\ }}
\let\s=\sigma\let\z=\zeta\let\L=\Lambda\def\*{{\vskip3mm}}\let\0=\noindent
\def\fra#1#2{{#1\over#2}}\let\io=\infty\def\\{\hfill\break}
\def\st{\scriptstyle}

\centerline{\bf Atomism, Boltzmann and nonequilibrium statistical
mechanics${}^*$}
\* 

\centerline{Giovanni Gallavotti}
\centerline{Mathematics Dept., Rutgers University, U.S.A., and}
\centerline{I.H.E.S, 91440 Bures/sur Yvette, France} \vskip0.5cm

We often try to infer reality of objects, or concepts, by assuming
their existence or consistence and deducing consequences so extreme in
appearance, with respect to the starting assumptions, to imply the
searched reality, once their truth can be be
checked.\footnote{\parindent=0pt}{\0\rm${}^*$% \nota\0Presented at the
Symposium {\it The genesis and development of ethernal thruths}, ETH
(Wissenschaftphilosophie und Wissenschaftforschung departement f\"ur
Humanwissenschaften), Z\"u\-rich, 4 nov 1996.}\parindent=0.5cm

This process has been behind many of the great advances in Science, in
Physics at least. As an example I am thinking of the "aristotelian" view
of motion (as composition of uniform circular revolutions, [Sc]) which
proved to the Ancient Greeks the reality of the quasi periodic planetary
motions.  Or of Saccheri's attempt, [S], to infer the non existence of a
non euclidean geometry by assuming the non validity of the fifth
Euclid's Axiom and proceeding to a breathtaking, almost endless, search
for a contradiction, which he thought to have found in the necessary
validity of some consequences that he regarded as repellent, failing to
see that having found no logical contradictions he had in fact almost
established the existence of non--euclidean geometry.  A fact recognized
later by Gauss, Bolyai, Lobachevsky.

I will concentrate, here, on Boltzmann's investigation and confirmation
of the atomic hypothesis from the analysis of its prime consequence,
statistical mechanics of equilibrium.  One reason this is worth
examining now is its relation to recent developments that, having
been ourselves witnesses to them, can be perhaps better understood via
their connection with similar past examples.

The aspect of Boltmann's work that I will examine is his {\it heat 
theorem}: he worked all his scientific life on this problem and one of 
his last active research works presents a complete solution
(a solution usually attributed to Gibbs, who in fact attributed 
it to Boltzmann, [Gi]), [B84].

The question is the following: if we suppose matter to consist of
many point like {\it atoms} interacting pairwise via forces and moving
according to Newton's second law, {\it can we deduce any macroscopic
consequences that}, through their test, {\it could confirm the assumption?}

Kinetic theory was clearly the key to the answer, as many predecessors
and contemporaries of Boltzmann had seen, [C]. Boltzmann asks the
question: since the quantities that are the object of investigation in
equilibrium thermodynamics can be regarded, in a atomistic view of
matter, as averages of mechanical quantities, is it then possible that
this, {\it by itself}, implies that the relation between mechanical
averages admitting a thermodynamical interpretation is necessarily such
that it implies the second law of thermodynamics? [G1], (A,C), [G3].

To clarify the idea Boltzmann curiously provides us with a rather
strange example, drawn from Helmoltz' discovery of the analogy between
mechanical properties of "monocyclic systems" and classical
thermodynamics, [H]; namely he considers the Moon on its orbit and
defines mechanical quantities that can be given formally the name of
"pressure" $p$, "volume" $V$, "energy" $U$ and "temperature" $T$: the
volume is associated with the orbit semiaxis, the energy is the
mechanical energy of the motion, the temperature is the average
kinetic energy and the pressure is suitably defined.  He then shows,
probably losing most of his readers and thus accounting for the fact
that the paper is still largely unknown (until recently, as I pointed
out repeatedly [G1],[G3], the only motivated reference to this paper
was in Brush's book, [Br1]), that the differential form $(dU+ p dV)/T$
is exact:

$${d U +pd V\over T}=\,{\rm exact}\eqno(1)$$
%

Thus the problem is "given an initial state $x_0$, can we define
mechanical quantities like $U,p,V,T$ associated with averages over the
motion so that their infinitesimal variation, when the initial state
is changed infinitesimally, verify eq.  (1)"? A positive anser to such
statement can be called a "heat theorem".  Its establishment opens the
way to several experimental checks, and eventually it made the
position of the opponents of the atomic hypothesis recognize that
their position was untenable.

