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\*
\0{\it Internet:
Authors' quoted preprints (latest version) freely downloadable at

\centerline{\tt http://chimera.roma1.infn.it}
\centerline{\tt http://www.math.rutgers.edu/~giovanni}

\0Mathematical Physics Preprints (mirror) pages.\\
\*
\sl e-mail: giovanni@ipparco.roma1.infn.it
}}

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\centerline{\titolo Breakdown and regeneration of time reversal
}
\centerline{\titolo symmetry in nonequilibrium Statistical Mechanics}
\*

\centerline{\it Giovanni Gallavotti\footnote{${}^*$}{Permanent address:
Fisica, Universit\`a di Roma, P.le Moro 2, 00185 Roma, Italia. This
paper is dedicated to Professor Raoul Gatto for his 65-th birthday,
retirement age in Switzeland: his support and example during my
scientific formation have been very influential and I have always
nostalgy of those days in Firenze, and regret for not having been really
able to reach anywhere close to imitate his masterful conception of
symmetries and their role.}}

\centerline{\it {\rm IHES},
91440 Bures s/Yvette, France} \*\*

{\it Abstract: A review and non technical exposition of recent studies
on the relevance of time reversal, on its spontaneous breakdowns, and on
its possible resurgence as a weaker symmetry that still anticommutes
with time evolution.}

\* \0{\bf 1. Time reversal and irreversibility.}  \numsec=1\numfor=1
\*

This review illustrates joint work with F. Bonetto ([BG]) and P. Garrido
([BGG]).

Time reversal symmetry is manifest in most fundamental equations. It is
very often associated with the property of motion reversibility and
considered the source of various difficulties into which run attempts at
fundamental explanations of easily observed irreversibility phenomena.

It is important to realize immediately that a microscopically
reversible system can exhibit irreversible behaviour even without
invoking thermodynamic limits or suitable {\it ansatzes}: this is well
established by now, see [EM], [CELS], [G4] for a list of
references. Consider an example (also well known), which is
paradigmatic and not really as special as it might look initially: a
system of $N$ particles (one or more) moving on a closed surface
$\Si$, interacting via conservative pair forces with potential $V$ and
subject to an external {\it field} of potential $E\,\F$ which
``drives'' the system (in the sense that $\F$ is not single valued on
the surface $\Si$, \ie the corresponding force is an ``electromotive
force'') and to the forces that keep the particles on $\Si$ and {\it
also} enforce as a {\it constraint} that the total energy $\EE=K+V$ is
an exact constant of motion ($K=\fra1{2m}\sum_i \V p_i^2$, if $m$ is
the particles mass).

Thus the equations of motion are:

$$m \ddot{{\V q_i}}=-\Dpr_{\V q_i} V(\V q)- E \Dpr_{\V q_i}\F+\V
R_i\Eq(1.1)$$
%
where the {\it constraints reactions} $\V R_i$ are determined by
imposing that the particles stay on the surface and keep constant total
energy $\EE$.  The constraint is anholonomous and it will be supposed
{\it ideal} in the sense that it verifies Gauss' least constraint
principle, see appendices of [G1], [BGM] (for instance).

It realizes a {\it thermostatted model}, in the cases of interest to us,
in which the geometry of $\Si$ is nontrivial and the external field is
locally conservative but not globally conservative. In the following I
present a general analysis of systems subject to a reversible dynamics
of which the above is just an example. I do not discuss under which
circumstances such equations might be a good model to describe some
physical phenomenon: therefore our discussion is {\it universal} for all
models governed by reversible equations and with motions confined to a
finite region in phase space.

The literature on the interest of gaussian models of thermostats (very
special and of a physical relevance which I think is still not
immediately transparent) is vast and equally vast is the literature on
the interest of reversible dynamics to model irreversible behaviour.
The subject is obviously of importance, and its discovery and
investigation (see for instance [HM], [HHP], [EM], [ECM1], [ECM2],
[DPH]) has had {\it at least} the fundamental role of presenting basic
new ideas in the field of nonequilibrium statistical mechanics and in
helping establishing a clear relation with the ideas developed in the
60's and early 70's in connection with dynamical systems and turbulence
(see [Si], [R1]).

The new ideas were consistently and mainly propounded by Hoover (to my
knowledge); they seem to have surfaced independently in various fields and
are, in my opinion, basic and go somewhat beyond what was considered
``established''.

This explains, in part, why they meet opposition (at times aprioristic
or even strong opposition); another reason could be traced to the lack
of links, in the seminal papers and until recently ([ECM1], [ECM2] are
among the first few exceptions), with the previous important theoretical
works on dynamical systems by Anosov, Smale, Sinai, Ruelle. Taking them
into account could not only have helped (probably) a more rapid
diffusion, at least of the substantial and non controversial parts, but
would have certainly attracted the active attention of many more
researchers. In fact, in spite of some recent claims that the subject
was perfectly understood via the first pioneering works, I feel that
there still is a lot of work to do, before a beginning of a theory of non
equilibrium statistical mechanics really emerges.  In the present
review, however, I cannot address such problems, and maybe the times are
not yet ripe for such a discussion which would necessarily point to too
many open questions and difficulties.

