Internat. J. Math. & Math. Sci.
VOL. 12 NO. 2 (1989) 397-402
397
ON COARSE-GRAINED ENTROPY AND MIXING IN STATISTICAL MECHANICS
C.G. CHAKRABARTI
Department of Applied Mathematics
Unlverslty of Calcutta
Calcutta
700009, INDIA
(Received January 7, 1985 and in revised form October 22, 1986)
The object of the paper If the study the roles of non-equillbrium entropy
ABSTRACT.
and
phases
of
mixing
the
in
of
characterization
statistical
the
coarse-grained
interpretatlon of the irreverslble approach to statlstlcal equlllbrlum of an isolated
system.
Probability space, o-algebra, observational states, sufficient
KEY WORDS AND PHRASES.
partition, Coarse-grained distribution, Ergodlcty, and Weak-mixing.
82A05.
1980 AMS SUBJECT CLASSIFICATION CODE.
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
The coarse-grainlng cannot be done arbitrarily and Giggs entropy
equations of motion.
on
based
relaxation
an
arbitrary
to
statlstlcal
does
distribution
coarse-grained
object of
The
equilibrium.
not
always
ensure
the present paper
is
the
to
introduce a non-equillbrium entropy after Goldstein and Penrose [I] and to study the
of
importance
ergodicity
and
mixing
in
the
statistical
characterization
of
the
irreversible approach to statistical equilibrium of a classical isolated system.
2.
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
is the family of tlme-evolutlon transformations (automorphisms)
phase-points and {T
.
defined
space
other
t}
generated by the dynamical equation of motion in phaseLet m be the invarlant Llouvllle’s measure of phase-space and let 9 be any
for
all
measure
real
t
absolutely
microstates p(to) is
defined
continuous
by
a
to
normalized
m.
The
density
probability
given
by
the
density
derivative
p(to)
dv
(to)
of
Radon-Nikodym
(2.1)
398
C.G. CHAKRABARTI
t(Tt
[t
consists of the family of measures
The tlme-evolutlon of the measure
)
defined by
().
(2.2)
A).
(2.3)
This implies that for any set A
The
statistical
structure
CA)
(T
(o r
model)
the
of
space (, A, {v t })where A is the o-algebra of subsets
is the family of probability measures on A.
and, {
the
by
system
of
probability-
itself,
including
t
0 we
To determine the observational states of the system at the initial time t
divide the time of observation into a countably infinite nmber of intervals of equal
Considering the length of each interval as the unit of the measurement of
length.
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
Two phase
0.
points that are indistinguishable by an observation made at time t
points ml and m2 are then observatlonally equivalent if and only if the time
which lle, for every non-negatlve integer t in the same set
and T
t ranlates T
t
I
t
m2
In other words,
from the partition P.
TtP.
partition
I
and
2
rest lle in the same set from the
Let us define the o-algebra
2.4)
=ffiv0 T_t
T_IP, T_2P,
as the smallest o-algebra which contains all the partitions P,
every set in
a;
is the image under of some set in
consists of image under T
Thus
that is, to say, the a-algebra T
of all sets belonging to and includes (among others) all
t
the sets of t itself
T
The
condition
(2.5)
is
the
(2.5)
a
t
of
condition
loss
of
observational
information
and
represents the asymmetry between past and future [I].
3.
IRREVERSIBILITY AND MIXING.
NOS-EQUILIBRIb ENTROPY:
The entropy (flne-gralned) of the classical dynamical system is defined by the
functional
S(t,
-k
The
entropy S(v
t’ A)
A) ffi-K
f
log
f pt()
(dvt/dm)
log
dv
pt()
dm()
tC).
() contains full
the fine gralned-denslty
For observation behavior of the system such detailed
defined over
infromatlon about the system.
(3.1)
information or description about the system is not necessary.
Pt
For this a coarse-
graining of mlcrostates is necessary [2].
Let
of
us
pt()
consider
the
coarse-gralned
density 0 as the conditional expectation
t
with respect to the o-algebra t (which is the consltlonal expectation for the
COARSE-GRAINED ENTROPY AND MIXING IN STATISTICAL MECHANICS
399
measure m/m()) [1].
t
The
entropy
for
E(P()Ia)
state
non-equilibrlum
the
(3.2)
the coarse-gralned
then defined by
is
entropy [I]
(t’
a)
-K
%t
f
log
t am(m).