In a mechanical system enclosed in a vessel with volume $V$ the
pressure exerted by its $N$ particles on the wall can be defined as
the average momentum transfer to the wall due to the collisions of the
particles with it per unit surface and time: this is in principle a
quantity that can be computed if we know the {\it statistics} of the
motion. The latter is a probability distribution $\m$, on the space of
the configurations $x$ with energy equal to that of the initial datum
$x_0$, with the property that the time average of an observable $F(x)$
is the integral of $F$ with respect to $\m$.  Hence the average
kinetic energy $T$ can also be computed by means of the same
distribution.\footnote{${}^1$}{\nota The existence of such a
distribution is sometimes called the {\it zero-th law of
thermodynamics}, see [UF], [GC2].}

It was quite well established that in kinetic theory (of equilibrium)
the average kinetic energy could be identified as proportional to the
absolute temperature, [C], [M].  Therefore the only missing item needed
to check the heat theorem was a method to evaluate the averages.

Boltzmann's discrete vision of the continuoum in which we (usually) think 
all material objects are immersed was of great help to him in
understanding what should have been the rule to evaluate the mechanical
averages.  

Namely he thought that the microscopic configurations could be
considered as susceptible of assuming several but finitely many ({\it
very many}) forms, [B68], [Ba].  The latter can be identified with small
cells in phase space, \ie as regions defined by constraining the
position coordinates $q_{i\a}$ ($i=1,\ldots, N$, $\a=1,2,3$) and the
corresponding momentum coordinates $p_{i\a}$ of the $i$--th (among $N$)
particle to lie within an interval of size $\d$ or, respectively, $\d'$
around the coordinates $q^0_{i\a}$, respectively $p^0_{i\a}$, of a phase
point $x^0$.  The family of such "centers" form a square lattice in
$R^{6N}$.

The time evolution could also be discretized and thought of as taking 
place at steps of (very small) size $t_0$, comparable to the time scale 
of an atomic collision, see [G1] for an attempt at analyzing this 
viewpoint.

Of course the size of $\d,\d', t_0$ (or of $t_0,h=\d\d'$, since for most
purposes only $t_0$ and the product $h$ of $\d,\d'$ matter) was regarded
by Boltzmann as {\it very} small so that when useful it might be let to
$0$ {\it as a calculational device}: contrary to what seems to have been
hinted, [Ba], one cannot regard (I think) Boltzmann's discrete vision of
the world as related to quantum mechanics, in any way.

Time evolution becomes then a simple permutation of the phase space
cells. If $\D_1,\D_2,\ldots, \D_{\NN}$ is an enumeration of the cells,
then the evolution is a permutation of $\NN$ objects.  It is a
permutation that it is unthinkable to identify, because this should
follow from a solution of the microscopic equations of motion, which is
an impossible task.  However Boltzmann, quite naturally, assumes that
the permutation is a {\it one cicle permutation of the $\NN$ cells} with
energy $U$ (being equal to the energy of the initial datum of interest,
see Fig.1).

\figini{figz1}
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\8<>
\8<0 D         M    5 D mul D       L>
\8<0 D 2 mul   M    5 D mul D 2 mul L>
\8<0 D 3 mul   M    5 D mul D 3 mul L>
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\8<7 D  mul 2 D mul translate Q>
\8<>
\8<3 D mul 0 translate Q>
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\figfin

\*

\eqfig{300pt}{100pt}
{\ins{29pt}{71pt}{$\st\D_1$}
\ins{29pt}{29pt}{$\st\D_2$}
\ins{69pt}{51pt}{$\st\D_3$}
\ins{149pt}{51pt}{$\st\D_1$}
\ins{179pt}{51pt}{$\to$}
\ins{209pt}{51pt}{$\st\D_2$}
\ins{232pt}{51pt}{$\to\cdots\to$}
\ins{289pt}{51pt}{$\st\D_1$}
}{figz1}{Fig.1}
\*\*