%%%%%%%%%%%%%%%%%%% 
%This explains, in part, why they meet opposition (at times aprioristic
%or even strong opposition); another reason is the consequence of the
%systematic neglection, in the seminal papers and until recently
%([ECM1], [ECM2] are among the first few exceptions), of the important
%works, preceding by far the numerical experiments, on dynamical
%systems by Anosov, Smale, Sinai, Ruelle. Taking them into account
%would not only have (probably) saved several computer experiments but
%would also have provided the right formalism for the analysis and
%directed it towards more relevant experiments. They would have also
%helped avoiding errors or misconceptions and misinterpretations that
%cloud a subject which, surprisingly, some still wish to think was
%perfectly understood via the first pioneering works: in fact the last
%three paragraphs have been written to try to appease the historical
%concerns of an angry referee. But such matters have no relevance for
%the present paper: here I do not address such problems {\it at all}.
%%%%%%%%%%%%%%%%%%%%%

Taking a model in the above class, \equ(1.1), just as a concrete
guide, the manifold $\Si$ could be a torus deprived of a few convex
regions, {\it obstacles}, and the picture will be like:

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\didascalia{Fig.1: Circular scatterers of radius $R_1$ and $R_2$ in a
periodic box ``horizontal electric field'' $\V E=E \V j$ along the axis
$\V j$ and $1$ particle.}

\0This can be considered a model for electric conduction in a crystal
with $N$ ($N=1$ in Fig.1) free electrons per cell.  After an
elementary analysis of the
gaussian constraint, if $V$ is (for
simplicity) supposed a hard core potential and
\vfill\eject
\noindent$\V p_i=m \dot{{\V
q_i}}$, the equations \equ(1.1) for the latter simple model become:

$$m \ddot{{\V q_i}}= E \V j-\a \V p_i, \qquad \a=\fra{\V E\cdot\sum_i \V
p_i}{\sum_i \V p_i^2}\Eq(1.2)$$
%
plus elastic collision rules (at particle--particle or
particle--obstacle collisions). The energy $\EE$ equals the kinetic
energy $K$, and if $K$ is written $K= dN\fra12{k_B \th}$ ($d$ being
the space dimension and $k_B$ the Boltzmann's constant) this is a model
for electric conduction at temperature $\th$.  The equation \equ(1.2)
should be compared with the one sometimes used in the simplest
electric conduction models (simplifications of {\it Lorentz' model} of
conductivity) at temperature $\th$:

$$m \ddot{{\V q_i}}= E \V j-\n \V p_i, \qquad \n=const\Eq(1.3)$$
%
where $\n$ is a phenomenological constant adjusted so that the average
kinetic energy is $dN\fra{k_B \th}2$.

The two models \equ(1.2) and \equ(1.3) should be essentially equivalent
(for a discussion see [G2],[G3],[G4]). But it is clear that they are
deeply different in various respects. Namely the first is {\it time
reversal invariant} in the sense that there is an isometry $I$ defined
on phase space, \ie the familiar map $I(\V q,\V p)=(\V q,-\V p)$,
which {\it anticommutes} with the time evolution operator $(\V q,\V
p)\to S_t(\V q,\V p)$:

$$I S_t=S_{-t} I\qquad {\rm for\ all} \ t\Eq(1.4)$$
%
while the second is manifestly not $I$--invariant.

{\it Nevertheless} \equ(1.2) shows an irreversible behavior in the sense
that if $E\ne0$:

$$\fra1T\ig_0^T\a(S_t x) dt \tende{T\to+\io} \media{\a}_+ >0\Eq(1.5)$$
%
for almost all choices of the initial datum $x$ (with respect to the
area distibution) and time reversal symmetry implies ``only'' that the
contraction of phase space in the backward direction is also
$\media{\a}_+>0$.

Since the phase space contraction rate is:

$$\s_E(x)\=(d N -1)\,\a(x)=\fra{dN-1}{\sum_i \V p_i^2}\V E\cdot\sum_i\V
p_i\Eq(1.6)$$
%
we see that $\media{\a}_+>0$ implies singularity of the probability
distribution $\m_+$ giving us the time averages of observables $F$
followed on the trajectory starting at a point $x=(\V q,\V p)$,
randomly chosen with respect to the area on the surface of energy $\EE$:

$$\fra1T\ig_0^T F(S_t x)dt\tende{T\to\io}\ig\m_+(dx) F(x)\Eq(1.7)$$
%

In other words the distribution $\m_+$ which, for the driven system, is
the analogue of the microcanonical distribution {\it cannot} be
expressed in the form (if $x=(\V q,\V p), dx=d\V p d\V q$):

$$\r_E(x)\,\d(K+V-\EE)dx\Eq(1.8)$$
%
for {\it any} $\r_E$, except when $E=0$ (in which case $\r_E$ is just
a constant and $\m_+$ is the microcanocial ensemble).

The property $\media{\a}_+>0$ is of course essential and one has to
make sure that it holds.  Fortunately this is a theorem in simple cases
(\ie $N=1$), see [CELS],[BGM].  It is also in general true that
$\media{\a}_+\ge0$, as proved by {\it Ruelle's H--theorem}, [R2].
Finally the property $\media{\a}_+>0$ can be verified numerically quite
easily in systems with few degrees of freedom, see for instance [BGG].

Calling $\m_-$ the analogue of $\m_+$ for the time averages towards the
past (clearly $\m_-$ is the time reversal image of $\m_+$: $\m_-=I
\m_+$) we can try to visualize (improperly, as it will be seen shortly)
the situation by thinking that there is a set $A_+$ of $0$ area such
that $\m_+(A_+)=1, area(A_+)=0$ and another set $A_-$ (time reversal
image of $A_+$: $A_-=I A_+$) with $\m_-(A_-)=1, area(A_-)=0$ such that
a randomly chosen initial datum $x$ evolves towards the closure of
$A_\pm$ as the time tends to $\pm\io$:

\figini{due}
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\*
\eqfig{100pt}{100pt}{
\ins{100pt}{15pt}{$A_+$}
\ins{100pt}{85pt}{$A_-$}
}{due}{Fig.2}
\*

\0and an initial $x$ randomly chosen (with respect to the uniform area
distribution) will evolve towards $\lis A_+$ as $t\to+\io$ and towards
$\lis A_-$ as $t\to-\io$, if as usual $\lis A$ denotes the closure of
the set $A$.

For very large $t$ the average of the values taken by an observable over
the trajectory of $x$, \ie the average of of $F(S_t x)$, will be the
same as that of $F(S_t x_+)$ where $x_+$ is a (suitably chosen) point on
the attracting set $\lis A_+$, while as $t\to-\io$ the history of $F$
will look the same as that of $F$ itself on the trajectory of a suitable
point $x_-\in \lis A_-$.

