\magnification 1200 %versione con spaziature maggiori, semplificata %def. matem. \def\reali{\hbox{\rm I\hskip-2pt\bf R}} \def\realipunto{\hbox{\rm I}\hskip-2pt \dot {\bf R}} \def\complessi{\hbox{\rm I\hskip-5.9pt\bf C}} \def\neRightarrowro{\hbox{\bf T\hskip-7.4pt\rm I}} \def\razionali{\hbox{\rm I\hskip-5.7pt\bf Q}} \def\interi{\hbox{\rm{Z\hskip-8.2pt Z}}} \def\naturali{\hbox{\rm I\hskip-5.5pt\bf N}} \def\dsec#1#2{{{\partial^2 #1 } \over {\partial #2^2}}} \def\dpar#1#2{{{\partial #1 } \over {\partial #2}}} \def\modulo#1{\left| #1 \right|} \def\norma#1{{\left\| #1 \right\|}} %spaziature \def\bsk{\bigskip} \def\msk{\medskip} \def\ssk{\smallskip} \def\ni{\noindent} % Fonti di caratteri \font\titfnt=cmbx10 at 14.40 truept \font\tentex=cmtex10 \font\ninerm=cmr9 \font\eightrm=cmr8 \font\sixrm=cmr6 % Definizione formato della pagina \newdimen\pagewidth \newdimen\pageheight \newdimen\ruleht \hsize=31pc \vsize=45pc \maxdepth=2.2pt \parindent=19pt \pagewidth=\hsize \pageheight=\vsize \ruleht=.5pt \abovedisplayskip=10pt minus 2pt \belowdisplayskip=11pt minus 2pt \abovedisplayshortskip=8pt minus 2pt \belowdisplayshortskip=9pt minus 2pt \baselineskip=15pt minus 0.5pt \lineskip=2pt minus 0.5pt \lineskiplimit= 2pt \mathsurround=1pt % Definizioni per l'output %(numeri in alto alternati) \nopagenumbers \def\testos{\null} \def\testod{\null} \headline={\if T\tpage{\gdef\tpage{F}{\hfil}} \else{\ifodd\pageno\rightheadline\else\leftheadline\fi} \fi} \gdef\tpage{T} \def\rightheadline{\hfil{\tensl\testod}\hfil{\tenrm\folio}} \def\leftheadline{{\tenrm\folio}\hfil{\tensl\testos}\hfil} %(numeri in alto al centro) %\nopagenumbers %\headline={\hss\tenrm\folio\hss} %\voffset=2\baselineskip % Registrazione referenze su file esterno e macro di conteggio \newcount\numref \global\numref=1 \newwrite\fileref \immediate\openout\fileref=ref.tmp \immediate\write\fileref{\parindent 30pt} \def\citaref#1{${[\the\numref]}$\immediate\write\fileref {\par\noexpand\item{{[\the\numref]\enspace}}}\ignorespaces \immediate\write\fileref{{#1}}\ignorespaces \global\advance\numref by 1\ignorespaces} \def\references{\immediate\closeout\fileref \par\vfill\eject \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 \, {*}}}} \def \Ca{{$C^{*}$ algebra}} \def \Cas{{$C^{*}$ algebras}} \def \Lum{{L^1 (X, d\mu)}} \def \Ldm{{L^2 (X, d\mu)}} \def \Lfm{{L^\infty (X, d\mu)}} \def \Lu{{L^1}} \def \Ld{{L^2}} \def \Lp{{L^p}} \def \Lf{{L^\infty}} \def \Luf{{L^1 \cap L^\infty}} \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}}} \def \Ut{{U^t}} \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 \def \K{{\cal K}} \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