In order that the above representation be possible one needs the cells
not to be too deformed by the time evolution. Since they {\it will},
in general, be squeezed in some directions and expanded in others
(keeping a constant volume by the nature of Newton's equations) it is
clear that some errors will be made in deciding (in principle) which
is the cell into which a given cell evolves. 
%*******
The volume preserving property of the hamiltonian evolution, \ie of
the evolution in absence of friction, can be used to show that 
%*******
the errors can be made small by taking the time unit $t_0$ over which
the evolution is observed small. It cannot be taken arbitrarily small,
however (``{\it Zeno's paradox}'': we must fix it so large that
``something happens''). For a discussion of the above consistency
problems in the equilibrium cases see [G1] (C,D).

In this way it is natural to think that the averages can be computed
by using as {\it statistics} (\ie as probability distribution over
phase space) the distribution giving equal probability to each
configuration (\ie to each cell). This has to be considered only as a
suggestion, at this point, because there is a rather important
question left open, and precisely how long will it take for the main
observables to attain their %*******% %
average value within a sensible accuracy; Boltzmann devoted much time
to elucidate this point but here I do not discuss the latter problem
and rather consider the one cycle hypothesis as {\it merely
suggesting} the use of the uniform distribution on the energy surface
as the correct statistics of a randomly chosen initial datum $x_0$.

The one cycle assumption was a very bold one as it allows us to think 
that the system is "monocyclic" in Helmoltz's sense, and to be guided
by his work on monocyclic systems to achieve an extension (surprising 
and far reaching) of the thermodynamic analogies of Helmoltz to a 
"general" system, {\it free} of the integrability hypothesis behind 
Helmoltz' work. 

It is certainly unclear that with this assumption (known as {\it ergodic
hypothesis}) one can achieve any result. But in fact Boltzmann was able
to formulate, and then derive, from it the {\it heat theorem}: given
$V,U$, we now know how to compute $p, T$, at least in principle, and
thus it makes sense to check whether (1) holds.

Boltzmann in fact went much further, [B84], and asked which families
$\EE$ of stationary probability distributions $\m$ on phase space or,
as he called them, which "{\it monodes}" could be defined so that they had
the property that eq. (1) held once $U,V,T,p$ were defined as the
appropriate $\m$--averages (with the kinetic energy always identified
with the absolute temperature) and $\m$ was varied infinitesimally
within $\EE$.  

This is a question that can, in great generality, be asked {\it even in
cases in which the distributions $\m\in \EE$ are not concentrated on a
single trajectory} ("cycle") and therefore the time evolution is not
"ergodic", \ie it is not a one cycle permutation of the phase space
cells with the same energy as the initial datum.

He called such families "{\it orthodic}" and shew that the "{\it
ergode}" (nowadays called the microcanonical ensemble, after Gibbs) and
the "{\it holode}" (now known as the canonical ensemble, after Gibbs)
are among the orthodic ensembles.\footnote{${}^2$}{\nota One can deduce
from this terminology the correct ethymology of the adjective "{\it
ergodic}": it appears from [B84] that ``ergode'' is an abbreviation of
"ergomonode" which means a monode whose elements are probability
distributions over configurations with a given energy.  Thus the
"official" ethymology, [EE], that claims that ergodic arises from
"ergon" and "hodos"
%****
(\ie \ergon\ and \odos, see also [Sw]) has to be replaced with "ergon"
and "eidos" (\ie \ergon\ and
\ifnum\greco=0\eidos\else \begingreek e>~idos\endgreek\fi), because
"monode" arises from "monos" (\ie \monos) and "eidos", see [G1] (A,C),
[G3].}

Summarizing this (part of) Boltzmann's work one can say that he assumed
that an ideally isolated gas (or other matter aggregate), in a perfectly
reflecting walls container, consisted of a collection of elementary
entities, atoms or molecules, whose microscopic configurations could be
regarded as consisting of the collection of all phase space cells of
given energy 
%*****
and
%
permuted by the time evolution to form, at fixed energy, a one cycle
permutation.  From this he was able to infer that the second law of
thermodynamics was a necessary consequence of the assumptions,
ultimately justifying the atomic hypothesis as well as some form of
the ergodic hypothesis, because the exactness of the form in eq. (1)
was considered extremely well established.