For very large positive $t$ the points $S_tx$ and $S_t x_+$ will not be
really different, while for very large negative $t$ the points $S_tx$
and $S_t x_-$ will be essentially the same.

But, {\it no matter how large a prefixed time $t_0$ is taken}, the
$t$-average of he function $F(S_{-t}(S_{t_0}x_+))$ {\it will still be}
$\ig F d\m_+$ (both forward and backward) and that of
$F(S_{t}(S_{t_0}x_-))$ (also both forward and backward){\it will still
be} $\ig F d\m_-$: this is a mathematically clean way of describing the
phenomenon of irreversibility.

We see that time reversal symmetry plays no role except in the fact
that it says that the forward attracting set $\lis A_+$ and the
backward attracting set (\ie the repelling set) $\lis A_-$ are linked by the
symmetry $I\lis A_+= \lis A_-$.

However the picture cannot be so simple: in fact if $E=0$ the two sets
$\lis A_+$ and $\lis A_-$ {\it coincide} if one assumes the ergodic
hypothesis.  Thus we can at most expect that the two sets $A_+$ and
$A_-$ are ``very close'' when $E\ne0$ is small.

In fact for driven systems the role of the ergodic hypothesis can be
taken by the chaotic hypothesis, [GC], which will imply more: namely
that the system is so chaotic that $\lis A_+\=\lis A_-=$ {\it whole
energy surface} not only for $E=0$ but {\it also} for all $E$'s small
enough.

For reference the chaotic hypothesis states:
\*
\0{\it Chaotic hypothesis: A many particle system can be supposed, for
the purposes of computing time averages of macroscopic observables, to
be a mixing Anosov system.}
\*

In spite of its apparent ``uselessness'', only partly due to the fact
that Anosov systems are not really familiar to most of us (se [G4] for
instance), it turns out to be, like the seemingly useless ergodic
hypothesis for equilibrium statistical mechanics, a hypothesis rich of
implications.

Anosov systems, also called {\it hyperbolic}, are systems with highly
unstable motions; {\it the motion in the vicinity of a point $x$ seen
from an observer that moves with $x$ is like the motion near an
hyperbolic fixed point}. So unstable that any other system close enough
to them shares the same property of showing highly unstable motions. In
particular if the system is ergodic and Anosov at $E=0$ it will remain
Anosov at small $E$ and its attracting set will be the whole phase
space.

This shows that the above picture was naive and {\it basically incorrect}
close to equilibrium where one cannot really distinguish $A_+$ from
$A_-$ (in a practical sense) because they are dense into each other
(being $\lis A_+=\lis A_-$)..

Thus reversibility and irreversibility can coexist in a far stronger
sense: namely $\lis A_+=\lis A_-$ coincide with the whole phase space
but $A_+\ne A_-, \m_\pm(A_\pm)$ $=1, \m_{\mp}(A_\pm)=0$.  And close to
equilibrium the reversible system described by \equ(1.2) behaves
irreversibly even though the attracting set is the full phase
space. This also means that the chaotic hypothesis implies a {\it
suitable} interpretation of the four properties stated in [HM]
(see p. 5324) at the end of a remarkable experimental analysis.

In [GC] it was pointed out that such property, rather than hindering the
theoretical interpretation, does lead to {\it observable} consequences
in the form of relations between observable quantities, and such
relations are {\it exact and parameter free}, hence they are general laws.
Except for their importance they can be compared to the relations that,
in equilibrium statistical mechanics, follow from the ergodic
hypothesis and express the exactness of the differential form $(dU+p
dV)/T$ (called the {\it heat theorem} by Boltzmann, [G5]).

In the above example, \equ(1.2), the consequence is a strong restriction
on the distribution of the current fluctuations.  Let $J(x)=\sum_i
\fra1m \V p_i\cdot \V j$ be the total current in the $\V j$ direction
(see Fig.1) and $J_\t(x)$ be its average over a time interval $\t$:
$J_\t(x)=\fra1\t\ig_{-\t/2}^{\t/2} J(S_tx)dt$. We observe this quantity
as a random variable with the distribution that is assigned to it by the
stationary state $\m_+$ and write it as $J_\t(x)= p(x)\, J_+$, where
$J_+$ is the stationary average current ($J_+\=\ig J\,d\m_+$) and $p$ is
a ``dimensionless average current''.

Then the distribution of $p$ will be written as $\p_\t(p)dp=e^{\z(p)\t}dp$,
with $\z(p)$ being an entropy function for the fluctuating variable $p$
and the {\it fluctuation theorem} of [GC] says:

$$\fra{\z(p)-\z(-p)}{\t J_+}=1, \qquad -p^*<p<p^*\Eq(1.9)$$
%
where $p^*\ge1$ is the maximum observable value of the current (note
that by our definitions $\media{p}_+=\ig p \,d\m_+\=1$).
\*

In general reversible systems the role of the current is plaid by the
phase space contraction rate $\s_E(x)$: as noted above the current in
the model \equ(1.2) is proportional to $\s_E(x)$ (because it is,
essentially, $\s_E(x)=\fra{E J(x)}{k_B \th}$, by \equ(1.6),\equ(1.2)).
The quantity $\s_E(x)$, however, can be defined {\it in general} as the
divergence of the equations of motion and in driven systems it is
expected to have a positive average $\s_+$: if $p$ and the average
contraction rate $\s_\t(x)\=\s_+ p$ are defined by
$\s_\t(x)=\t^{-1}\ig_{-\t/2}^{\t/2} \s_E(S_t x)dt$ then its stationary
distribution verifies \equ(1.9) whatever the meaning of the driving
force parameter actually is.  This is important for later developments
even if one is just interested in the system \equ(1.2).  \*
\*

The above relation holds as long as $\lis A_+=\lis A_-$, [GC]. However
it has long been known since the observation of the so called {\it
string phase} in similar systems, [EM], (and it is easy to check in
various cases, see for instance \S6 of [BGG]) that for large $E$ such
property cannot be generally true and $\lis A_+\cap \lis A_-=\emptyset$
so that the attracting set $\lis A_+$ becomes strictly smaller than the
phase space (\ie it has an open complement in it).

