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Intrinsic irreversibility of Kolmogorov dynamical systems
Physica 122A (1983) 459482
North-Holland Publishing Co.
INTRINSIC IRREVERSIBILITY OF KOLMOGOROV DYNAMICAL SYSTEMS M. COURBAGE* Laboratoire de Probabilitb,
Tour 56, 4, Place Jussieu, 75230 Paris Cedex 05, France
Received 21 June 1983
We give the mathematical details and various extensions of the results stated in previous work of Courbage and Prigogine. “Intrinsic random systems” are deterministic and conservative dynamical systems for which we can associate two dissipative Markov processes through a one-to-one “change The of representation”, the first leading to equilibrium for t -++cc and the second for r-+-co. microscopic formulation of the second principle of thermodynamics permits to lift the degeneracy by the exclusion of all states that do not approach equilibrium for t + + cc. The set of admitted initial conditions g+ is then characterized by a non-equilibrium entropy functional which is infinite for rejected initial states and takes finite values for admitted initial conditions. Thus, rejected initial states correspond to an infinite amount of information. To realize this selection rule we consider general probability measures on phase space that are not necessarily absolutely continuous and we extend the theory of transition to Markov processes to such measures. Owing to the non-invariance of s+ under the time inversion, the evolution of these states in the new representation can only be given by one of the two possible Markov processes.
0. Introduction More than ever, the problem of the microscopic interpretation of irreversibility remains in the forefront of physics. This is due to the impact of the recognition of the fundamental role played by irreversible processes in widely different fields such as elementary particles, cosmology and self-organization processes in chemistry and biology’). Moreover, it is of fundamental importance to elucidate the connection between the reversible and deterministic laws of physics and the second law of thermodynamics. According to a widely held view, irreversibility is a result of a “coarse-grained” reduction of the (deterministic) dynamics, and, recently, it has been argued that irreversibility appears when the physical observations realise this reduction in a non-symmetric way with respect to time inversion (see e.g. the works of Goldstein and Penrosez13) and Misra and Prigogine14)). Another approach to the microscopic interpretation of irreversibility due to Prigogine and coworkers’) starts from a different viewpoint. Let us first of all recall * Supported in part by Instituts Intemationaux
037%4371/83/0000-0000/$03.00
de Physique et de Chimie Solvay, Brussels.
0 1983 North-Holland
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M. COURBAGE
that the functional
s
P,(O) logp,(w) d/do)
9
(0.1)
where p, is a Gibbs distribution function of a system of particles in phase space, remains constant in time. This can be interpreted as a preservation of the “order” or the “information” associated to the state P, under the representation of the dynamics by trajectories. Moreover, the constancy of (0.1) can be interpreted in terms of the existence of two opposite processes the first leading to equilibrium distribution of velocities and the second corresponding to the creation of correlations between particles resulting from their interactions. Boltzmann, by neglecting the second process (for physical reasons) was able to derive his #-theorem. The viewpoint of Prigogine et al. proposes to redefine the distribution function P, by a transition to a new representation. This change of representation is interpreted as a redefinition of units in terms of collective modes for highly complex systems involving also a redefinition of the correlations between units. Under this change of representation the time evolution leads, under a probabilistic process, to the damping of the correlations and to equilibrium. The transition to this new representation should be given by a non-unitary transformation /l:p --+ (&I)(O) = p’(o) in such a way that the functional
s
P”,(o)log i%(o) G(o)
(0.2)
describes the information associated to fir and decreases monotonically towards a minimum in equilibrium. The existence and the construction of such a transformation n leading from a deterministic dynamics the Markov process is not always possible. The dynamical systems where such a transition is possible have been studied by Misra et al.‘) and called “intrinsic random systems”. But we are still faced with a degeneracy: intrinsic random systems may be mapped by two distinct Markov processes the first leading to equilibrium for t -+ + co and the second for t + - co. On the macroscopic level we have also a similar situation. For example, to the Fourier equation for t 2 0 leading to the uniform distribution of heat for t -+ + co corresponds an (anti)-Fourrier equation leading to equilibrium for t + - co. As well known, the Fourier equation for t >, 0 cannot admit solution leading to equilibrium for t < 0 and only a distinct equation may describe such an evolution. Therefore, to lift the degeneracy between the two Markov processes we have to discuss the problem of the intrinsic physical distinction (if any) between the two
IRREVERSIBILITY
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possibilities. The selection between the two possibilities is a result of the restriction of the transformation A to a class of admitted initial states which is not symmetric under the time inversion and which is compatible with dynamics as it is propagated in the future. Moreover, in this work we have shown that the entropy acts as a selector of initial conditions, for, the rejected initial conditions correspond to an infinite entropy (0.2) while admitted initial states correspond to a finite entropy. These concepts have been introduced in our paper without proofs. Some of the theorems concerning non-equilibrium entropy and the extension of n have only been stated. For the readers interested in this problem we shall now give the mathematical proofs. But first, to make the presentation self-contained we shall recall the mathematical formulation of the basic concepts previously introduced.