It is here important to stress that eq.  (1) can be regarded as a
(surprising) {\it parameterless} prediction of the atomic and ergodic
hypotheses.  The equality
$(\fra\dpr{\dpr U}\fra{p}{T})_V=(\fra\dpr{\dpr V}\fra1T)_U$ in fact
contains {\it no free parameters} and can therefore be experimentally
checked and provides a very stringent test of the validity of the
hypotheses.  \*

I attempt now at a comparison between the above fundamental chain of
ideas with some recent developments that I find similar in spirit, if
somewhat less ambitious and less far reaching in their consequences, so
far.  The preceding discussion illustrated the {\it genesis} of an {\it
ethernal truth}: the following may, perhaps, illustrate the {\it
development} of an {\it ethernal thruth} (but of course it is too early
to judge as we can recognize thruths only when they are fully
established).

The problem of {\it nonequilibrium statistical mechanics} is not really 
touched by the above approach. But it deals with very familiar phenomena 
like stationary electric conduction (in wires or gas tubes), stationary 
heat conduction and other nonequilibrium stationary phenomena for which 
it is clearly desirable and important to build a microscopic theory.

Although it may look surprising the problem of a theory of such
phenomena, even at the level of generality of the second law in
equilibrium, is still a major open problem.

This can be seen from the fact that the volume of a phase space cell not
only is not constant but it even usually contracts because of the
dissipation occurring in the system (which is sometimes modeled by
viscous forces that imply a constant rate decrease of phase space
volume). This makes it inconsistent to think that the statistics of
randomly chosen initial data is a uniform distribution on the surface of
constant energy, as in the equilibrium case.

In '73, in a talk that was presented at ETH in Z\"urich and very likely
even earlier, Ruelle proposed a viewpoint which I liberally reinterpret
in the present day language and which remarkably proposes a solution to
the problem, far more general than that of Boltzmann's quest for the
microscopic source of the macroscopic heat theorem.

Ruelle's proposal is based on an extension, at least such if properly 
looked at, of Boltzmann's ergodic hypothesis. In systems in which 
dissipation occurs one can think that the motion develops asymptotically 
on an {\it attractor} smaller,
%*****
in general,
%
than phase space but that, on it, motion will not sensibly differ from
a motion subject to conservative forces.

This means that the attractor can be divided into cells so that the 
evolution is a one cycle permutation of them and the statistics, \ie the 
probability distribution to use in computing time averages, is 
therefore a uniform distribution on the attractor.

The motion can be still looked upon as a cell permutation. However non
conservation of the cells volume under the time evolution makes it
difficult to represent the motion as a cell permutation, so that it is
necessary to change point of view in search of a representation which
adapts to the nonequilibrium as well as to the equilibrium cases. The
cells will {\it no longer} be regarded as {\it drawn on the full phase
space, but rather on the attractor} on which motion eventually
concentrates (because of the contraction of phase space along some
directions).

The attractor has to be thought of as a surface folding in
phase space and filling it,
%***
 uniformly in the case of no dissipation and nonuniformly in the
dissipative cases: the dimension of the surface will be that of the
expanding directions, which is essentially half the dimension of the
phase space (\ie $\sim3N$ except under extreme forcing).

The cells $\D$ of equal volume of Boltzmann's representation will be
``cut'' by the attractor. The intersection of the attractor with the
cells will consist of many "parallel" surfaces $\b$ (which are the
intersection of a single very large connected surface ``filling'' the
part of phase space covered by the attractor).  The cells which receive
equal weight will be {\it small surface elements} $\g_\b$, lying on the
attractor, which partition each $\b$.  Their number is inversely
proportional to the expansion of the attractor surface under the action
of $\t$ steps, with $\t$ large, evaluated at the considered surface
element: if $x$ is the "center" of the element $\g$ the expansion rate
can be denoted $\L_\t(\g)^{-1}$.

