ON COARSE-GRAINED ENTROPY AND MIXING IN STATISTICAL MECHANICS

The object of the paper If the study the roles of non-equillbrium entropy 
and mixing of phases in the statistical characterization of the coarse-grained 
interpretation of the irreversible approach to statistical equilibrlum of an isolated 
system.


I. INTRODUCTION.
The process of coarse-graining which is equivalent to the statistical averaging of the mlcro-states over the various phase-cells plays a significant role in the study of the macroscopic property of irreversibility from the reversibility of dynamical equations of motion.The coarse-grainlng cannot be done arbitrarily and Giggs entropy based on an arbitrary coarse-grained distribution does not always ensure the relaxation to statlstlcal equilibrium.The object of the present paper is to introduce a non-equillbrium entropy after Goldstein and Penrose [I] and to study the importance of ergodicity and mixing in the statistical characterization of the irreversible approach to statistical equilibrium of a classical isolated system.

DYNAMICAL SYSTEM AND OBSERVATIONAL STATES.
Let us consider a classical dynamical system whose dynamical state if given by (R, T t) where R is the phase-space consisting of all possible phase-points and {T t} is the family of tlme-evolutlon transformations (automorphisms)   defined for all real t generated by the dynamical equation of motion in phase- space .Let m be the invarlant Llouvllle's measure of phase-space and let 9 be any other measure absolutely continuous to m.
The probability density of microstates p(to) is defined by a normalized density given by the Radon-Nikodym derivative dv p(to) (to) (2.1) The tlme-evolutlon of the measure consists of the family of measures [t defined by t(Tt ) (). (2.2) This implies that for any set A CA) (T A). (2. 3) The statistical structure (o r model) of the system by the probability- space (, A, {v })where A is the o-algebra of subsets of including itself, t and, { is the family of probability measures on A. t To determine the observational states of the system at the initial time t 0 we divide the time of observation into a countably infinite nmber of intervals of equal length.
Considering the length of each interval as the unit of the measurement of time, the evolution of a subset A in the course of time is given by the series TtA (t-...-2,-1,0,1,2...).
Let us define a partition P of phase-space into sets of points that are indistinguishable by an observation made at time t 0.
Two phase points ml and m2 are then observatlonally equivalent if and only if the time t ranlates T t m2 I and T t which lle, for every non-negatlve integer t in the same set from the partition P.
In other words, I and 2 rest lle in the same set from the partition TtP.Let us define the o-algebra =ffiv0 T_t

2.4)
as the smallest o-algebra which contains all the partitions P, T_IP, T_2P, Thus every set in is the image under of some set in a; that is, to say, the a-algebra T consists of image under T of all sets belonging to and includes (among others) all t the sets of t itself T a (2.5) t The condition (2.5) is the condition of loss of observational information and represents the asymmetry between past and future [I].
The entropy (flne-gralned) of the classical dynamical system is defined by the functional S(t, A) ffi-K f pt() log pt() dm() -k f log (dvt/dm) dv tC).
The entropy S(v A) defined over the fine gralned-denslty Pt () contains full t' infromatlon about the system.
For observation behavior of the system such detailed information or description about the system is not necessary.
For this a coarse- graining of mlcrostates is necessary [2].
Let us consider the coarse-gralned density 0 t as the conditional expectation of pt() with respect to the o-algebra t (which is the consltlonal expectation for the measure m/m()) [1].
t E(P()Ia) (3.2) The entropy for the non-equilibrlum state is then defined by the coarse-gralned entropy [I] (t' a) -K f %t log t am(m).

3.3)
Since the measure m used in defining the entropy is invariant, we have, S (t' 6) S (v, T_t. (3.4) From (2.5), we have T [ C , (t > 0). (3.5) -t As a consequence of (3.5), it is easy to prove that S (v, T a) > (v,a) or by (3.4), we have S (t,a) > S (,a), (t > O) (3.7) which proves the non-decreasing property of the entropy S (v ) with time; that is, t' the H-theorem.
The equality in (3.7) corresponds to the stationary state of statistical equilibrium of the system at the initial time t 0. Mathematically this holds for The different non-null atoms of the sufficient o-algebra t represent the different macrostates of statistical equilibria of the system at the initial time t 0. This is a significant result.In an earlier paper [4], we have in fact shown that the sufficiency of the o-algebra a for statistical equilibrium results from the ergodicity of the system.
The non-decreasing property of the entropy S (v a) is however, not sufficient to ensure the relaxation to equilibrium over the phase-space (energy-shell) .For this a more broad assumption, namely the assumptlonof mixing of phases is necessary.
That the coarse-grained distribution generated by the o-algebra a corresponds to the process of mixing results from the relatlon: Ttc D .For measure-preservlng automorphlsm Tt, the sequence {Tta} forms a monotonically increasing sequence of o-algebra and let where a =tV=0 T t (3.10) be its limit in the sense that Ttt + t(R).Note the t being the smallest n-algebra which includes all sets belonging to Tt (t 0,1,2...) is, therefore, equal to the o- algebra {@,l}, consisting of the null-set @ and the phase-space (ergodlc set) .Then by Doob's convergence theorem [5] lim [A Tta} {AI}   (3.11)   which is the condition of weak-mlxlng or relaxation to statistical equilibrium [6].
To express it in a more familiar form we note that the o-algebra to-{,} comprises of all sets of measure 0 and m().The mixing condition (3.11), then implies the convergence of the coarse-grainded density Pt to the statistical equilibrium (mlcrocanonical) density I/m(): llm Pt i/m() (3.12) or lira S (t,a) K log m() (3.13)where the r.h.s is the thermodynamic equilibrium entropy.Thus, while the ergodiclty corresponds to the states of statistical equilibria over the various phase-cells (nonnullatoms of t at the initial time t 0, the mixing of phases ensures the limiting case of relaxation of the system to statistical equilibrium over the whole of phase- space of the system. 4. CONCLUSIONS. The paper aims to stress the importance of the properties of ergodicity and mixing in the coarse-gralned interpretation of the irreversible approach to statistical equilibrium.The analysis is based on a measure of entropy defined for the non-equillbrium states of an isolated system.The invarlance of the o-algebra t under measure-preserving automorphism T t corresponds to the statistical equilibria over the various phase-cells (including the whole phase-space R also) at the initial time t 0. The sufflclency-a reduction principle of statistics, plays a significant role in the statistical characterlzatlon of statistical equilibria at the initial time.
In the case of initial non-equilibrium distribution, it is, however, the assumption of phase-mixlng which ensures the relaxation to statistical equilibrium over the whole of phase-space [4].
in (3.6), which is a consequence of the relation (3.5), holds for the invariance relation: that is, to the condition of ergodicity of the system.The equality has also an important statistical significance.This, in fact corresponds to the sufficiency of the o-algebra or to the sufficient partitioning of mitt.states(or phase-space) into equivalent class of macrostates of the system [3.4].The oalgebra a is sufficient for the family of probability measures {vt if the conditional expectation of any dynamical variable, say Hamlltonian X(m) given the o-algebra , that is, if E {X()I} is the same for all {t[3].The sufficiency of the o- )la} which by definition is our coarse-grained density under consideration.