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\null\msk
\centerline{\bf References}
\bsk
\input ref.tmp}
%Fine delle def. generali
\def \qed{{\it q.e.d.}}
\def \Pf{{\bf Proof.}}
\def \eps{{\varepsilon}}
\def \Gt{{\Gamma^t}}
\def \Gtn{{\Gamma^t_n}}
\def \Gmt{{\Gamma^{-t}}}
\def \Gmtn{{\Gamma^{-t}_n}}
\def \CX{{C^0(X)}}
\def \CXz{{C^0_0(X)}}
\def \CXs{{C^0 (X)^{*}}}
\def \at{{\alpha^t}}
\def \atm{{\alpha^{-t}}}
\def \as{{\alpha^s}}
\def \asn{{\alpha^s_n}}
\def \atps{{\alpha^{t+s}}}
\def \ats{{\alpha^{t \, {*}} }}
\def \atn{{\alpha^t_n}}
\def \atns{{\alpha^{t \, {*}}_n}}
\def \atnm{{\alpha^{-t}_n}}
\def \atnms{{\alpha^{-t \, {*}}_n}}
\def \atms{{\alpha^{-t \, {*}}}}
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\def \Lu{{L^1}}
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\def \Lf{{L^\infty}}
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\def \Lup{{L^1 \cap L^p}}
\def \Luc{{L^1 \cap C^0}}
\def \o{{\omega}}
\def \F{{\cal F}}
\def \tF{{\tau_{\cal F}}}
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\def \Utn{{U_n^t}}
\def \tuk{{ t_1 \ldots t_k }}
\def \Auk{{ f_1 \ldots f_k }}
\def \Aukt{{ f_1 (x(t_1)) \ldots f_k (x(t_k)) }}
\def \Atnuk{{ \alpha^{t_1}_n f_1 \ldots \alpha^{t_k}_n f_k }}
\def \Atnukk{{ \alpha^{t_1}_{n_1} f_1 \ldots \alpha^{t_k}_{n_k} f_k }}
\def \dro{{d\rho_\o}}
\def \droxt{{d\rho_\o (x(t))}}
\def \dmu{{\; d\mu}}
\def \ntf{{ n \to \infty}}
\def \Supf{{\sup_{f \in \CX \ ||f||_\infty = 1}}}
%def mie
\def \red{{\reali ^2}}
\def \x{{\bf x}}
\def \si{{\sigma}}
\def \al{{\alpha}}
\def \be{{\beta}}
\def \De{{\Delta}}
\def \no{{n^{-1}}}
\def \nt{{n^{-2}}}
\def \ns{{n^2}}
\def \xt{{\x^\prime}}
\def \yt{{y^\prime}}
% def mie
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\def \H{{\cal H}}
\def \lb{\langle}
\def \rb{\rangle}
\def \Us{{U^s}}
\def \Usn{{U_n^s}}
\hfill IFUP--TH 5/98
\bsk\bsk\bsk
\centerline{\titfnt
Irreversible weak limits}
\bsk
\centerline{\titfnt
of classical dynamical systems }
\bsk\bsk\bsk
\centerline{F. Gentili}
\msk
\centerline{\it Dipartimento di Fisica dell'Universit\`a and
INFN, Bologna, Italy}
\bsk
\centerline{G. Morchio}
\msk
\centerline{\it Dipartimento di Fisica dell'Universit\`a and INFN,
Pisa, Italy}
\bsk\bsk\bsk\bsk
\ni {\bf Abstract}.
Limits of classical dynamical systems depending on a parameter are
considered, leading in general to irreversible time evolutions
and stochastic maps.
The appearence of irreversibility is characterized in terms of
convergence properties of the corresponding automorphisms of
the observable algebra.
The resulting maps are shown to be
automatically \lq\lq bistochastic\rq\rq .
Necessary and sufficient conditions are given for the limits to
define a time independent Markov process.
A model is studied for a particle interacting with fixed
configurations of external obstacles.
\bsk\ni
KEY WORDS: irreversibility, weak limits, classical dynamical
systems.
\vfill\eject
\ni
{\bf 1. Introduction}
\bsk
Irreversible time evolution plays a major and generic r\^ole in
the phenomenological description of macroscopic systems.
Its derivation from underlying reversible dynamical systems is
however under control only in special cases
\citaref{P.A.Martin, {\it Mod\`eles en M\'ecanique Statistique des
Processus Irr\'eversibles}, Lectures Notes in Physics {\bf 103},
Springer 1979}
\citaref{H.Spohn, {\it Large Scale Dynamics of Interacting
Particles}, Springer, Berlin 1991},
and it is not clear whether one may characterize a unique set of
conditions leading to irreversible behaviour,
or one should rather think in terms of different
schemes and limit procedures, applying to different
physical situations
\citaref{J.L.Lebowitz, Supplement of the Progress in Theoretical
Physics {\bf 64}, 35 (1978)}
\citaref{D.Ruelle, {\it Hasard et chaos}, Jacob, Paris 1991}
\citaref{J.L.Lebowitz, {\it Microscopic Reversibility
and Macroscopic Behaviour: Physical explanation and
Mathematical Derivations},
in {\it 25 years of Non--Equilibrium
Statistical Mechanics}, Lectures Notes in Physics {\bf 445},
Springer 1995, and references therein;
Physica {\bf A 194}, 1 (1993)}
\citaref{J.R.Dorfman,
{\it From molecular chaos to dynamical chaos},
Institute for Physical Science and Technology,
University of Maryland}.
\ssk
Since the classical work of Boltzmann, it is in fact well known
that the presence of a large (infinite)
number of degrees of freedom plays a crucial r\^ole
for the appearence of irreversible behaviour, and that other
conditions are necessary, involving at least dynamical instabilities
which grow suitably with the number of degrees of freedom, toghether
with restrictions on the states appearing \lq\lq at time zero\rq\rq\
\citaref{L.Boltzmann, Wien. Ber. {\bf 66}, 275 (1872), in
{\it Wissenschaftliche Abhandlungen}, Chelsea 1968, I Band, p.316}
\citaref{S.G.Brush, {\it The kind of motion we call heat}, vol.2,
North Holland 1976}
\citaref{H.Grad, in {\it Handbuch der Physik} {\bf 12},
Springer 1964}
\citaref{O.E.Lanford, {\it Time Evolution of Large Classical
Systems}, Lectures Notes in Physics {\bf 38},
Springer 1975}
\citaref{C.Cercignani, R.Illner,
M.Pulvirenti, {\it The mathematical theory of diluite
gases}, Springer 1994}.
Moreover, a distinction between \lq\lq macroscopic\rq\rq\
and \lq\lq microscopic\rq\rq\ observables plays an important
r\^ole in the general discussion, and statistical arguments enter
in an essential way, either through a classification of the
states in terms of macroscopic and microscopic variables
\citaref{L.Boltzmann, Wied. Ann. {\bf 57}, 773 (1896), in
{\it Wiss. Abhandl.}, III Band, p.567}
[5], or in their very description [1][2][10][11].
\ssk
The restriction to a suitable subset of observables is also
at the basis of the discussion of irreversible behaviour
for a fixed subsystem of an infinite system, and
an irreversible time evolution has in fact been derived in
general by Davies in the \lq\lq weak coupling\rq\rq,
or \lq\lq diffusive\rq\rq\ limit,
for finite subsystems of infinite quantum systems
in pure states
\citaref{E.B.Davies, Comm. Math. Phys. {\bf 33}, 171 (1973);
{\bf 39}, 31 (1974)}.
The resulting scheme is, however, only indirectly related to the one
advocated in the Boltzmann--Grad limit, and, at least, the problem
arises of identifying common structures behind different
constructions and limit procedures.
\msk
Since in all cases irreversible behaviour is obtained as a limit
description in the variation of some parameter, it is worthwhile to
investigate whether this ingredient may also be sufficient, i.e. whether
{\it irreversible time evolution may be the generic result of taking
such limits}. The r\^ole of the other conditions would then be
interpreted as related to the description of different classes of
physical situations, in which different parameters and limits
become relevant, in relation with specific structures.
In models, the above mentioned parameters appear in the description of
the structure of the system (the number of particles in the
Boltzmann--Grad case), of the states (the correlation functions at time
zero) and of the dynamics (the rescaling of the particle radius);
however, since dynamical instabilities seem to be an essential
ingredient if irreversibility, it is reasonable to concentrate the
attention on parameters which enter in the definition of the time
evolution, leaving for the moment the system and its states invariant.
Such a possibility is not far from the situation in the
Boltzmann--Grad limit since, in the spatially homogeneous case,
that limit can be formulated in terms of an infinite system
of hard spheres, in a fixed time zero configuration,
in terms of a suitable rescaling of the radius ($ r \to \lambda r$)
and of the time scale ($t \to \lambda^{-2} t$
in three space dimensions).
\ssk
We are therefore led to investigate limits of sequences of
reversible dynamical systems.
More precisely, on the basis of the general interpretation of dynamical
systems in terms of observables and states, the most general relevant
limit is the limit for the result of measurements of fixed observables,
given any state in some relevant class.