1. Mathematical formulation of the basic concepts An abstract classical dynamical system is a one parameter group of transformations S, acting on o-points of the phase space r and leaving invariant some distinguished measure p on r. This dynamical system can equivalently be represented by a group of unitary operators U, on Li preserving the set of positive functions, U, is related to S, by the Koopman lemma:
P(~)-w,P)(~) = P(StQJ) *
(1.1)
U, can also be defined by (1 .l) as an operator on LL. On the other hand, a stochastic Markov process on the phase space r, preserving p, is given by transition probability P(t, co, E) from o E r to an element E of the a-algebra a of measureable sets in time t. As well known, to any Markov process one can associate a semi-group of operators on Li:
f-W’J-)(a)
=
f WW,
0, do’),
s r
(1.2)
W, is a semi-group of contracting operators on L: called a semi-group of a Markov process. This Markov process defines a group of automorphisms on the space of all probability measures on (r, a) given by: v -+( w’:v)(E)
=
s
P(t, co, E) dv(o) .
E
(1.3)
462
M. COURBAGE
It is easy to see that if v is absolutely continuous w.r.t. p with square integrable density @, then @TV is also absolutely continuous with a density b, given by pl,= VP, where WY
t-cc
for
all probability distributions MEL:, p # 1. In other words, intrinsic random dynamical systems are systems for which the deterministic dynamics for t b 0 can be considered as “equivalent” to a Markov process, for /i can be interpreted as a change of representation. The conceptual relation between physical irreversible processes and the semi-group W: consists in 1) the construction of an H-function (see property vi) which implies that -
1p”,1s p”,dp
(1.5)
J r
increases monotonically to a maximum value, zero, only reached by the function 1 and 2) the non-invariance of W: by the time inversion, that is, W: cannot be extended for t ,< 0 to a semi-group oja Markovprocess (in fact, AU, A -’ cannot be positivity preserving for t < 0, see refs. 5 and 8). This second point means in particular that one should associate by a couple of transformations A and A’ a couple of semi-groups of Markov process WY = AU,A-’ and W;* = A’U,A’-’ for t > 0 and t < 0, respectively, the first approaching the equilibrium when t + + cc monotonically (in the sense of vi) and the second for t + - co. This scheme corresponds to a well known irreversible process like heat conduction etc. where the time inversion t + - t corresponds to the passage from one process to a distinct one. The interesting question of a necessary condition that should be satisfied by a dynamical system in order to be intrinsically random has been discussed by Misra’) who showed that the existence of a Liapounov function (property vi) implies that the system is mixing. As expected, the motion of the system should be highly unstable. In fact it has been shown that all B-systems are intrinsically random5~‘0) and these results have been extended to all K-systems”).
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Nevertheless, the time symmetry breaking of the dynamical group U, into WY and W;* for t 2 0 and t < 0, respectively,
having finite entropy represents the set of admissible states. Now a sysem is intrinsically irreversible if there exists a subset of physically admissible initial states CP tending to equilibrium in future but not in the past. As a consequence, the velocity inverse Vg- of the states % + cannot tend to equilibrium in the future and are therefore exluded by the microscopic formulation of the second principle. It is clear that for mixing systems such a requirement implies that one should go beyond L$ for L:-states tend to equilibrium for both t--r + co and t + - cc. This is also true for all absolutely continuous states. For mixing systems one should consider states which are not absolutely continuous. In the case of unstable hyperbolic systems, it is possible to consider states with broken time symmetry. Examples are measures concentrated on dilating fibers and contracting fibers. In physical models like the Lorentz gas dilating and contracting fibers may have physical interpretation. In the Lorentz gas for example, measures concentrated on dilating fibers correspond to an ensemble of particles which have formed a parallel beam of particles in the distant past before undergoing collisions, while contracting fibers correspond to a set of particles which will be parallel after executing many collisions. The selection rule expresses the unrealizability of experiences in which a set of particles that undergo several collisions will asymptotically emerge with parallel velocity.