3.3)
Since the measure m used in defining the entropy is invariant, we have,
(t’
S
6)
From (2.5), we have
T
-t
[
S (v,
,
C
(t
>
T_t.
(3.4)
(3.5)
0).
As a consequence of (3.5), it is easy to prove that
S (v, T
-t
or by
a) >
(v,a)
(3.6)
(3.4), we have
S
> S (,a),
(t,a)
(t
>
O)
(3.7)
which proves the non-decreasing property of the entropy S (v
t’ )
with time; that is,
the H-theorem.
The
(3.7)
in
equality
corresponds
the
to
equilibrium of the system at the initial time t
the measure-preserving automorphism T
stationary
0.
state
of
statistical
Mathematically this holds for
t
(3.8)
or
(3.6), which
Also the equality in
is a consequence of the relation
(3.5), holds for
.
the invariance relation:
T a
-t
(3.9)
Thus the inequality in (3.7) for statistical equilibrium at the initial time t
corresponds
to
the
invariance
of
the o-algebra
a
under
0
measure-preserving
automorphism T
The equality
that is, to the condition of ergodicity of the system.
t
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
if the conditional
expectation of any dynamical variable, say Hamlltonian X(m) given the o-algebra
that
{vt
{X()I} is
is, if E
algebCa a implies
density
E{p()la}
the
same
the
which
for
{t
all
time-variance
by
definition
of
is
[3].
the
our
The
,
sufficiency
conditional
coarse-grained
of
the o-
probability
density
under
400
C.G. CHAKRABARTI
consideration.
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
0.
t
In an earlier paper [4], we have in fact shown
This is a significant result.
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
to ensure
a) is
.
however, not sufficient
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:
measure-preservlng
automorphlsm
Tt,
increasing sequence of o-algebra and let
a =tV=0
be its limit in the sense that
T
Ttt
{Tta}
sequence
the
Ttc D
forms
.
a
For
monotonically
where
(3.10)
t
+ t(R).
Tt
Note the
t
being the smallest n-algebra
.
0,1,2...) is, therefore, equal to the oalgebra {@,l}, consisting of the null-set @ and the phase-space (ergodlc set)
Then
by Doob’s convergence theorem [5]
[A Tta}
lim
(3.11)
{AI}
which includes all sets belonging to
(t
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
the
to
statistical
equilibrium
(mlcrocanonical) density I/m():
llm
Pt
lira
S
(3.12)
i/m()
or
(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
0, the mixing of phases ensures the limiting
at the initial time t
case of relaxation of the system to statistical equilibrium over the whole of phaseof the system.
space
4.
CONCLUSIONS.
The paper aims
mixing
in
the
to
stress
coarse-gralned
statistical equilibrium.
the importance of
interpretation
measure-preserving
the
irreversible
approach
to
The analysis is based on a measure of entropy defined for
the non-equillbrium states of an isolated system.
under
the properties of ergodicity and
of
automorphism T
The invarlance of the o-algebra t
corresponds
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
t
to
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].
time.
COARSE-GRAINED ENTROPY AND MIXING IN STATISTICAL MECHANICS
401
REFERENCES
1.
2.
GOLDSTEIN, S. and PENROSE, C. J. Star. Phys. 24(1981), 325.
FARQUHAR, F.
Erodlc Theory in Statistical Mechanlcs Intersclence Publ., New
York, 1964.
3.
4.
5.
6.
KULLBACK, S. Information Theor and Statlstlcs Wiley and Sons, 1959.
J. Math. and Phys. Scl. 9(1975), 381.
CHAKRABARTI, C.G.
Report Math. Phys..
16(1969), 143. Z. Phys. B51(1983), 265. Z. Phys. B61(1985), 353.
BILLINGSLEY, P. Ergodlc Theory and Information, Wiley and Sonc, New York,1965.
HOBSON, A. Concepts of Statistical Mechanics Gordon and Breach, San Francisco,
1970.