Time reversibility of the stationary state is {\it spontaneously
broken} in a quite spectacular way; irreversibility becomes obvious
(as the above described naive analysis based on Fig. 2 now applies);
{\it but the fluctuation theorem seems to be gone}.

Nevertheless there is some experimental evidence, [BGG], that a
relation like \equ(1.9) continues to hold even at large $E$, with
possibly $1$ replaced by another constant (still
parameterless).\footnote{${}^1$}{Time reversal symmetry and density of
attractor and repeller imply equal number of positive and
negative Lyapunov exponents: therefore a very sensitive test of the
density of $A_\pm$ is that the number of positive Lyapunov exponents is
less than that of the negative ones, see [BGG].  The parameter values
where the density of the attractor no longer holds are values
corresponding to which the system shows a ``non trivial'' {\it
vanishing} Lyapunov exponent: thus they are values in correspondence of
which the chaotic hypothesis does not hold.  They are however special
isolated parameter values, see [BGG].}

Therefore a natural idea is that there might still be a kind of time
reversal symmetry for the system, a symmetry in which the phase space is
exactly the attracting set $\lis A_+$: if there was {\it another} map
$I^*$ transforming the invariant set $\lis A_+$ into itself and {\it
anticommuting} with time reversal, then a relation ``like'' the \equ(1.9)
could hold. Indeed the derivatiion of such relation is, for the model
\equ(1.2), essentially based on:
\*

\0(1) the assumption of time reversal symmetry and that the system is a
mixing Anosov flow on the {\it whole }phase space.

\0(2) the proportionality between the phase space contraction $\s_E$ and
the electric current, see \equ(1.6) recalling that $K=\fra1{2m}\sum_i \V
p_i^2$ is constant.  \*

\0and at least (1) would be verified (see below for what concerns (2)).

Thus we look for a {\it local time reversal}, \ie for a map $I^*$ such
that:

$$I^* S_t=S_{-t}I^*,\qquad I^*\lis A_+=\lis A_-\Eq(1.10)$$
%
and the idea is that {\it under very general settings} one can find a third
map $\tilde I$ commuting with $S_t$ and transforming $\lis A_-$ to $\lis A_+$
so that $I^*=\tilde I\cdot I$ anticommutes with $S_t$ and changes $\lis
A_+$ into itself (recall that $I$ exchanges $\lis A_+$ and $\lis A_-$).
\*

The above reminds us (or at least me) very much of the well known fact
that in our Universe time reversal $T$ is not a symmetry but $TCP$ is
such: hence $I$ plays the role of $T$, $\tilde I$ that of $CP$ and $I^*$
that of $TCP$.
\*

In [BG] we discuss a condition (see below) ensuring that every time the
time reversal symmetry $I$ is spontaneously broken because the
attracting set becomes smaller than phase space, and is therefore
mirrored by a repeller set, a new symmetry $I^*=\tilde I\cdot I$
anticommuting with the evolution is spawned.

This does not yet make the \equ(1.9) apply immediately because, since
only one of the two conditions mentioned before \equ(1.10) is fulfilled,
it can be shown to ``only'' give a relation like \equ(1.9) valid for the
fluctuations of the average contraction rate of the area element of
$\lis A_+$ (see also the comments following \equ(1.9)): we saw
that such rate was, {\it in the small $E$ case}, proportional to the
current but there seems to be no \ap reason to be so once one compares
the current with the area contraction on the attracting set $\lis A_+$,
when the latter is smaller than the whole phase space.

{\it However} in [BG] use is made of {\it another} recently discovered
remarkable property of thermostatted hard core systems like \equ(1.2),
the {\it pairing rule} of the Lyapunov exponents, [ECM1],[DM],[WL]
whereby the (non trivial) Lyapunov exponents arranged in decreasing
order of size are paired so that the sum of the exponents
equidistant\footnote{${}^2$}{In systems of particles subject to
gaussian constraints there are $2k$ degrees of freedom and at least
two trivial Lyapunov exponents: the one associated with the flow
direction and the one associated with the constrained variable (the
energy in the case \equ(1.1)).  Therefore, in absence of other trivial
exponents and if the exponents are ordered as $\l_1\ge \l_2\ge\ldots
\l_{k-1}\ge \l_k\ge\ldots\l_{2k-2}$, the pairing rule concerns the
pairs $(\l_{k-1},\l_k) \ldots(\l_1,\l_{2k-2})$.  Under the chaotic
hypothesis in equilibrium the sum of each pair is $0$ and one element
is positive and one negative; in presence of driving force this
remains true for small enough force altough the sum of the pairs is
now negative, but eventually there may appear pairs consisting of two
negative exponents.} from the two central ones is a constant equal to
the {\it average friction} $\media{\a}_+$.  Arguments are given, in [BG],
leading to a proportionality of the total current to the area
contraction rate of $\lis A_+$.  If $\NN$ is the number of pairs of
Lyapunov exponents and $\MM$ is the number of pairs with {\it two}
negative exponents (which never exist, by time reversal symmetry, when
$\lis A_+$ is the whole phase space) then the fluctuation theorem
\equ(1.10) is modified into:

$$\fra{\z(p)-\z(-p)}{\t J_+}=1-\fra\MM\NN, \qquad -p^*<p<p^*\Eq(1.11)$$
%
a surprising result because one might naively expect a r.h.s. {\it greater}
than $1$, se [BGG], [BG].
\*

I conclude by stating in qualitative form the assumptions that insure
that time reversal is {\it an unbreakable symmetry} in the sense that
at every spontaneous breakdown it respawns a new (``smaller'', \ie
defined on an invariant set much smaller than the whole phase space)
symmetry which still anticommutes with time evolution.