Mathematically, in the Hamiltonian case, observables may be identified
with regular functions on the phase space, states with measures on
the phase space,
and the problem is to take limits of expectation values
of observables over states evolved in time under groups
of transformations defined by Hamiltonians depending on a parameter.
More generally, i.e. for quantum systems and infinite systems,
observables may be identified (see e.g. \citaref{O.Bratteli,
D.W.Robinson, {\it Operator Algebras and Quantum Statistical
Mechanics}, Springer, Berlin, 1979})
with the hermitean elements of a C* algebra, states with positive
linear functionals, time evolution with a group
of automorphisms of the observable algebra, and the relevant limit
is the weak limit of the time evolution automorphisms,
taken with respect to a subspace of the state space, interpreted
as the set of states of interest.
\ssk
In this form, the problem is similar to the one that appears
in the framework of the algebraic
formulation of Quantum Field Theory and Statistical Mechanics
\citaref{R.Haag, {\it Local Quantum Physics}, Springer 1993},
in relation with the construction of the time evolution
of infinite quantum systems in presence of
long range interactions. A general analysis of limits of
automorphisms arising in that context
has been developed has in fact been developed in refs.
\citaref {D.W.Robinson, Commun. Math. Phys. {\bf 7}, 337 (1968)}
\citaref{D.A.Dubin, G.L.Sewell, J. Math. Phys. {\bf 11}, 2990 (1970)}
\citaref{G.L.Sewell, Commun. Math. Phys. {\bf 33}, 43 (1973) }
\citaref{G.Morchio, F.Strocchi,
Comm. Math. Phys. {\bf 99}, 153 (1985);
J. Math. Phys. {\bf 28}, 622 (1987)}.
The relatively strong assumptions which had to be required in that
analysis in order that the limit still give rise to a group of
automorphisms, i.e. to a reversible dynamical system, leave open the
possibility that the result of such limits may be interpreted, in some
generality, in terms of irreversible time evolution.
\ssk
The purpose of the present paper is to show that this is the case,
i.e. that maps having all the properties usually assumed to characterize
irreversible time evolution appear quite in general as weak limits of
reversible dynamical systems, on the basis of only two ingredients:
\ni
i) existence of a limit in a parameter, which describes
{\it growing dynamical instability}
\ni
ii) statistical {\it regularity} of the states involved in the
limit procedure.
\ssk\ni
The first requirement seems to be essential in order to obtain
the loss of information which is implicit in irreversible evolution.
The second one has a clear mathematical r\^ole in defining the kind of
limit which is taken, and can be physically interpreted (see below)
in terms of the
statistical description of the states of finite subsystems arising,
through a space ergodic mean, from a pure state of an infinite system.
It must be stressed that the present scheme for the origin of
irreversibility is {\it not} incompatible with the usually advocated
ones, and that our purpose is rather to focus the attention on an
abstract mechanism which {\it by itself} gives rise to irreversibility
{\it in a generic way}, and may therefore be a good candidate to explain
the generality in which irreversible behaviour appears in the
description of nature.
In particular, the fact that irreversible behaviour may appear
also in finite systems does not contradict the idea that
irreversibility comes from the loss of information \lq\lq
at infinity\rq\rq\ in infinite systems.
First, it is clear that finite systems
{\it with a continuous phase space} have room enough, in the
small, in order to let information become \lq\lq inaccessible on
a given scale\rq\rq , and this can be interpreted as a generalization
of the Boltzmann mechanism.
More important, the application of our approach
to relevant infinite systems should yield irreversible behaviour
exactly when the range of the interactions (or the change of the time
scale) is made to grow fast enough in order to give rise to
\lq\lq loss of information at infinity\rq\rq ;
this follows from the fact that the condition that
the limit still give an automorphism of a (\lq\lq quasi--local\rq\rq)
observable algebra has in general the interpretation of allowing
for a sufficiently local description of the dynamics, i.e. of
excluding loss of information at infinity
(see Sect. 2 for the relation between automorphism property and
reversible interpretation); the r\^ole
of such condition is particularly clear in quantum spin systems,
where the phase space can be thought as locally discrete, and
time evolution remains in fact reversible if the range of the
interaction remains suitably bounded [16].
\msk
>From a general mathematical point of view, the essential point
of our results is that all the features that seem to characterize the
description of irreversible time evolution
(see Sect.2), and are sometimes even {\it assumed as an
independent source of information} on the behaviour of physical systems
\citaref{R.F.Streater, {\it Statistical dynamics},
Imperial College Press 1995}, arise
{\it generically}, from reversible dynamical systems,
as a consequence of taking limits in some parameter which enters
in the dynamics.
\msk
We restrict our attention in this paper to classical dynamical systems,
under assumptions which are typically satisfied in Hamiltonian models
with a finite number of degrees of freedom. In this sense, our results
do not directly apply to infinite systems in general.
However, if a finite system is interpreted as a (non-interacting)
subsystem of an infinite system, the latter being obtained, e.g.,
by taking copies of the
system, indexed by an integer, then any statistical state, i.e. any
probability measure $\mu$ on its state space,
defines a product measure $\rho$ on the
state space of the infinite system, which is ergodic under translations
of the parameter (see e.g. \citaref{V.I.Arnold, A.Avez, {\it Problemes
ergodiques de la mecanique classique}, Gauthier--Villars 1967}).
The original state is therefore automatically interpreted in terms of the
statistics defined by the ergodic mean with respect to the
integer parameter indexing the subsystems,
for $\rho$--almost all fixed configuration of the infinite system
(see \citaref{D.Ruelle, {\it Statistical mechanics: rigorous results},
Benjamin, 1969, cap.6}, [5]).
Our results can therefore be interpreted as holding in a class of
infinite systems, for which
\ssk\ni
i) the presence of an infinite number of subsystems is only
used as the basis for a statistical description of the
states of each of them;
\ssk\ni
ii) the origin of growing dynamical instability is separated
from the increasing of the number of components, and is assumed to
be obtained from the variation of a parameter in a finite dynamical
system.
\msk
No restriction on the observables of the system is
involved in our analysis, even if a relation with the notion of
\lq\lq macroscopic\rq\rq\ observables may be regarded as implicit in
the interpretation of the statistical description in terms
of the subalgebra of observables generated by the ergodic
means over a parameter indexing the subsystem of an extended system.
\ssk\goodbreak
In Sect.2 we make our framework more precise and give the
main results; Sect.3 contains the proofs; in Sect.4 a model is
discussed, bearing some similarity to the situation arising in the
Boltzmann--Grad limit, giving rise to an irreversible time
evolution, and reproducing the general structures
characterized in Sect.2.
\bsk\bsk\goodbreak\ni
{\bf 2. Assumptions and results}
\bsk
We consider classical dynamical systems, defined as in
\citaref{R.V.Kadison, Topology, {\bf 3}, 177 (1965)} \ by groups of
continuous transformations $ \Gamma_t$, $t \in \reali$ of a locally
compact Hausdorff topological space $X$. We will discuss continuous
transformations $\Gtn$ depending on a parameter $n \in \naturali$, and
their limits as $n \to \infty$.
The observables of the system are identified with the $C^{*}$ algebra of
the continuous functions on $X$ vanishing at infinity, $C^0(X)$, i.e.,
the completion, in the sup norm, of $\CXz$, the space of continuous
functions of compact support.
The transformations $\Gtn$ define as usual groups of
automorphism $\atn$
of $\CX$, $\atn f (x) = f(\Gmtn (x))$.
Conversely, by Gelfand's isomorphism \citaref{M.Takesaki {\it Theory
of operator algebras I}, Springer 1979}, any group of
automorphisms of a commutative \Ca \ defines a group of continuous
transformation of its spectrum, a locally compact Hausdorff space (the
set of multiplicative linear functionals being clearly left invariant by
the transpose of an automorphism).
The states of the system are the positive normalized linear functional
$\omega$ on $ \CX $, i.e., by the Riesz--Markov theorem, the
regular Borel probability measures on $X$.
\msk
The only restrictions on $\atn$ that we will assume are the existence of
a common invariant measure, and a condition which keeps the time
evolution \lq\lq far from infinity\rq\rq\ uniformly in $n$;
more specifically, we will assume
\msk\ni
$M)$ a $\sigma$ finite regular Borel measure $\mu$ is
defined on $X$, and left invariant by all $\Gtn$, $n \in \naturali$.
\msk\ni
$F)$ for all real $t$ and compact set $K \subset X$
there is a compact set $M (K,t) \subset X$ such that
$$ \Gtn (K) \subset M \ \
\ \ \ \ \ \ \ \forall n \in \naturali $$
\msk
Given the invariant measure $\mu$, a distinguished class of states
is given by the set of probability measures $\nu$ which are continuous
with respect to $\mu$: $\nu (x) = h(x) \mu (x)$, $h(x) \in L^1 (X,
d\mu)$. The set $\F$ of such states, which will be called \lq\lq
regular\rq\rq, is the positive part of a closed subspace of the dual of
$\CX$ [19], and defines a \lq\lq full folium\rq\rq\
of states in the sense of \citaref{R.Haag, N.M.Hugenholz, M.Winnink,
Commun. Math. Phys. {\bf 5}, 215 (1967)}.