Whenever the arrow of time is given by %+, the second principle become a selection principle which eliminates non-admissible initial states. The entropy
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functional plays the role of selector of initial states displaying the amount of information necessary for their preparation, and the admissible states are all states with$nite entropy. This set, that will be denoted 9+, has a time broken symmetry as it contains the V+ set. The initial conditions selected by the second law correspond to a finite amount of information while the initial conditions rejected correspond to an infinite amount of information. In this note, we show that such an entropy can be constructed for unstable systems of K-type, where the class GP is the set of probability measures uniformly concentrated on a dilating fiber. This entropy is a Liapounov function over the set of states. Whenever the set of initial states 9?+ is selected we show that there exists an afine transformation on this set under which the reversible deterministic evolution of these states is transformed into an irreversible Markovian evolution. This transformation extends the operator constructed in refs. 5, 10 and 11 for L:-states. The non-invariance of .9+ under the time inversion lifts the degeneracy between @‘: for t 2 0 and tit:* for t < 0 as the set 9- on which act ,4 ’ and @I: contains the set V- which has been rejected by the selection rule. Consequently, the K-systems are intrinsically random and intrinsically irreversible when the invariant measure p represents equilibrium state.
2. Intrinsic irreversibility of Kolmogorov dynamic systems First we recall the definition of intrinsic irreversibility of dynamical systems introduced in ref. 7. An abstract dynamical system is a Lebesgue space12) (r, a, ,u), where p is a normalized positive measure on the Bore1 o-algebra a of measurable subsets of f, on which acts a group of measure preserving transformations S,, f E R. For simplicity, we suppose that r is compact. The system is reversible if it has a time velocity inversion, I, introduced in section 1, also called a time inversion transformation. This transformation defines the operator on the space of measures:
v-+vv=v’:v’(E)=v(ZE).
(2.1)
All dynamical systems considered below will be of this type. The states of the system are all probability measures on (r, a) and we denote this set by Y. S, induces on Y a dynamical automorphisms group a, = v E~+U,V E Y (ct,V)(E) = v,(E) = V(S_,E) .
(2.2)
Let us define the approach to equilibrium of a state v. We say that v, approaches
IRREVERSIBILITY
OF KOLMOGOROV DYNAMICAL SYSTEMS
the equilibrium as t + + co if v,+p in the W*-topology
ji
dvt-
465
i.e. if
jt?-cb
(2.3)
r
r
for all continuous functions J More precisely, a reversible dynamic system is called intrinsically irreversible and intrinsically random if: 1) There exists a set of initial states W CY that is affine and not invariant by the time inversion V, i.e. .9+ contains a subset V+ of probability measures, such that its time inverse W cannot tend to p as t--f + co, i.e. Vv E YV+, vr+p, t-++CO.
2) c@+ C 9+, Vt. 3) VVE~+, v,+p, ast-++oo. 4) There exists an affine transformation /1”on W that realizes an “equivalence” between 01,and some irreversible Markov process in the following sense: i) /i maps a state v into a state, denoted G, and /ip = p. ii) /1”is one-to-one. iii) Aa,v = FP:fTv for any v E9+, where @‘: is a semi-group of automorphisms of a Markov process associated to a transition probability P(t, co, E) i.e.
dv”(o)P(t, CO,E) .
(I@‘:+)(E) =
(2.4)
s
r
iv) v”is absolutely continuous w.r.t. p with a density fiOsuch that
(2.5)
v) xv, is absolutely continuous with a density p”,verifying an &‘-theorem that is,
s
Q(v,)= Ptlogp”,dp
(2.6)
r
decreases monotonically
and tends to zero (= G(p)) as t + + CO.
A set of initial states Qt satisfying the above conditions will be called a set of admissible initial states.
466
M. COURBAGE
Clearly, the property 1) asserts that the class %+ allows to display the arrow of time. This property is not a consequence of the dynamics, it represents a supplementary property of the physical system. Evidently G!Z+does not determine the class of states 9+ physically admissible, this class of states should be characterized by the “amount of information” relatively to equilibrium and necessary for their realization. Such a functionnal of states should satisfy to the conditions: 1) to be finite for %+ but infinite for V, 2) to tend monotonically to zero when the state tend to equilibrium. We present here the construction of such /i’ for K-systems. We shall show that this construction coincides with the operator II previously constructed for densities in L:. We characterize the set 59’ and we define 1 for all states in LB+. We recall that the “canonical” method to construct the operator /1 on Lz permits to associate with the unitary group U, (1.1) two distinct Markov processes for t > 0 and t < 0. We will not recall this method which introduces II as a function of an internal time operator5). All proofs of this property for K-systems can be found in ref. 11. Here we present directly 1 and show that all states of 9!+ transformed by /1”,evolve under a Markov process. This shows, when the dilating fibers are chosen as a class %?+, that a K-system is intrinsically random and intrinsically irreversible. We recall first of all, that a K-system is a dynamic system having a distinguished sub-a-algebra a,, such that (1) S,a, = a,ca,, if t
IRREVERSIBILITY
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(2.7)
We recall that a sequence of finite partitions is said decreasing if 9,+, is a refinement of Y’,(Y,,+, > 9,,). It is well known’*) that if (r, a,,, p) is a Lebesque (separable) measure space, there exists a decreasing sequence of finite partitions 9, which tend to the partition to that generates q. We have the following theorem: Theorem 1 (with the above notations)
1) sZ(v, 9,) is a positive and non-decreasing Q(v, a,) as n+co:
function of n, which tends to
Q(v, a,) = lim Q(v, Y’,) .