The assumptions amount at the requirement that the system should verify
a geometric property called in [BG] {\it axiom C}. To introduce the
property we note that the chaotic hypothesis has to be interpreted, in
the cases in which the attracting set is not the whole phase space, as
saying that the attracting set is a smooth surface and the flow
restricted to it is a Anosov flow. Such attractors are special cases of
the {\it axiom A attractors} (which essentially are more general only
because the attracting set is not supposed to be a smooth surface).

A difficulty with axiom A attractors is that the axiom A property is a
{\it local property} formulated in terms of motions near the
attractor. And this is not satisfactory in systems in which what happens
near an attractor is related to what happens far from it (because, for
instance, there is a time reversal symmetry mapping the attracting set
into a repelling one).

This was probably perceived by Smale, [Sm], when he introduced the notion of
{\it axiom B} systems which basically are systems with attracting sets
verifying axiom A {\it and} with some relation between the stable
manifolds of the attracting sets and the unstable manifolds of the
repeller and of other possibly existing invariant sets.

Smale's axiom B is too weak for our purposes (and it has the
disadvantage of {\it not} being structurally stable, \ie such that a
small modification of the equations of motion does not substantially
change the nature of the system and keeps it still veryfying axiom B).

The axiom C of [BG] is a strengthening of the axiom B property, which is
stable under perturbations. Informally the axiom C property in simple
systems with only one attracting set and one repelling set, different
from each other, is that the stable manifold of the points on $\lis A_+$
reaches the repeller $\lis A_-$ approaching it transversally (of course
without ever reaching it) and viceversa the unstable manifolds of the
repeller reach the attractor transversally, see the schematic picture in
Fig.3 below.

In the figure the first picture illustrates a point $x\in \lis A_+$ and
a local piece of its stable manifold, the second picture a point $x'\in
\lis A_-$ with a local piece of its unstable manifold; the third picture
shows an intersection between the unstable manifold of a point $x\in
\lis A_+$ and the stable manifold of $\tilde I x$, which is a manifold
(one--dimensional in the picture but in general of dimension equal to
the codimension of $\lis A_\pm$): the intersection generates a
correspondence between $\lis A_+$ and $\lis A_-$ which in fact defines
$\tilde I$. The points ``between the two sets'' represent most of the
phase space points and are {\it wandering} (or nonrecurrent) points.

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\*
\eqfig{280pt}{90pt}{
\ins{26pt}{9pt}{$x$}
\ins{106pt}{85pt}{$x'$}
\ins{186pt}{9pt}{$x$}
\ins{183pt}{89pt}{$\tilde I x$}}{tre}{Fig.3}
\*
Let $\d$ be the dimension of the surfaces $\lis A_\pm$ and $u$ and $s$
be the dimensions of the relative (\ie relative to the dynamics
restricted to $\lis A_\pm$) stable and unstable manifolds of the points
in $\lis A_\pm$ so that $u+s=\d$.  In the case of Fig.3 it is $u=s,
\d=2$.

Then the complete (\ie relative to the dynamics in the full phase
space) stable manifolds of the points in $\lis A_+$ have dimension
$s+m$ for some $m>0$ (because the stable manifolds of the points of
$\lis A_+$ ``stick out'' of $\lis A_+$ by the attractivity of $\lis
A_+$; in Fig.3 it is $m=1$).

Time reversal symmetry implies that $u=s$ and that the dimension of the
unstable manifolds of $\lis A_-$ is also $u+m$: hence $u+s+m$ is the
dimension of the phase space; and the stable manifolds of points of
$\lis A_+$ intersect the unstable manifolds of the points in $\lis A_-$
locally on manifolds of dimension $m$.  The latter intersect $\lis A_+$
and $\lis A_-$ in single points and can be regarded as {\it threads}
linking $\lis A_+$ and $\lis A_-$ and establishing the correspondence
$\tilde I$ (which is defined only on $\lis A_\pm$ and {\it not} outside
them, so that $I^*$ also is defined only on $\lis A_\pm$).  This shows
that the Fig.3 represents correctly, if symbolically, the general case.

The axiom C excludes the possibility that the stable manifold out of $x
\in \lis A_+$ in the first figure {\it wraps around} $\lis A_-$ instead
of ``cutting through it'', and also it excludes the corresponding
possibility for the unstable manifold of $x'\in \lis A_-$.

Hence one can view the above axiom as a maximal simplicity
assumption. It is not clear whether the axiom C property is really
relevant for nonequilibrium statistical mechanics since the
spontaneaous breakdown of time reversal symmetry may occurr at
extremely large driving fields: at least this seems to be so for
\equ(1.2) regarded as a conduction model, [BGG], as well as in the
pioneering experiment of [ECM2]), source of all the above theoretical
ideas.

But the above philosophy applies also to the theory of fluid motions,
[G2], [G3], and leads to properties that might be checkable
experimentally (in real or numerical experiments): and in such cases it
is certain that the attractors, being finite dimensional, will be much
smaller than the total phase space (which is infinite dimensional).

Therefore in [BG] we proposed to modify the Anosov property in the
formulation of the chaotic hypothesis into the axiom C property, at
least in the cases of reversible systems.

Finally one may object that {\it in reality} systems are dissipative in
the naive sense of the word, like in the example \equ(1.3), and
irreversibilty (due to some constant friction coefficients) is always
accompanied by an explicit time reversal violation (\ie the equations
of motion are not time reversal invariant).

The latter question is examined, giving among other considerations
arguments for the effective equivalence between \equ(1.2) and
\equ(1.3), in [G2],[G3],[G4].  For irreversible equations that are
phenomenological macroscopic descriptions of evolutions
microscopically based on reversible equations it is argued that the
usual {\it irreversible} macroscopic equations (like the
Navier--Stokes equations) should be equivalent, for practical
purposes, to other {\it time reversible} macroscopic equations:
practical purposes should include taking the average of ``local
observables'' in some limiting situation (like the thermodynamic limit in
the case of particles systems, or the limit of infinite Reynolds number
for fluids). Here local means depending only on the coordinates of
particles that are in a fixed volume (while the container volume tends
to $\io$) in the case of particle systems or depending only on
finitely many Fourier modes in the case of a fluid (while the
Kolmogorov momentum, or equivalently the Reynolds number tend to
$\io$).  And in fact this is used to suggest a natural extension of
the notion of {\it "statistical ensemble"} to nonequilibrium
statistical mechanics.