Regular states will be identified in the following
with functions in $\Lum$.
The invariance of $d\mu$ implies that $\F$ is left invariant by the
transpose $\atns$ of $\atn$, defined by $\atns \o (f) = \o ( \atn f )$.
It also implies that time evolution can be formulated
\lq\lq a la Koopman\rq\rq\
\citaref{M.Reed, B.Simon, {\it Methods of modern mathematical
physics}, Academic Press 1975} \ in $\Ldm$:
$\forall g \in L^1 \cap L^2$, $\Utn g$ is defined by
$\atns g$, and extends by continuity to a unitary group in $\Ldm$.
\msk
The above general structure covers in particular the case of
Hamiltonian systems, for which $X$ is the phase space, $\Gt$ the
Hamiltonian time evolution, $d\mu$ the (infinite) Liouville measure and
the regular states are the probability measures on the phase space with
well defined ($L^1$) density.
We do not assume that $d\mu$ is a finite measure; in fact,
even in the Hamiltonian case, a sequence of groups of
transformations does not in general leave invariant
a common compact subspace, and there is no common
invariant finite measure. For the same reason, $X$ is assumed to be
only locally compact.
\msk
The advantage of the above formulation is to make explicit the r\^ole of
the observables and of the states of the system, so that the
discussion of the limit $n \to \infty$ can be done with a clear physical
interpretation, once the preceding remarks on the origin of the
statistical description of the states of a finite system are taken into
account.
The existence of a {\it limit description} of the dynamical
systems $(X,\Gtn)$ as $n \to
\infty$, given a time zero state $\o$, amounts in fact to
convergence of all the mean values $\o (\atn f)$, $f \in \CX$. As we
will see (Proposition 1), this is also equivalent to convergence of
$\o (\atn \chi)$, for all characteristic functions $\chi$ of $\mu$
measurable subsets of $X$, i.e. convergence of the frequence of
answers for all yes--no experiments based on $\mu$ measurable
subsets of $X$, given the state $\o$.
Since we will admit as time zero states all regular states, we will
investigate the limit of $\atn$ in the weak topology $\tF$ defined by
$\F$ on $\CX$. $\tF$ can also be described as the ultraweak topology
\citaref{J.Dixmier, {\it Les algebres d'operateurs dans l'espace
hilbertien}, Gauthier Villars 1969 }
\ associated to the representation of $\CX$
as multiplication operators in $\Ldm$, and the $\tF$ closure of $\CX$
can be identified with $\Lfm$.
\msk
In the following, we will also use the strong and ultrastrong topologies
defined on $\CX$ and $\Lfm$ by the same representation; for bounded
sequences $f_n$ the strong and ultrastrong topology coincide, and
$f_n $ converge to $f $ iff
$$ \int |f_n (x) - f(x)|^2 \;
h(x) d\mu (x) \to 0 \ \ \ \ \ \forall h(x) \in \Lum $$
We will assume that the $\tF$ weak limit of $\atn$ exists for all $t$,
and investigate under which condition it can be interpreted as an
irreversible time evolution. More detailed results,
which do not assume convergence for all times,
are given in
\citaref{F.Gentili, Tesi, Dipartimento di Fisica, Pisa 1996}.
The following result holds for any $\tF$ convergent sequence
of automorphisms of $\CX$:
\bsk\ni
{\bf Proposition 1}
\ssk\ni
{\it If a sequence of automorphisms $\atn$ of $\CX$, satisfying
M, converges in the weak topology $\tF$, for all
$t \in \reali$, then
\ssk\ni
i) the limit defines
$\tF$ continuous linear positive maps $\at$ of $\CX$ into $\Lfm$,
with norm at most one.
\ssk\ni
ii) the transpose $\ats$ of $\at$ leaves $\F$ invariant;
\ssk\ni
iii) $\at$ extends by continuity to a linear positive mapping
of all $\Lfm$into itself.
\ssk\ni
iv) $\ats = \atm$ on $\Luf$;
\ssk\ni
v) $\forall p \in [1, \infty) \; \forall h \in L^p \cap L^1$
$ || \ats h ||_p \leq || h ||_p $;
\ssk\ni
If the sequence $\atn$ also satisfies property F, then
\ssk\ni
vi) $\ats$ preserve the $L^1$ norm on $(\Luf)^{+}$, and
$\at$ leaves the identity invariant;
\ssk\ni
vii) the $\tF$ continuous extension of $\atn$ to $\Lfm$ also converges
$\tF$ for $n \to \infty$ to $\at$ on $L^\infty$}.
\bsk
Positive maps are well known candidates for the description
of irreversible evolution. Moreover, property $vi)$
characterizes {\it bistochastic maps}, and it is
generally assumed as fundamental
\citaref{P.M.Alberti, A.Uhlmann,
{\it Stochasticity and partial order}, Reidel 1982} \ [20]
for an axiomatic description of irreversible processes.
In the present framework, $vi)$ {\it automatically holds}
for all maps which
arise from weak limits of classical dynamical systems.
All the results of Proposition 1 hold, however, even for limits
which still describe groups of automorphisms of $\Lfm$.
In order to characterize limits describing irreversible
evolution, one must ask whether the mapping $\at$ are
invertible, and whether the group law $\at \; \as = \atps$ is satisfied.
A priori, as candidates for the description of irreversible time
evolution arising in the above limit we have therefore:
\ssk\ni
i) Failure of the group law for $\at$
\ssk\ni
ii) failure of invertibility of $\at$.
\ssk\ni
It is also clear that
\ni
iii) failure of the morphism property of $\at$
\ssk\ni
gives a good notion of irreversible behaviour.
In fact, the transpose $\ats$ of a linear positive map $\at$ of any
abelian \Ca, with invariant identity, sends pure states
into pure states if and only if
$\at$ is a morphism, as it is easily seen by
using the Gelfand construction. Property iii) has therefore in
general the interpretation of a loss of information on the system,
corresponding to a transformation of pure into mixed states.
More concretely, a transformation of $\Lfm$ is a morphism iff
it maps characteristic functions of measurable sets into
themselves, i.e. iff all yes--no questions are transformed
into yes--no questions, equivalently, no information is lost in the
time evolution.
Moreover, in the same generality, both $ii)$ and $iii)$ imply $i)$.
Only $iii) \Rightarrow i)$ is in fact non--trivial, and follows from the
fact that if the $\at$ satisfies the group law, then
$\atm$ inverts $\at$ and is a positive map, so that $\atms$
sends mixed states into mixed states, and therefore invertibility
implies that $\ats$ sends pure states into pure states.
\msk
The following Proposition shows that in the present framework the
notions i)--iii) coincide, so that {\it there is a unique notion of
irreversibility for weak limits of classical dynamical systems};
it also provides necessary and sufficient conditions for the
appearence of irreversibility.
\bsk\ni
{ \bf Proposition 2}
\ssk\ni
{\it The following properties are equivalent for the weak limit $\at$
of a sequence of automorphisms $\atn$ of $\CX$, satisfying M:
\ssk\ni
i) $\forall t \in \reali$, $\at$ is an automorphism of $\Lfm$;
\ssk\ni
ii) $ \at $, $t \in \reali$, is a one parameter group of automorphisms
of $\Lfm$;
\ssk\ni
iii) $\forall t \in \reali $, $\atn$ converges in the ultrastrong
topology associated to the representation of $\CX$ as multiplication
operators in $\Ldm$, and $\at(1) = 1$;
\ssk\ni
iv) For some $ p \in (1, \infty)$ $\at : \CXz \mapsto \Lfm $
preserves the $L^p$ norm, for all $t$,
$$ || \at f ||_p = || f ||_p \ \ \ \ \forall f \in \CXz \ \ ,
\ \ \ t \in \reali \ \ \ ; $$
\ssk\ni
v) For some $ p \in (1, \infty)$ $\ats$ preserves the $L^p$ norm,
for all $t$:
$$ || \ats h ||_p = || h ||_p \ \ \ \ \forall h \in \Lup \ , \ \ \
t \in \reali \ \ \ ; $$
\ssk\ni
vi) For all $ p \in [1, \infty]$, for all $t$,
$\ats$ preserves the $L^p$ norm. }
\bsk
Similar results hold for the convergence of the unitary group
$\Utn$ which implement the time evolution in $\Ldm$:
\bsk\goodbreak\ni
{\bf Proposition 3}
\ssk\ni
{\it Assuming M, weak $\tF$ convergence of $\atn$ is equivalent to
convergence of $\Utn$ in the weak operator topology in $\Ldm$.