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\8<C T CC { 3 H mul punto} for>
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\*\*

\eqfig{150pt}{40pt}
{\ins{132pt}{25pt}{$\st\g$}
\ins{86pt}{18pt}{$\st\to$}
\ins{185pt}{19pt}{$\st\b$}
\ins{81pt}{36pt}{$\st\b$}
\ins{81pt}{9pt}{$\st\b$}
}{figom1}{Fig.2}

\*\*
See Fig.2 where a cell $\D$ is depicted togheter with the local
parts of the attractor $\b$ that intersect it: the separate pieces of
$\b$ (five in the figure) are in fact connected because they are part of
a large surface folded inside the attractor. Each of the $\b$ layers is
partitioned into small cells $\g$ as illustrated in Fig.2 for one of the
$\b$ layers.

Representing 
%****
motion as a permutation of the new type of cells has
the advantage of not making reference to the conservative or dissipative
nature of the system. In the first case the attractor fills uniformly
$\D$ (as hinted by Fig.2) and the total number of cells falling in $\D$
will be the same for all $\D$'s thus showing the compatibility of the
new viewpoint with the previous one; in the second case the filling of
$\D$ by the attractor layers $\b$ will be unequal allowing for a fractal
structure of the attractor. {\it It is remarkable that with such a
description of motion there is no substantial difference between
conservative and dissipative motions}: one can say that on the attractor
motions are always conservative.

The possibility of such a representation is not at all obvious: one has
to analyze several consistency properties, whose collection leads to
the notion due to Sinai, [Si], of {\it markovian partition} of phase
space.  The advantage is that it {\it does not require} that the cell
$\D$ nor the surface elements $\g$ have the same volume or surface. 
Thus it has the immense advantage that it applies in principle to all
systems, whether in equilibrium or out of equilibrium and in a
stationary state.

Ruelle was suggesting that assuming the possibility of such
representation might have been a reasonable hypothesis, [R1], and
he supported his view by exhibiting one class of examples in which the
consistency properties could be checked to hold rigorously, [R2].  This
was his theory of {\it axiom A}, or {\it strange}, attractors to which
he was able to extend the results of Sinai on Anosov systems, [Si].

Ruelle's original formulation was not explicit enough for practical
consequences to be deduced from it.  Recently his view has been
reformulated so that it could be shown to lead to concrete
consequences, [GC1], [GC2].  Namely it has been formulated as the {\it
chaotic hypothesis} whereby {\it for the purposes of computing
"relevant" macroscopic averages" one can suppose that the system is an
Anosov system}.

Even so formulated the hypothesis is not applicable in an obvious way to
any physical sistuation. But it has been shown to be a reasonable
theoretical explanation of, by now several, experiments on nonlinear
evolution.

Consider systems subject to a reversible dissipation, \ie described by
differential equations

$$\dot x= f(x)\eqno(2)$$
%
where $f(x)$ is such that the system motion is confined in a finite
volume region in phase space (so that a stationary state can be
attained and the equations can be called "thermostatted equations") and
the solution flow $S_t$ to the above equations is "time reversible" ,
\ie such that $i S_t= S_{-t}i$ where $i$ is a map of phase space with
$i^2=$ identity ({\it time reversal}).

The chaotic hypothesis can be shown to lead, in systems in which
reversible dissipation occurs, to a property of the stationary
distribution of the average phase space contraction $p$.  The latter
can be defined in terms of the divergence $\s(x)=\sum_j\dpr_{x_j}
f_j(x)$ of the r.h.s.  of the equations of motion and in terms of its
time average $\s_+=\lim_{T\to\io}\fra1T\int_0^T\s(S_t x)dt$, by

$$p=\fra1{\t \s_+} \int_{-\t/2}^{\t/2} \s(S_tx)dt\eqno(3)$$
%
If one writes the probability, in the stationary state,for finding $p$
between $p$ and $p+dp$ as $e^{-\t\z(p)}dp$, then the function $\z(p)$
verifies, for $\s_+>0$, a {\it fluctuation theorem}:

$$\fra{\z(-p)-\z(p)}{\s_+ p}=1\eqno(4)$$
%
for all $p$ in some range $(-p^*,p^*)$ (with $p^*\ge1$ if $\s_+>0$).  It
is here important to quote {\it Ruelle's H-theorem} that shows that, in
general, $\s_+$ cannot be negative: $\s_+\ge0$ (valid for Axiom A
systems and in fact in considerably greater generality), see [R3].