Basically one can think of the phase space of a system described by
irreversible equations and with a small attractor as being ``half''
the phase space of {\it another} system which is reversible, verifies
axiom C and has {\it the same attractor}. Being reversible it must,
however, have (somewhere in its phase space) also a repeller: hence
the axiom C will guarantee that there is a time reversal symmetry
valid on the attractors of the second system and, therefore,
properties that hold for the second system as a consequence of the
existence of a time reversal symmetry will also hold for the first. In
other words the motion on the attractor of such irreversible systems
will still look like a reversible motion.

It is not clear that this is enough for establishing a fluctuation
theorem for interesting irreversible systems: doubts have been expressed
in [G2],[G3],[G4]. But the arguments against do not seem really
compelling and surprises are possible. Independently on the fluctuation
theorems the assumption that for instance \equ(1.2) and \equ(1.3) are in
some sense equivalent leads, when combined with the chaotic hypothesis,
to experimentally testable predictions. And
the same is true in the case of the Navier--Stokes equations, see [G2],
[G3]. Work on these questions is in slow progress.
%\1

\*
\0{\it Acknowledgement:} This review illustrates joint work with
F. Bonetto ([BG]) and P. Garrido ([BGG]). I am indebted to F. Bonetto
and A. Scotti for many discussions and suggestions.  It is part of the
research program of the European Network on: "Stability and Universality
in Classical Mechanics", \# ERBCHRXCT940460.
\*

\vglue0.5cm

\centerline{\bf References}\*

\0[BG] Bonetto, F., Gallavotti, G.: {\it Reversibility,
coarse graining and the chaoticity principle},  mp$\_$arc@ math. utexas.
edu, \#96--50, to appear in Communications in Mathematical Physics.

\0[BGG] Bonetto, F., Gallavotti, G., Garrido, P.: {\it Chaotic
principle: an experimental test}, mp$\_$arc @math. utexas. edu,
\# 96-154, in print on Physica D.

\0[BGM] F.Bonetto, G.Gentile, V.Mastropietro. ``Electric
fields on a surface of constant negative curvature.'' Preprint, 1996.

\0[CELS] Chernov, N. I., Eyink, G. L., Lebowitz, J.L., Sinai, Y.:
{\it Steady state electric conductivity in the periodic Lorentz gas},
Communications in Mathematical Physics, {\bf 154}, 569--601, 1993.

\0[DPH] Dellago, C., Posch, H., Hoover, W.: {\it Lyapunov instability in
system of hard disks in equilibrium and non-equilibrium steady states},
Physical Review, {\bf 53E}, 1485--1501, 1996.

\0[ECM1] Evans, D.J.,Cohen, E.G.D., Morriss, G.P.: {\it Viscosity of a
simple fluid from its maximal Lyapunov exponents}, Physical Review, {\bf
42A}, 5990--\-5997, 1990.

\0[ECM2] Evans, D.J.,Cohen, E.G.D., Morriss, G.P.: {\it Probability
of second law violations in shearing steady flows}, Phys. Rev.
Letters, {\bf 71}, 2401--2404, 1993.

\0[DM] Dettman, C.P., Morriss, G.P.: {\it Proof of conjugate pairing
for an isokinetic thermostat}, N.S. Wales U., Sydney 2052, preprint,
1995

\0[EM] Evans, D.J., Morriss, G.P.: {\it Statistical Mechanics of
Nonequilibrium fluids}, Academic Press, New York, 1990.

\0[G1] Gallavotti, G.:{\it Chaotic hypothesis: Onsager reciprocity
and fluctuation--dis\-si\-pa\-tion theorem}, J. Statistical Physics,
{\bf 84}, 899--926, 1996.

\0[G2] Gallavotti, G.: {\it Equivalence of dynamical ensembles
and Navier Stokes equations}, Physics Letters A, {\bf A 223}, 91--95,
1996.

\0[G3] Gallavotti, G.: {\it Dynamical ensembles equivalence in
statistical mechanics}, in mp$\_$arc@ math. utexas. edu \#96-182, in
print on Physica D.

\0[G4] Gallavotti, G. {\it New methods in nonequilibrium gases and
fluids}, Maribor meeting lectures, june 1996-- 5 july 1996, {\rm
mp$\_$arc@ math. utexas. edu} \#96-533, in print in Open Systems \&
Information Dynamics. And Gallavotti, G.: {\it Topics in chaotic
dynamics}, Lectures at the Granada school 1994, Ed. P. Garrido,
J. Marro, in Lect. Notes in Phys., Springer Verlag, {\bf 448},
p. 271--311, 1995.

\0[G5] Gallavotti, G.: {\it Ergodicity, ensembles, irreversibility
in Boltzmann and beyond}, Journal of Statistical Physics, {\bf 78},
1571--1589, 1995.

\0[GC] Gallavotti, G., Cohen, E.G.D.: {\it Dynamical ensembles in
nonequilibrium statistical mechanics}, Physical Review Letters, {\bf74},
2694--2697, 1995. And Gallavotti, G., Cohen, E.G.D.: {\it Dynamical
ensembles in stationary states}, Journal of Statistical Physics, {\bf
80}, 931--970, 1995. See also, for a more mathematical analysis:
Gallavotti, G.: {\it Reversible Anosov maps and large deviations},
Mathematical Physics Electronic J., MPEJ, (http:// mpej.unige.ch),
{\bf 1}, 1--12, 1995.