Assuming convergence and denoting
by $\Ut$ the weak limit of $\Utn$, conditions i) - vi) of Proposition 2
are equivalent to each of the following:
\ssk\ni
i) $\forall t \in \reali$ the operators $\Utn$ converge strongly
in $\Ldm$;
\ssk\ni
ii) $\forall t \in \reali$ the operators $\Ut$ are unitary;
\ssk\ni
iii) The operators $\Ut$ form a one--parameter unitary group. }
\bsk
The following result shows that dynamical instabilities growing
with $ n $ (at fixed time) are necessary for irreversible
behaviour to take place in the limit $ n \to \infty$:
\bsk\ni
{\bf Proposition 4}
\ssk\ni
{\it Assume that $X$ is a metric space, with distance $d(x,y)$, and that
the maps $\Gtn$, satisfying M and F, are stable, at fixed time,
uniformly in $n$, except
possibly on \lq\lq locally small subsets\rq\rq ; i.e. assume:
$\forall$ compact $K \subset X$, $ \eps > 0$,
$ t \in \reali$, $\exists M (K,\eps,t) \in \reali$ such that
$$d ( \Gtn (x) , \Gtn (y) ) < M \, d(x,y) \ \ \ \ \
\forall n \in \naturali, \ \forall x,y \in K \setminus S $$
$ S = S(K,\eps,t) $ an open subset of $K$ of $\mu$ measure
smaller than $\eps$.
\ssk\ni
Then, if the sequence $\atn$ converges in the weak $\tF$ topology,
it also converges strongly and $\at$ is a group of
automorphisms of $\Lfm$. }
\bsk
In general, the map $\at$ has an immediate probabilistic interpretation:
given any state $h \in \Lum$ and denoting by $\chi_B$ the characteristic
function of a Borel set $B \subset X$, the mean value
$$\int h(x) \, \at \chi_B (x) \; d\mu (x)$$
gives by definition the probability that a point distributed
with $h(x) d\mu (x)$ at time $0$ ends in $B$ at time $t$.
$\at \chi_B (x)$ is therefore interpreted as a probability $P_t (x,B)$
that a point starting at $x$ at time $0$ falls in $B$ at time $t$.
Since $\at \chi_B$ is a characteristic function of a
measurable set for all $B$ if and only if $\at$ is an automorphism of
$\Lfm$, $P_t (x,B)$ takes values in $ \{ 0, 1 \} $ ($d\mu$ a.e.) if
and only if there is no irreversibility.
\msk
The situation is therefore very close to that of a stochastic
process, as it also follows by a closer examination of
limits at different times. Consider in fact finite sequences of
measurements at different times, $\tuk$,
of observables $ \Auk $, and assume that, given a regular state
$\o \in \F$, the mean values
$$ \o ( \Atnuk ) $$
converge for $n \to \infty$.
The following Proposition shows that, if such a limit exists, it
defines a stochastic process, i.e. a measure $ d\rho_\o (x(t))$
on trajectories in $X$, and gives necessary and sufficient condition
for the process to be Markov.
\bsk\ni
{\bf Proposition 5}
\ssk\ni
{\it Let $\atn$ satisfy M and F and converge weakly
for $n \to \infty$;
assume that for a fixed $\o \in \F$, and
for all finite sequences $\Auk \in \CX$ and
$0 \leq t_1 < t_2 < \ldots < t_k \in \reali $
$$ \o ( \Atnuk ) $$
converge for $ n \to \infty$.
\ssk\ni
Then, there exists a unique regular Borel probability measure
$ \dro $ on
$\prod_{t \in \reali^+} {\dot X}_t$, ${\dot X}$ the one
point compactification of $X$,
such that
$$ \lim_n \; \o ( \Atnuk ) = \int \droxt \; \Aukt $$
For $\o (x)$ of compact support $K$, the measure $\dro$ is supported
on a product of compact sets $K_t$, $ t > 0 $
\ssk\ni
The process defined by $\dro$ is Markov, with time independent
transition function
$P_t (x,B)$ = $\at \chi_B (x)$, if
$$ \lim_n \; \o ( \Atnuk ) = $$
$$ = \lim_{n_k} \ldots
\lim_{n_2}
\lim_{n_1}
\o (\alpha^{t_1}_{n_1}(
f_1\alpha^{t_2-t_1}_{n_2}
( \ldots (
f_{k-1}\alpha^{t_k-t_{k-1}}_{n_k} f_k ) \ldots )))
$$
This condition is also necessary for the
time independent Markov property if
$\o (x)$ is a.e. different from zero}.
\bsk
In Sect.4 a model will be discussed leading to an irreversible
time evolution, actually defining a time independent Markov
process.
\bsk\bsk\goodbreak\ni
{\bf 3. Proofs}
\bsk
As mentioned above, states in $\F$ will be identified with their
representative function in $\Lum$. If the state $\o \in \F$ is
represented by the function $h \in \Lum$, we will write $\atns h$
for the representative function of $\atns \o$.
It is immediate to see that in $\F$ the norm of the dual $\CXs$ of $\CX$
coincides with the $ \Lum $ norm. The $\F$ notation will be preferred to
that based on $\Lu$ in arguments where the dual
$\CXs$ of $\CX$ is involved.
Since $d\mu$
is sigma--finite, the continuous functions with compact support belong
to $\Lum$ and are dense there; $\Luc$ is therefore dense in $\Lu = \F$.
By weak and strong $\tF$ topology we will denote respectively the weak
topology defined on $\CX$ by the linear functionals in $\F$ and the
ultrastrong topology defined as above by the representation of $\CX$ in
$\Ldm$, identical with the strong topology for bounded sets.
By $\tF$ topology and $\tF$ continuity we will refer to the weak one.
The following Lemma is preliminary:
\bsk\ni
{\bf Lemma 6}
\ssk\ni
{\it i) $ \forall t \in \reali $, $ \atns$ maps $\F$ into $\F$ and
$$\atns h(x) = h(\Gtn x) \eqno(3.1) $$
\ssk\ni
ii) $\atn$ is continuous in the weak (and in the strong) $\tF$
topology, and extends therefore by continuity to an automorphism
of the Von Neumann algebra $\Lfm$;
\ssk\ni
iii) $ \forall f \in \Luf \ \ \atns f = \atnm f$;
\ssk\ni
iv) $ \forall p \in [1, \infty], \ \ \forall f \in \Lup \ \
|| \atns f ||_p = || f ||_p$}.
\bsk\ni
\Pf
\ssk\ni
Eq. (3.1) follows immediately from the invariance of $d\mu$:
$\forall h \in \Lum$, $h(\Gtn x) \in \Lum$ and
$$ \int h(\Gtn x) \, f(x)
\; d\mu = \int h(x) \, f(\Gmtn x) \; d\mu = \int h(x) \, \atn f \dmu \ \
\ \ \ \ \forall f \in \CX \eqno(3.2) $$
Invariance of $d\mu$ implies
conservation of all $\Lp$ norms.
Invariance of $\F$ implies (see e.g.
[19]) weak $\tF$ continuity
of $\atn$, which extends by continuity to the weak closure $\Lf$ of
$\CX$.
\bsk\goodbreak\ni
{\bf Proof of Proposition 1}.
\ssk\ni
{\it i)}: For all $f$ in $\CX$, the weak $\tF$ limit of $\atn f$ defines a
continuous linear functional on $\Lum$, with norm bounded by
$\sup | f |$, i.e. an element of $\Lfm$;
the limit $\at$ of $\atn$ defines therefore a linear
map of $\CX$ into $\Lfm$, of norm at most one;
its transpose $\ats$ a priori maps $\F$ into $\CXs$,
and is continuous in the norm
of $\CXs$, denoted by $|| \cdot ||_{*}$: $\forall h \in \Lu$
$$ || \ats h ||_{*} =
\Supf \ | \lim_n \int h \, \atn f \dmu | \leq ||h||_1 $$
Weak $\tF$ continuity of $\at$ is equivalent [17] to
stability of $\F$ under $\ats$, i.e. point {\it ii)};
for positivity see {\it iii)} below.
\ssk\ni
{\it ii)}: Consider
first states represented by functions $h \in \Luc$;
by Lemma 1, $\atns h
= \atnm h$, which $\tF$ converges for $\ntf$ to $\atm h$. This means
that $\forall f \in \CXz $
$$ \int \ats h \, f \dmu = \lim_n \int \atns h \, f \dmu =
\lim_n \int \atnm h \, f \dmu = \int \atm h \, f \dmu \eqno(3.3) $$
so that
$$ \ats h = \atm h \ \ \ \ \ \forall h \in \Luc \ \ \ \ .
\eqno(3.4)$$
\ni
Moreover,
$$ || \atm h ||_1 = || \ats h ||_1 = || \ats h ||_{*} \ \ \ .
\eqno(3.5) $$
Eq. (3.5), continuity in $\CXs$ norm of $\ats$ and density of $\Luc$
in $\F$ imply stability of $\F$ under $\ats$.
\ssk\ni
{\it iii)}: $\at$ extends by $\tF$
continuity to the Von Neumann algebra $\Lf$.
Positivity of $\at$ is equivalent to
$$ \int h(x) \, \at f(x) \; d\mu \geq 0 \ \ \ \ \
\forall f \geq 0 \ , \ \ f \in \Lf \ , \ \ h \geq 0 \ , \ \
h \in \Lu \eqno(3.6) $$
For $f \in \CX$, eq. (3.6) follows from the positivity of
the same expression with $\atn$ replacing $\at$, for
$f \in \Lf$ by definition of continuous extension of $\at$.