The eq. (4) is a {\it parameterless} prediction that has to hold in all 
stationary states of a reversibly thermostatted system, {\it if the 
chaotic hypothesis} is assumed, [GC1], [GC2].

Experiments in which the function $\z(p)$ has been measured are
available, [ECM2], [BGG], (and more will be soon available): the first
in fact prompted the research leading to eq. (4), while the second was
an {\it a posteriori} test.  They concern, so far, only numerical
simulations; note however that the prediction being "absolute" one needs
unusually accurate numerical experiments for its testing: which explains
why sometimes it has been necessary to perform the experiments very
accurately to see that the law was satisfied.  One cannot accept any
deviation from it as it should hold in general.

Note that the law applies also (and naturally) to situations in which
one has reasons to believe that the system {\it is not} Anosov (in
Anosov systems the above result would be a {\it mathematical theorem},
[G2], [G1] (D), [Ge], so that there would be nothing to check): 
%%%%****just as 
this ia parallel to what happens when we apply 
%*********
the ergodic hypothesis in equilibrium to systems which we have reasons
to think that {\it do not verify} it, or that we know do not verify it
(like free gases or harmonic chains).

It is not here the appropriate place for a discussion of the
consequences of eq.  (4), see [G5],[G6], and of the chaotic
hypothesis: they may be far reaching and lead to a general theory of
nonequilibrium statistical mechanics, [G4], [G7].  The rather strong
time reversibility assumption needed to derive eq.  (4) may prove, in
the end, to be {\it not} so restrictive, [BG], [G4], [G7].  But
clearly we have to learn how to derive more properties from the
chaotic hypothesis, because so far the only consequences deal with
numerical experiments with few particles, [ECM2], [BGG], or with the
derivation 
%******
of general and "well established" facts like Onsager's reciprocity or
Green--Kubo's formulas, [G4],[GR].

\*
{\bf References.}
\*
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also Ruelle, D.: {\it Chaotic motions and strange attractors}, Lezioni
Lincee, notes by S.  Isola, Accademia Nazionale dei Lincei, Cambridge
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\item{[R2]} Ruelle, D.: 
{\it A measure associated with axiom A attractors}, American
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Ergodic theory of differentiable dynamical systems}, Publications
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equilibrium statistical mechanics}, Journal of Statistical Physics
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\item{[Sc]} Schiaparelli, G.: {\it Le sfere omocentriche di Eudosso, di 
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eliocentrico presso i greci}, Memorie del R. Istituto lombardo di 
scienze e lettere, XVIII, 1898; {\it I precursori di Copernico nell' 
antichit\`a}, Hoepli, 1873, Milano.

\item{[Si]} Sinai, Y.G.: {\it Gibbs measures in ergodic theory}, Russian
Mathematical Surveys, {\bf 27}, 21--69, 1972. Also: {\it Lectures in
ergodic theory}, Lecture notes in Mathematics, Prin\-ce\-ton U. Press,
Princeton, 1977.

\item{[Sw]} Schwartz, J.: {\it The Pernicious Influence of Mathematics on
Science}, in "Discrete thoughts: essays in Mathematics, Science,
and Phylosophy", M. Kac, G. Rota, and J. Schwartz, eds., Birkhauser, Boston,
1986, p. 19--25.

\item{[UF]} Uhlenbeck, G.E., Ford, G.W.: {\it Lectures in Statistical
Mechanics}, American Mathematical society, Providence, R.I.,
pp. 5,16,30, 1963.

\*
\0{\it Internet:
Author's preprints can be also freely downloaded
(latest postscript version with misprints/errors corrections) from:
 
\centerline{\tt http://chimera.roma1.infn.it}
\centerline{\tt http://www.math.rutgers.edu/$\sim$giovanni}
 
\0Mathematical Physics Preprints (mirror) pages.\\
\sl e-mail: giovanni@ipparco.roma1.infn.it
}

\bye

%