\0[HM] Hoover, W.G., Moran, B.: {\it Phase-space singularities in
atomistic planar diffusive flow}, Physical Review {\bf A40}, 5319--
5326, 1989.

\0[HHP] Holian, B.L., Hoover, W.G., Posch. H.A.: {\it Resolution of
Loschmidt's paradox: the origin of irreversible behavior in reversible
atomistic dynamics}, Physical Review Letters, {\bf 59}, 10--13, 1987.

\0[R1] Ruelle, D.: {\it Chaotic motions and strange attractors},
Lezioni Lincee, notes by S.  Isola, Accademia Nazionale dei Lincei,
Cambridge University Press, 1989; see also: Ruelle, D.: {\it Measures
describing a turbulent flow}, Annals of the New York Academy of
Sciences, {\bf 357}, 1--9, 1980.

\0[R2] Ruelle, D.: {\it Positivity of entropy production in
nonequilibrium statistical mechanics}, J. Statistical Physics,
{\bf 85}, 1--25, 1996.

\0[S] Sinai, Y.G.: {\sl Gibbs measures in ergodic theory},
Russian Math. Surveys, {\bf 27}, 21--69, 1972.  Also: {\it
Introduction to ergodic theory}, Prin\-ce\-ton U.  Press,
1977.

\0[Sm] Smale, S.: {\it Differentiable dynamical systems}, Bullettin of
the American Mathematical Society, {\bf 73 }, 747--818, 1967.

\0[WL] Wojtkowski, M.P., Liverani , C.: {\it Conformally Symplectic
Dynamics and Symmetry of the Lyapunov Spectrum},
mp$\_$arc@math.utexas.edu, \#97-111.

\*
\FINE
\bye

\def\aAA{ Arnold, V., Avez, A.: {\it Ergodic problems of classical
mechanics}, Benjamin, 1966.}
\rif{AA}{}{\aAA}{0}

\def\aBG{ Bonetto, F., Gallavotti, G.: {\it Reversibility,
coarse graining and the chaoticity principle},  mp$\_$arc@ math. utexas.
edu, \#96--50.}
\rif{BG}{}{\aBG}{0}

\def\aBGG{ Bonetto, F., Gallavotti, G., Garrido, P.: {\it Chaotic
principle: an experimental test}, mp$\_$arc @math. utexas. edu,
\# 96-154.}
\rif{BGG}{}{\aBGG}{}

\def\aBJPV{Bohr, T., Jensen M.H., Paladin, G., Vulpiani, A.:
{\it Dynamical systems approach to turbulence}, Cambridge Nonlinear
series, 1996.}
\rif{BJPV}{}{\aBJPV}{0}

\def\aCELS{ Chernov, N. I., Eyink, G. L., Lebowitz, J.L., Sinai, Y.:
{\it Steady state electric conductivity in the periodic Lorentz gas},
Communications in Mathematical Physics, {\bf 154}, 569--601, 1993.}
\rif{CELS}{}{\aCELS}{0}

\def\aDM{ Dettman, C.P., Morriss, G.P.: {\it Proof of conjugate pairing
for an isokinetic thermostat}, N.S. Wales U., Sydney 2052, preprint,
1995.}
\rif{DM}{}{\aDM}{0}

\def\aDGM{ de Groot, S., Mazur, P.: {\it Non equilibrium thermodynamics},
Dover, 1984.}
\rif{DGM}{}{\aDGM}{0}

\def\aDPH{ Dellago, C., Posch, H., Hoover, W.: {\it Lyapunov instability in
system of hard disks in equilibrium and non-equilibrium steady states},
Physical Review, {\bf 53E}, 1485--1501, 1996.}
\rif{DPH}{}{\aDPH}{0}

\def\aEM{Evans, D.J., Morriss, G.P.: {\it Statistical Mechanics of
Nonequilibrium fluids}, Academic Press, New York, 1990.}
\rif{EM}{}{\aEM}{0}

\def\aECM{ Evans, D.J.,Cohen, E.G.D., Morriss, G.P.: {\it Viscosity of a
simple fluid from its maximal Lyapunov exponents}, Physical Review, {\bf
42A}, 5990--\-5997, 1990.}
\rif{ECM}{1}{\aECM}{0}

\def\bECM{ Evans, D.J.,Cohen, E.G.D., Morriss, G.P.: {\it Probability
of second law violations in shearing steady flows}, Physical Review
Letters, {\bf 71}, 2401--2404, 1993.}
\rif{ECM}{2}{\bECM}{0}

\def\aFP{Frisch, U., Parisi, G.: in {\sl Turbulence and predictability},
p. 84--87, ed. M. Ghil, R. Benzi, G.Parisi, Noth Holland, 1984.}
\rif{FP}{}{\aFP}{0}

\def\aGa{ Gallavotti, G.: {\it Ergodicity, ensembles, irreversibility
in Boltzmann and beyond}, Journal of Statistical Physics, {\bf 78},
1571--1589, 1995.}
\rif{Ga}{1}{\aGa}{0}

\def\bGa{ Gallavotti, G.: {\it
Reversible Anosov diffeomorphisms and large deviations.},
Ma\-the\-ma\-ti\-cal Physics Electronic Journal, {\bf 1}, (1), 1995,
(http:// www. ma. utexas. edu/ MPEJ/ mpej.htlm).}
\rif{Ga}{2}{\bGa}{0}

\def\cGa{ Gallavotti, G.: {\it Topics in chaotic dynamics}, Lectures
at the Granada school 1994, Ed. P. Garrido, J. Marro,
in Lecture Notes in Physics, Springer Verlag, {\bf 448}, p. 271--311,
1995.}
\rif{Ga}{3}{\cGa}{0}

\def\dGa{ Gallavotti, G.:{\it Chaotic hypothesis: Onsager reciprocity
and fluctuation--dis\-si\-pa\-tion theorem}, J. Statistical Physics,
{\bf 84}, 899--926, 1996. See also: {\it Chaotic principle: some
applications to developed turbulence}, archived in {\it
mp$\_$arc@math.utexas.edu}, \#95-232, and {\it chao-dyn@ xyz.lanl.gov},
\#9505013, in print in J. of Statistical Physics.}
\rif{Ga}{4}{\dGa}{0}