\ssk\ni
{\it iv)}: Eq. (3.5) extends to all $ h \in \Luf $, since,
$\forall h \in \Luf $, $ f \in \Luc $,
$$ \int \ats h \, f \dmu = \int h \, \at f \dmu =
\int h \, \atms f \dmu = \int \atm h \, f \dmu \eqno(3.7) $$
and $\Luc$ is dense in $\CX$.
\ssk\ni
{\it v)} follows from H\"older inequality:
$$ \int \atns h \, f \dmu \leq || h ||_p \; || f ||_q \ \ \ \ \
\forall h \in \Lup \ , \ \ f \in C^0 \cap L^q \ ,
\ \ n \in \naturali $$
which implies
$$ \int \ats h \, f \dmu \leq || h ||_p \; || f ||_q $$
so that, by density of $C^0 \cap L^q$ in $L^q$
$$ || \ats h ||_p \leq || h ||_p \eqno(3.8) $$
\ssk\ni
{\it vi)}: In order to prove
$$ || \ats h ||_1 = || h ||_1 \ \ \ \
\forall h \in L^{1 +} \eqno(3.9) $$
consider first $h \in \CXz$.
Denote by $\chi_K$ the characteristic function of a compact set
$K \subset X$; using Lemma 1, condition $F$ implies
$$ \int \chi_K \; \atnm h \dmu = \int \chi_K \; h \dmu
\ \ \ \ \forall n
\eqno(3.10) $$
if $K$ is sufficiently large, depending only on $h$ and $t$.
Given the support $L$ of $h$, it is in fact enough to take
$K \supset L$ such that $\Gtn (L) \subset K \ \forall n$.
Using eqs.(3.4), (3.10) it follows
$$ \int \ats h \dmu = \lim_K \int \chi_K \; \atm h \dmu = $$
$$ = \lim_K \lim_n \int \chi_K \; \atnm h \dmu =
\lim_K \int \chi_K \; h \dmu = \int h \dmu $$
Eq.(3.9) then follows from density in $\Lu$ of $\CXz$
and $\Lu$ continuity of $\ats$.
Eq.(3.9) also implies $\at 1 = 1$.
\ssk\ni
{\it vii)}: Using Lemma 1 and eq.(3.4), $\forall h \in \Luf ,
\ f \in \Luc$
$$ \int \atn h \, f \dmu = \int h \, \atns f \dmu =
\int h \, \atnm f \dmu \to $$
$$ \to \int h \, \atm f \dmu
= \int h \, \ats f \dmu = \int \at h \, f \dmu \eqno(3.11)$$
and $\tF$ convergence of $\atn h$ to $\at h$ follows from
density of $\Luc$ in $\Lu$ and uniform boundedness of $\atn h$ in $\Lf$.
\ni
To obtain $\tF$ convergence of $\atn h$ to $\at h$ for all
$h\in\Lf$ consider a sequence
$\{K_i\}$ of compact sets with $\chi_{K_i}$ converging $\tF$ to $1$
(it exists because $\mu$ is $\sigma$-finite); then,
$ \forall h \in \Lf^+ \ , \ \forall l \in \Lu^+ $
$$ \int \atn (h-h\chi_{K_i} ) \, l \dmu \leq
\|h\|_\infty \int (1-\atn \chi_{K_i}) \, l \dmu \eqno(3.12) $$
but $\tF$ convergence of $\atn$ on $\Luf$, $\tF$ continuity of $\at$ and
$\at 1=1$ imply
$$\lim_i \lim_n \hbox{ RHS of (3.12)}=0 \ \ \ \ ;$$
on the other hand, by $\tF$ continuity of $\at$,
$$\lim_i \lim_n \int \atn (h \chi_{K_i}) \, l \dmu =
\int \at h \,l \dmu$$
and therefore
$$ \lim_n \int \atn h \, l \dmu =
\int \at h \, l \dmu$$
which extends by linearity to $h \in \Lf$ and $l \in \Lu$.
\bsk\ni
{\bf Proof of Proposition 2}.
\ssk\ni
$vi) \Rightarrow v)$ and $ ii) \Rightarrow i)$ are obvious;
$ iv) \iff v)$ and $ i) \Rightarrow vi) $ immediatly follow from
Prop.1, (for the latter implication, see the proof of Lemma 6);
it is therefore enough to prove
$v) \Rightarrow i)$ and $vi) \Rightarrow iii) \Rightarrow ii)$.
\ssk\ni
$v) \Rightarrow i)$:
For any Borel set $B$ of finite measure $d\mu$,
$\ats \chi_B$ is real, non negative, and
$$ || \ats \chi_B ||_\infty \leq || \chi_B ||_\infty \leq 1 \ \ , $$
so that
$$ (\ats \chi_B)^p \leq \ats \chi_B \ \ . \eqno(3.13) $$
$v)$ implies that for some $p \in (1,\infty)$
$$ \int (\ats \chi_B)^p \, d\mu = \int \chi_B^p \, d\mu =
\int \chi_B \, d\mu \geq \int \ats \chi_B \, d\mu \ \ , \eqno(3.14) $$
the last equality following from Prop.1, $v)$.
Eqs. (3.13) and (3.14) imply
$$ \ats \chi_B = (\ats \chi_B)^p $$
$d\mu$ a.e., so that
$ \ats \chi_B $ is a measurable function taking values in $\{ 0,1 \}$,
i.e. it is the characteristic function of a measurable set
$B^\prime \subset X$, and by eqs.(3.13), (3.14),
$\mu (B^\prime) = \mu (B)$.
\ni
>From linearity and the fact that $L^\infty$ norms do
not increase (Prop.1, $i$), it
follows that disjoint sets $B,C$ have disjoint images $B^\prime ,
C^\prime$; $\at$, coinciding with $\atms $ on $\Luf$,
(Prop.1, $iv)$, is therefore a morphism of the algebra of
simple functions based on sets of finite measure, and, by
a density argument, of $\Lfm$.
It also follows that $\ats$ preserves the $L^1$ norm on
$L^{1 +}$, which implies invertibility
of $\at$, actually $\atm \at = \at \atm = 1$;\ in fact,
for all positive simple functions $f,g$,
$$ \int f \atm \at g \, d\mu = \int \at f \at g \, d\mu
= \int \at (fg) \, d\mu = \int f g \, d\mu \ \ . $$
The same holds for $\at \atm$, and invertibility of $\at$
follows by a density argument.
\ssk\ni
$vi) \Rightarrow iii)$:
convergence in the strong and ultrastrong topologies is
equivalent for a bounded sequence [25], and amounts to
$$\int |\atn f - \at f|^2 \, h \, d\mu \to 0
\ \ \ \ \ \ \forall h \in \Lum \ \ . \eqno(3.15)$$
By a density argument, it is enough to obtain eq.(3.15)
for all $f$ in $ \CXz \subset \Luf $; in this case,
$\atn f = \atnms f$ converges to
$\at f = \atms f$ strongly in $L^2(X, d\mu)$, as a consequence
of weak $L^2$ convergence and of conservation of the
$L^2$ norms in the limit. $\at 1 = 1$ immediately follows from
conservation of the $L^1$ norm.
\ssk\ni
$iii) \Rightarrow ii)$: Strong convergence implies in general
that $\at$ is a morphism. The group property follows since
for all $f,g$ in $\CXz$,
$$ \int f \atps g \, d\mu = \lim_n \int f \atn \asn g \, d\mu =
\lim_n \int \atnm f \asn g \, d\mu \ \ \ \ ; \eqno(3.16) $$
strong convergence in $L^2$ of $\atnm f$ and $\asn g$ imply
in fact
$$ \lim_n \int \atnm f \asn g \, d\mu = \int \atm f \as g \, d\mu
= \int f \at \as g \, d\mu \ \ . \eqno(3.17) $$
\bsk\ni
{\bf Proof of Proposition 3}.
\ssk\ni
The algebra $\CXz$ of continuous functions of compact
support is dense in $\CX$ and in $\Lum$ in the respective norms.
Moreover,
the sequence $\atn h$ is uniformly bounded
in $\Lu$ and in $C^0$.
$\tF$ convergence of $\atn$ forall $f \in C^0$
is therefore equivalent to convergence on $\CXz$.
\ni
In the same way, since $\CXz$ is dense in $\Ld$
and $\Utn$ preserves the $L^2$ norm,
$\tF$ convergence is equivalent to
$$ \forall h_1,h_2 \in \Ld \ \ \ \exists \lim_n \int h_1^*\,\Utn h_2
\dmu $$ i.e. to convergence of $\Utn$ in the weak topology $w_{op}$
for operators in $L^2$
\ni
Denoting $\Ut \equiv w_{op}-\lim_n \Utn$, $\forall
h_1,h_2 \in \CXz$,
$$\int h_1^*\,\Ut h_2 \dmu=\lim_\ntf \int h_1^*\,\atn
h_2 \dmu=\int h_1^* \, \at h_2 \dmu \eqno(3.18) $$
which implies $\Ut h = \at h$ for $h \in \CXz$ and,
by $L^2$ continuity of both $\Ut$ and $\at$ (Prop.1, $v$), for
$h \in \Luf $.
The rest of the proof of Prop.3 is then reduced to the
following simple fact:
\ssk\ni $\forall n \in \naturali$ let $\Utn$ be a one parameter group
of unitary operators on a Hilbert space $\H$,
converging weakly for $n \to \infty$, $\forall t \in \reali$,
$ w_{op} - \lim_n \Utn \equiv \Ut$.
Then the following are equivalent:
\ssk\ni
$a)$ $\forall t \in \reali , \ \Utn$ converges strongly to $\Ut$,
\ssk\ni
$b)$ $\forall t \in \reali \; \forall h \in \H \;\; \| \Ut h \| =
\| h\|$,
\ssk\ni
$c)$ $\forall t \in \reali \;\Ut$ is unitary,
\ssk\ni
$d$) $\Ut$ is a group of unitary operators.
\ssk\ni
In fact, $a) \Leftrightarrow b$ is well known, and
$d) \Rightarrow c) \Rightarrow b) $ are trivial.
\ssk\ni
To prove $a) \Rightarrow d)$,
we have, from Schwarz inequality
$$ | ( h , (\Utn \Usn - \Ut \Us ) h) | =
| (h , (\Utn \Usn - \Utn \Us + \Utn \Us -\Ut \Us) h) | \leq $$
$$ \leq | ( h , \Utn (\Usn - \Us) h ) | +
| ( h , (\Utn -\Ut) \Us h) | \leq $$
$$ \leq \| h \| \; \| (\Usn - \Us) h \| +
| ( h , (\Utn -\Ut) \Us h) | \ \ , $$
which implies the group property.
In particular $\Ut$ is invertible, and unitarity follows since
$ a) \Rightarrow b)$.
\bsk\ni
{\bf Proof of Proposition 4}.
\ssk\ni
For fixed compact $K \subset X , \ \eps > 0 ,
\ t \in \reali, \ f \in \CXz$ with
support in $K$, by property $F$ all the functions
$\atn f (x) = f(\Gmtn (x))$ have support contained in a compact
set $K^\prime$.
By assumption, there exists an open set $S$,
of measure smaller than $\eps$, such that
$$ d ( \Gmtn (x) , \Gmtn (y) ) < M (K^\prime, \eps, -t) \, d(x,y) $$
for $x,y \in K^\prime \setminus S$.
Since $f$ is of compact support and therefore
uniformly continuous, it follows that the functions
$\atn f (x) $ are equicontinuous
on $K^\prime \setminus S$.
\ssk\ni
Since $K^\prime \setminus S$ is closed and therefore
compact, by the Ascoli--Arzel\`a theorem there exists a subsequence
$\alpha^t_{n_k} f $ which converges there uniformly;
its limit coincides, on $K^\prime \setminus S$
with $\at f$, as a consequence
of weak convergence of $\atn f (x) $ to $\at f$.
On compact sets, uniform convergence implies convergence in $L^2$;
moreover, the functions $\atn f (x)$ are bounded by $\sup |f|$, and
therefore their $L^2$ norm coincides with the norm of their
restriction to $K^\prime \setminus S$ up to an error of
order $\eps$, and
the same is true for $\at f(x)$; this implies
$$ | \, || \at f ||^2_{L^2} - || f ||^2_{L^2} \, | \;
\leq \; 2 \, \sup |f|^2 \, \eps \ \ \ \ \ \ \
\forall \eps > 0$$
It follows that $\at$ preserves the $L^2$ norm on $\CXz$, for
all $t$, and therefore, by Proposition 2, {\it iv)},
it is a group of automorphisms of $\Lfm$.
%def mie
\def \sp{{C^{sp}}}
\def \an#1{\alpha^{t_#1}_n}
\def \ro{\rho_\o}
\def \tuk{{\{t_j\}_{1,k}}}
\def \tukm{{\{t_j\}_{1,]m[,k}}}
\def \tuku{{\{t_j\}_{1,k-1}}}
\bsk\ni
{\bf Proof of Proposition 5}.
\ssk\ni
We will use the shorthand
$\tuk$ for $0\leq t_1 < t_2 < \ldots From compatibility it also follows $\ro^\tuk (1) = \o (1) = 1$.
\ssk\ni
As regards the Markov property, let us indicate by
$$ p(t_1,x_1; \ldots ; t_{k-1},x_{k-1} | t_k, f_k ) \in L^\infty
(\dot X^\tuku,\ro^\tuku) $$
the conditional expectations of the process, characterized by
$$
\forall k \geq 2, \ \forall j = 1, \ldots ,k, \ \forall f_j \in
L^\infty (\dot X_{t_j},\ro^{t_j}) \ \ \ \ \
\int f_1(x_1) \ldots f_k(x_k) \, d\ro^\tuk = $$
$$ =
\int f_1(x_1) \ldots f_{k-1}(x_{k-1}) \,
p(t_1, x_1; \ldots ; t_{k-1}, x_{k-1}|t_k , f_k) \, d\ro^\tuku
\ \ \ . $$
The process is Markov iff
$$p(t_1,x_1,\ldots ,t_{k-1},x_{k-1} |t_k , f_k) =
p (t_{k-1},x_{k-1} | t_k ,f_k)$$
and it is time independent Markov iff it is Markov and
$$p(t_1, x_1 | t_2 ,f_2) = p(0,x_1 | (t_2 - t_1) ,f_2) \ \ \ .
$$
Hence, for a time independent Markov process, by iteration we obtain
$$\int f_1(x_1)\ldots f_k(x_k)\,d\ro^\tuk =$$
$$ =
\int p(0,x_0|t_1,
f_1 \, p(0,\cdot \,|t_2 - t_1, f_2 \ldots f_{k-1} \,
p(0,\cdot \,|t_k - t_{k-1},f_k)
\ldots )) \, \omega (x_0) \, d\mu \eqno(3.21)
$$
Notice now that, since the $\atn$ are automorphisms, it is
$$ \lim_{n_k} \ldots
\lim_{n_2}
\lim_{n_1}
\o (\alpha_{n_1}^{t_1} f_1 \ldots
\alpha_{n_k}^{t_k} f_k ) = \eqno(3.22) $$
$$ = \lim_{n_k} \ldots
\lim_{n_2}
\lim_{n_1}
\o ( \alpha^{t_1}_{n_1} (
f_1 \, \alpha^{t_2-t_1}_{n_2} ( \ldots (
f_{k-1} \, \alpha^{t_k-t_{k-1}}_{n_k} f_k ) \ldots )))
$$
which, by definition and continuity of $\at$, converges to
$$\o (
\alpha^{t_1}(
f_1 \, \alpha^{t_2-t_1} ( \ldots (
f_{k-1} \, \alpha^{t_k-t_{k-1}} f_k ) \ldots ))) \ \ \ . \eqno(3.23)
$$
So, if the limit (3.19) coincides with (3.22), then
the stochastic process $\rho$ is Markov
and time independent, with $p(0,x|t,f)=\at f\,(x)$.
\ni
The converse follows if $\o(x)$ only vanishes on a set of zero measure;
in fact, for $\o (x) \neq 0$,
$p(0,x|t,f) = \at f \,(x)$, and therefore,
if the process is time independent Markov,
i.e. if (3.21) holds, then it is also given by the limit
(3.22), (3.23).
\bsk\bsk\goodbreak\ni
{\bf 4. A model}
\bsk \ni
We consider a sequence of dynamical systems defined by a
particle moving horizontally in $\red$
with speed 1, and interacting with fixed vertical \lq\lq rods\rq\rq ,
of equal height; the only effect of the interaction is assumed to be the
change of the \lq\lq spin\rq\rq\ of the particle, taking values $\pm 1$.
The space $X$ of the configurations of the system is thus
$\red \times \{-1,1\}$,
on which we take the measure $\mu= d^2\x \; \otimes \; d\sigma$,
with $\sigma(\{1\}) = \sigma(\{-1\}) = 1$.
As space of states we will take $L^1 (\red \times \{-1,1\}, \mu)$,
i.e. the set of states defined by measures on the particle configuration
space which are absolutely continuous with respect to the
Lebesgue measure.
The dynamics will depend on a parameter $n$ through the change of the
positions and height of the rods, which will be given by a suitable
scaling of the positions and of the height of the rods {\it in one given
and fixed configuration} in a set to be specified in the following.
\ssk
In general, a configuration $\xi$ of rods is determined by
their height $h$ and by the set of their centers, $\xi \equiv
\bigcup_{i\in\naturali} \{(x_i,y_i)\}$; we will always consider only
configurations with a finite number of rods in any compact subset of
$\red$, and denote by $\Xi$ the set of all such configurations.