\def\eGa{ Gallavotti, G.: {\it Extension of Onsager's reciprocity to
large fields and the chao\-tic hypothesis},
mp$\_$ arc 96-109; or chao-dyn 9603003, in print in Physical Review Letters.}
\rif{Ga}{5}{\eGa}{0}

\def\fGa{ Gallavotti, G.: {\it Equivalence of dynamical ensembles
and Navier Stokes equations}, mp$\_$ arc 96-131; or chao-dyn
9604006, in print on Physics Letters A.}
\rif{Ga}{6}{\fGa}{0}

\def\gGa{ Gallavotti, G.: {\it
Dynamical ensembles equivalence in statistical mechanics}, in
mp$\_$arc@ math. utexas. edu \#96-182,
chao-dyn@xyz.lanl.gov \#9605006, in print on Physica D.}
\rif{Ga}{7}{\gGa}{}

\def\aGC{ Gallavotti, G., Cohen, E.G.D.: {\it Dynamical ensembles in
nonequilibrium statistical mechanics}, Physical Review Letters,
{\bf74}, 2694--2697, 1995.}
\rif{GC}{}{\aGC}{}

\def\bGC{ Gallavotti, G., Cohen, E.G.D.: {\it Dynamical ensembles in
stationary states}, Journal of Statistical Physics, {\bf 78}, ,1995.}
\rif{GC}{}{\bGC}{}

\def\aHM{ Hoover, W.G., Moran, B.: {\it Phase-space singularities in
atomistic planar diffusive flow}, Physical Review {\bf A40}, 5319--
5326, 1989.}
\rif{HM}{}{\aHM}{0}

\def\aHHP{ Holian, B.L., Hoover, W.G., Posch. H.A.: {\it Resolution of
Loschmidt's paradox: the origin of irreversible behavior in reversible
atomistic dynamics}, Physical Review Letters, {\bf 59}, 10--13, 1987.}
\rif{HHP}{}{\aHHP}{0}

\def\aLL{ Landau, L., Lifshitz, V.: {\it M\'ecanique des fluides}, MIR,
Moscow, 1966.}
\rif{LL}{}{\aLL}{0}

\def\aN{ Nelson, E.: {\sl Dynamical theories of brownian motion},
Mathematical Notes, Princeton U. Press, 1967.}
\rif{N}{}{\aN}{}

\def\aR{ Ruelle, D.: {\it Chaotic motions and strange attractors},
Lezioni Lincee, notes by S.  Isola, Accademia Nazionale dei Lincei,
Cambridge University Press, 1989; see also: Ruelle, D.: {\it Measures
describing a turbulent flow}, Annals of the New York Academy of
Sciences, {\bf 357}, 1--9, 1980.}
\rif{R}{1}{\aR}{}

\def\bR{ Ruelle, D.: {\it A measure associated with Axiom A
attractors}, American Journal of Mathematics, {\bf98}, 619--654, 1976.}
\rif{R}{2}{\bR}{}

\def\cR{ Ruelle, D.: {\it Positivity of entropy production in the
presence of a random thermostat}, in mp$\_$arc@ math. utexas. edu,
\# 96-167. See also {\it Positivity of entropy production in
nonequilibrium statistical mechanics}, in mp$\_$arc@ math. utexas. edu,
\# 96-166.}
\rif{R}{3}{\cR}{}

\def\dR{ Ruelle, D.: {\sl Elements of differentiable dynamics and
bifurcation theory}, Academic Press, 1989.}
\rif{R}{4}{\dR}{}

\def\aS{ Sinai, Y.G.: {\sl Gibbs measures in ergodic theory},
Russian Mathematical Surveys, {\bf 27}, 21--69, 1972.  Also: {\it
Introduction to ergodic theory}, Prin\-ce\-ton U.  Press, Princeton,
1977.}
\rif{S}{}{\aS}{}

\def\aSm{ Smale, S.: {\it Differentiable dynamical systems}, Bullettin
of the American Mathematical Society, {\bf 73 }, 747--818, 1967.}
\rif{Sm}{}{\aSm}{}

\def\aWL{ Wojtkowski, M.P., Liverani , C.: {\it Conformally Symplectic
Dynamics and Symmetry of the Lyapunov Spectrum},
mP$\_$arc@math.utexas.edu, 97-111.}
\rif{WL}{}{\aWL}{}

\raf{Ga}{1}{\aGa}{1}
\raf{S}{}{\aS}{2}
\raf{Sm}{}{\aSm}{3}
\raf{AA}{}{\aAA}{4}
\raf{R}{4}{\dR}{5}
\raf{BG}{}{\aBG}{6}
\raf{GC}{2}{\bGC}{7}
\raf{Ga}{6}{\fGa}{8}
\raf{N}{}{\aN}{9}
\raf{Ga}{7}{\gGa}{10}
\raf{LL}{}{\aLL}{11}
\raf{R}{2}{\bR}{12}
\raf{GC}{1}{\aGC}{13}
\raf{BGG}{}{\aBGG}{14}
\raf{DGM}{}{\aDGM}{15}
\raf{R}{3}{\cR}{16}
\raf{Ga}{2}{\bGa}{17}
\raf{Ga}{3}{\cGa}{18}
\raf{ECM}{2}{\bECM}{19}
\raf{CELS}{}{\aCELS}{20}
\raf{HHP}{}{\aHHP}{21}
\raf{EM}{}{\aEM}{22}
\raf{ECM}{1}{\aECM}{23}
\raf{DM}{}{\aDM}{24}
\raf{FP}{}{\aFP}{25}
\raf{BJPV}{}{\aBJPV}{26}
\raf{Ga}{4}{\dGa}{27}
\raf{Ga}{5}{\eGa}{28}
%