For {\it fixed} $\xi \in \Xi$, and $h > 0$, we will study the limit
$\ntf$ of the dynamics $\atn$ defined by the interaction with rods of
height $\nt h$ in the scaled configuration $\no \xi \equiv
\bigcup_{i\in\naturali} \{(\no x_i,\no y_i)\}$. The limit will be taken
in the weak topology defined by the state space $\F \equiv L^1 (\red
\times \{-1,1\}, \mu)$.
We will prove the following:
\bsk\ni
{\bf Proposition 7}
\ssk\ni
{\it For almost any
choice of the configuration $\xi$ with respect to the measure $\rho$
defined below, the dynamical systems defined as above
by the fixed configuration $\xi$ converge weakly for
$n \to \infty$, with respect to the set $\F$ of states,
to a positive map, and give rise, as in Proposition 5, to
a time independent Markov process}.
\bsk
We stress that for each finite $n$ the dynamics of the model is
{\it deterministic and reversible}, and only in the limit $n \to \infty$
the result for any given observable, in any given state in $\F$, is
described by a Markov process.
\msk
The r\^ole of the measure $\rho$ is to select the configurations
of rods which appear \lq\lq when their centers are given without
correlations and with density one\rq\rq ; the construction of
such a measure is necessarely a little absctract,
but it can be done in a relatively straightforward way,
following the procedure outlined in ref.[13], Sect.7.1.2.
\ssk\ni
In fact, (full details are given in [28]) the family of
measures on $\Lambda ^j$ $$d\rho_\Lambda^j \equiv
{e ^{- | \Lambda |} \over j
!} \; d\x_1\ldots d\x_j$$ with $j \in \naturali$ and $\Lambda$ bounded
Borel$\subset \red$, defines a bounded positive linear functional
$$\rho
(f) \equiv \sum_{j=0}^\infty {e ^{- | \Lambda |} \over j !}
\int_{\Lambda ^j} d\x_1\ldots d\x_j \, f( \xi ) \ \ \ , \eqno(4.1) $$
where
$f :\Xi \rightarrow \complessi$ depends only on the points,
exactly $\x_1 \ldots \x_j$,
of $\xi$ inside $\Lambda$,
over a suitable algebra $\cal B$ of functions on $\Xi$. The
spectrum $\cal E$ of the $\|\;\|_\infty$ closure of $\cal B$
turns out be the $*$-$w$ closure of $\Xi$
(the $*$ weak topology being the weakest one which makes
the functions in $\cal B$ continuous).
\ssk\ni
By the Riesz--Markov theorem, the positive functional
$\rho$ defines a Borel regular measure on ${\cal E}$, denoted
for simplicity again by $\rho$, and it is not difficult to
see that $\rho$ is in fact concentrated on $\Xi$, i.e.,
$\rho (\Xi ) = \rho ({\cal E}) = 1$ [28].
For the following, the essential point is that,
for functions depending only on a finite number of points,
integration with the measure $\rho$ is simply given by eq. (4.1).
\bsk\msk\goodbreak\ni
{\bf Proof of Proposition 7}.
\msk\ni
The time evolution of $\x \equiv (x,y) $ is a
translation, denoted by $\x + t \equiv (x+t,y)$.
For the spin $\si$, the effect of collisions is given by
$$ \atn \si =
\si \, (-1)^{ |\no \xi \,\cap\, B_{\x,t,\nt h}| } =
\si \, (-1)^{ | (\xi \,\cap\, B_{n \x,n t,\no h}| } \ \ , \eqno(4.2)$$
$|I|$ being the number of points in the finite set $I$, and
$$B_{\x,t,h} \equiv \{
(\al,\be) \in \red \,|\, \be \in [y-h/2,y+h/2), $$
$$ \al \in [x,x+t) \hbox{
for } t > 0, \ \al \in [x+t,x) \hbox{ for } t < 0 \}$$
For the states we have therefore:
$$ \forall l \in \Lum \ \ \ \atns l(x,y,\si)=
l(x+t,y, \si \cdot (-1)^{|\xi \,\cap\, B_{n \x,n t,\no h}| } )$$
and for the observables
$$ \forall f \in \CX \ \ \ \atn f(x,y,\si) =
f(x-t,y,\si\cdot(-1)^{|\xi \,\cap\, B_{n (\x-t),n t,\no h}| } ) \ \ .$$
Clearly, $\mu$ is left invariant by $\atn$, for all $t$ and $n$.
The choice of a discontinuous interaction makes $\atn$ not to map
$C^0$ into itself. However, the $\atn$ map $C^0$ into
the $\|\;\|_\infty$ closure of $\Luf$, and define automorphisms
of this space, which is enough for the proof of the results of Sect.2.
\ssk\ni
First of all we want to express the request of existence of a limit
dynamics, obtained by scaling the obstacles, in terms of observables of
the system
which are directly related to the configuration $\xi$ of rods.
Denoting by $g_{\pm 1,\x,t,h}(\xi)$ the characteristic functions of
the sets
$$\{\x \in \red \, : \, |\xi\,\cap\, B_{\x,t,h}| \ \hbox{ is} \
{\displaystyle {\hbox{even} \atop \hbox{odd}}}\} \ \ ,$$
from the fact that
$\forall f \in \CX ,\; \forall l \in \Lum$
$$\int_X \atn f\,l \dmu =
\sum_{\tau = \pm 1} \sum_{\si = \pm 1}
\int g_{\tau, n(\x-t), n t, \no h}(\xi) f(\x-t, \tau \cdot \si)\,
l(\x,\si) \, d^2\x $$
and since the characteristic
functions of bounded Borel subsets of $\red$ are dense in
$L^1(\red,d^2\x)$, it follows that the condition for the existence of a
limit dynamics, i.e.
$$ \forall f \in \CX ,\ \ \forall l \in \Lum \ \ \ \
\exists \lim_n \int \atn f \, l \dmu
\equiv \int \at f \, l \dmu $$
is equivalent to the existence of the limit
$$\exists \lim_n \int \chi_B(\x) \,
g_{\si ,n(\x -t),n t, \no h}(\xi) \, d^2\x \ \ . \eqno(4.3) $$
$ \forall B$ bounded Borel subset of $\red$,
and $ \si = \pm 1 $.
\ssk\ni
Moreover, calling $G_{\si, \x,t,h}(\xi)$ the $\tF$ limit of
$g_{\si,n \x,n t,\no h}(\xi)$,
for the limit dynamics $\at$ we may write
$$\sum_{\si=\pm 1} \int \at f(\x,\si)\, l(\x,\si)\, d^2\x = $$
$$ =
\lim_n
\sum_{\tau = \pm 1} \sum_{\si=\pm 1}
\int g_{\si_1,n (\x-t) ,n t, \no h}
(\xi) \, f(\x -t, \tau \cdot \si ) \, l(\x,\si) \, d^2\x = $$
$$ = \sum_{\tau = \pm 1} \sum_{\si=\pm 1}
\int G_{\si_1,\x-t , t, h}(\xi) \, f(\x -t, \tau
\cdot \si) \, l(\x,\si) \, d^2\x \;\; .$$
\ssk\ni
In the same way it turns out that the existence of a limit
for the measurement of $k$ observables at $k$ successive
times $0 \leq t_1 < t_2 < \ldots < t_k$ is
equivalent to $\tF$ convergence of the products $\prod_{i=1}^k\,
g_{\si_i,n (\x-t_i), n t_i, \no h} (\xi)$. Moreover,
denoting $t_0 \equiv 0$ and $\rho_i \equiv
\prod_{i\le j} \si_j$, it follows
$$
\prod_{i=1}^k\, g_{\si_i,n (\x-t_i), n (t_i-t_{i-1}), \no h} (\xi)
= \prod_{i=1}^k\, g_{\rho_i,n (\x-t_i), n t_i, \no h} (\xi) \ \ \ , $$
and we may write the request of convergence in a form that will be more
suitable to demonstrate the Markov time independent property for the
limit stochastic process:
$$\forall B \hbox{ bounded Borel set } \subset
\red , \forall k=1,2,\ldots \; , \
\forall \si_i = \pm1 \ \ , \ \ i=1,\ldots ,k $$
$$ \exists \lim_n \int \chi_B(\x) \,\prod_{i=1}^k g_{\si,n(\x
-t_i),n (t_i-t_{i-1}), \no h}(\xi) \, d^2\x \ \ . \eqno(4.4) $$
\ssk\ni
Now, from the construction of $\rho$ it follows that the
characteristic functions of the
sets $\{\x \in \red \, : \, | \xi\, \cap \, B_{\x,t,h}| = n \}$,
denoted by $\tilde g_{n,\x,t,h}(\xi) $,
are in $L^1 (\Xi, \rho)$, and
their integral is given by
$$\int _\Xi \tilde g_{n,\x,t,h}(\xi)
\, d\rho(\xi) = \sum_{j=0}^\infty {e ^{- | \Lambda |} \over j !}
\int_{\Lambda ^j} d\x_1\ldots d\x_j \,
\tilde g_{n,\x,t,h}(\xi ) =
e^{-t h} {|t h|^n \over n !} \ \ .$$
Moreover, if $\forall i \neq j \;\;
B_{\x_i,t_i,h_i} \cap B_{\x_j,t_j,h_j} = \emptyset$, then
$$ \int_\Xi
\prod_{i=1}^k \tilde g_{n_i,\x_i,t_i,h_i} (\xi)\, d\rho (\xi)\, =
\prod_{i=1}^k
\int_\Xi \tilde g_{n_i,\x_i,t_i,h_i} (\xi)\, d\rho (\xi)$$
(which expresse the fact that the positions of the
rods of $\rho$-almost each $\xi \in \Xi$ are uncorrelated).
\ssk\ni
Expressing the functions $g_{ \si ,\x,t,h}$
in terms of the $ \tilde g_{n,\x,t,h}$, we obtain
$$ \int_\Xi g_{\pm1,\x,t,h} (\xi)\, d\rho (\xi)\, =
{1 \pm e^{-2|t| h} \over 2}$$
and for $B_{\x_i,t_i,h_i} \cap B_{\x_j,t_j,h_j} = \emptyset$
$$ \int_\Xi \prod_{i=1}^k g_{\si_i,\x_i,t_i,h_i} (\xi)\,
d\rho (\xi)\, = \prod_{i=1}^k \int_\Xi g_{\si_i,\x_i,t_i,h_i}
(\xi)\, d\rho (\xi)\, = \prod_{i=1}^k {1 + \si_i
\cdot e^{-2|t_i| h_i} \over 2}\;\; .$$
\ssk\ni
The aim is now to show that
the conditions in eqs. (4.3) and (4.4) are fulfilled
for $\rho$-almost all $\xi \in \Xi$.
We will show that those limits exist in the $L^1(\Xi,\rho)$ norm,
and hence pointwise $\rho$-almost everywhere on $\Xi$
for a subsequence.
\ssk\ni
Consider for simplicity the single time case: $\forall B$
bounded Borel $\subset \red$:
$$ \int_\Xi d\rho(\xi) \left| \int_B d^2\x
\, g_{\si,n(\x-t),n t,\no h}(\xi) \, -|B| \, {1+\si\cdot e^{-2|t| h}
\over 2} \right| ^2 \, = $$
$$= \int_\Xi d\rho(\xi)
\int_B d^2\x \, g_{\si,n(\x-t),n t,\no h}(\xi) \,
\int_B d^2\xt \, g_{\si,n(\xt-t),n t,\no h}(\xi) \, + $$
$$ - \, \int_\Xi d\rho(\xi) \;
( |B| \, {1+\si\cdot e^{-2|t| h} \over 2} )^2 $$
where we applied Fubini's theorem to
$g$, which is measurable over
$(\red \times \Xi , d^2\x \otimes d\rho)$,
since it is pointwise approximated by a sequence of functions of the
form $$\sum_{\hbox{finite}} \phi_i(\x) \cdot \chi_i(\xi)$$ with
$\phi_i(\x)$ and $\chi_i(\xi)$ measurable.
\ssk\ni
Since the points of almost all $\xi$ are uncorrelated, in the
above integration no contribution comes from the set
$$Q \equiv \{
(\x,\xt)\in B \times B \,|\, n |y - \yt | \ge \no h \} \;\; .$$
As $\ntf$, $Q$ becomes all of $B \times B$: more precisely, taken a
rectangle in $\red$ of base $\De_1$ and height $\De_2$ which contains
$B$, it is
$$| (B \times B)\setminus Q | \le \int_B d^2\x \,\De_1
\int_{y- \nt h}^{y+ \nt h} \, d \yt
\le \, \nt \, 2 h \, \De_1^2 \, \De_2 $$
and therefore, since
$\|g\|_\infty =1$, we conclude $$ \int_\Xi d\rho(\xi) \left| \int_B
d^2\x \, g_{\si,n(\x-t),n t,\no h}(\xi) \, -
|B| \, {1+\si\cdot e^{-2|t|
h} \over 2} \right| ^2 \,
\le \, \nt \, 2 h \, \De_1^2 \, \De_2 \ \ . \eqno(4.5)$$
\ssk\ni
The same
reasoning and the same inequality apply for functions of the form
$$\prod_{i=1}^k g_{\si_i,n (\x-t_i),n (t_i-t_{i-1}),\no h} (\xi)\ \ ,
\hbox{ with } \ t_0 = 0 < t_1 < \ldots < t_k \ \ , $$
which describe all measurements at $k$ successive times.
Moreover, since the $g$'s
in the above product depend on $\xi$ only through points in
disjoint sets, and since the points of
almost all the $\xi$'s are uncorrelated, it follows that
$$\int_B d^2\x \; \prod_{i=1}^k
g_{\si_i,n (\x-t_i),n (t_i-t_{i-1}),\no h} (\xi)$$
converges, for the moment in $\|\;\|_2$ over $(\Xi,\rho)$,
to the product of the limits,
i.e. to
$$|B| \ \prod_{i=1}^k {1 + \si_i
\cdot e^{-2|t_i| h } \over 2}\ \ \ .$$
\ssk\ni
Since $\rho (\Xi)=1$, convergence also takes palce in the $L^1$ norm.
Now, if a sequence $\{f_j\}$
converges in $\|\;\|_1$ to $f$ and $\sum_{j=1}^\infty \|f_j - f_{j-1}
\|_1 \le \infty$, then $f_j$ converge pointwise a.e. to $f$
([26], vol.1, pag.18), and
in our case it is sufficient to take a subsequence
with $n_j=2^j$ to assure
$\rho$--a.e. pointwise convergence.
\ssk\ni
We can thus state that
for $\rho$-almost all $\xi \in \Xi$,
for any denumerable family ${\cal R}$ of bounded Borel
sets $B \subset \red$,
$\forall k=1,2,\ldots \;,\;\forall \rho_i = \pm1 $, $i=1,\ldots,k$,
$\forall t_0 =0 < t_1 < \ldots < t_k$ rational it is
$$\lim_n \int_B d^2\x \; \prod_{i=1}^k g_{\si_i,2^n (\x-t_i),2^n
(t_i-t_{i-1}),2^{-n} h}(\xi) \, = |B| \;
\prod_{i=1}^k {1 + \si_i \cdot
e^{-2|t_i-t_{i-1}| h} \over 2} \ \ .$$
\ssk\ni
By taking ${\cal R}$ such that the linear combinations of characteristic
functions $\chi_B$ are
dense in $L^1(\red , d^2 \x)$, and using the
$\|\;\|_\infty$ boundedness of the integrand,
uniform in $n$, we extend convergence to
any $B$ bounded Borel $\subset \red$,
which is the condition of existence of
a limit time evolution (the case $k=1$),
and in general of a limit stochastic
process, with the restriction of the times $t_i$ to a denumerable
subset of $\reali$.
\ni
In particular, the limit dynamics is given by
$$\at f(\x,\si)\;=
\sum_{\rho=\pm 1} f(\x -t, \rho \cdot \si) {1+ \rho \, e^{-2|t|h}
\over 2} \eqno(4.6) $$
for almost all configurations of rods and a denumerable set of
times $t$.
\ssk\ni
The convergence of $\prod_{i=1}^k g$ to $\prod_{i=1}^k G$
implies that measurements at $k$ successive times define a
Markov and time independent process,
with transition function
$$ P_t((\x,\si),B) = \at \chi_B (\x,\si) \ \ \ \ \ ; $$
in fact,
$$\atn f(\x,\si) = \sum_{\rho=\pm 1}
f(\x-t, \rho\cdot\si) g_{\rho,n(\x-t),n t, \no h}(\xi)$$
so that
$$
\al^{t_1}_{2^n} (f_1 \al^{t_2-t_1}_{2^n} (f_2 \al^{t_3-t_2}_{2^n} \ldots
( f_{k-1} \al^{t_k-t_{k-1}}_{2^n} f_k)\ldots))(\x,\si) =$$
$$=
\prod_{i=1}^k \sum_{\rho_i =\pm 1} \;
f_i ( \x-t_i, \si \cdot \prod_{j=1}^i \rho_j ) \
g_{\rho_i, 2^n(\x-t_i), 2^n (t_i-t_{i-1}), 2^{-n} h}(\xi) $$
which $\tF$ converges as $\ntf$ to
$$\prod_{i=1}^k \sum_{\rho_i =\pm 1}
f_i ( \x-t_i, \si \cdot \prod_{j=1}^i \rho_j ) \
{1+\rho_i e^{-2(t_i-t_{i-1})h} \over 2} = $$
$$ = \al^{t_1} (f_1 \al^{t_2-t_1}
(f_2 \al^{t_3-t_2} \ldots ( f_{k-1} \al^{t_k-t_{k-1}}
f_k)\ldots))(\x,\si) \ \ \ , \eqno(4.7) $$
which is the condition for the stochastic process
to be Markov and time independent.
\ssk\ni
The process is defined by the l.h.s. of eq.(4.7) for all
positive times, and the restriction to a denumerable set
of times is only involved in the construction of a fixed
set of configurations of rods (actually of $\rho$--measure one)
for which weak convergence holds.
\references
\bye