(2.8)
n-cc
2) WV, a01= ;yg WV, 2) .
(2.9)
).
3) If sZ(v, a,) < + co, then the measure vlOO restriction of v to a, is absolutely continuous with respect to p. In this case:
WV, a,> =
s
p.log p. +
(2.10)
,
r
where p. is the Radon-Nykodim density of vlaOwith respect to p. 4) Conversely, if vl%is absolutely continuous w.r.t. p with density p. and if { p. log p. dp < + 00 then Q(v, eo) = jr p. log PO GL. This theorem follows straightforwardly from the Martingales theory. We recall in the appendix some basic results of this theory and the proof of theorem l*. In the case of distributions vp with density p satisfying l p2 dp < + co Q(v,, ao) is identical to the entropy introduced in the work of Misra and Prigogine14)
WV,, a,> =
Pop
(2.11)
log Pop dm ,
where POis the projection onto Li(a,)c * We have learned from F. Ledrappier by Perez, see ref. 13.
Lt. This results from the above theorem.
that this theorem
has been proved
by Dobrushin
and later
M. COLJRBAGE
468
A functionnal such as (2.11) has been also considered by Goldstein and Penrose’) as a model of entropy for absolutely continuous measures v,. Theorem 2. Let v be an initial probability
1) a(~,. a,) decreases monotonically t-*+a, 2) VI--+/&t++cQ.
measure such that sZ(v, a,J < + cc, then: with t and tends to Q(p, a,,) = 0, when
1) From the theorem I, ~1,~is absolutely continuous with density po. For t > 0, a_, c a, and VI,_,is absolutely continuous with density E” i(p,J. (Definition and properties of the conditional expectation E”f are given in the appendix.) E”~~(p,) is a martingale which converges Lh to 1 and E” ‘(p,,) log E”~ r(pJ is a semi-martingale which converges LL to 0. Now, from (2.7), and 4) of theorem 1 we get:
Proof.
Q(v,, P’,) = Q(v, S-,.9’,)
1
(2.12)
lim 52(v, S-,9,> = Q(v, S-,a,), n>7 and therefore Q(v,, a”) = WV, S-,a,) = j E”ml(p,)log E”m(p,)dp \
0.
2) let A ~a,. For I sufficiently great S-J v,(A) = v(S,A) =
t-cc
.
E a,, then
s
~(1dp
.S-,A
(2.13)
here qE is the characteristic
function of a set E.
Any mixing system, S,, has the following property:
s
.f(~k(S,o) d/do) -
r
gh. r
(2.14)
I
if f6 Lh and g is a bounded measurable function Putting in (2.14) g = qA and f = p,, we get: v,(A)-
cl(A),
r-r+%’
VA Ea,.
The same property extends easily for any A E a, by c/3 approximation
argument.
IRREVERSIBILITY
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SYSTEMS
Although Q(v, a,J can be a selector of initial conditions it cannot represent the total information of the state v relatively to I(. It corresponds only to the information of vlaOrelatively to pl%, and, for all states such that v(,,~= pl(.,, Q(v, a,,) = 0. However, such states are not in equilibrium and their information with respect to finer sub-c-algebras can be non-vanishing. Then we shall take into account the whole information with respect to all ai, sZ(v, a,). Such an entropy is the result of a transformation /1 of v into 5 given on a and not contracted to a sub-g-algebra. Under this “change of representation” a, is transformed into a Markov process with an &‘-theorem associated to states v having finite relative information Q(v, ai) with respect to all ai. We want now to define the transformation 1 and its domain 9+. This will be presented as a result of four propositions. 1. Let 6i be a sequence of positive numbers summing to 1: X:_+zai = 1 and let v be a state such that vi,,; is absolutely continuous w.r.t. u/08 with density pi then C 6,p, converges, a.e. and L:, to an integrable function p”such that jr p”dp = 1. Proof. By the Beppo-Levy
theorem jSN= P,
Applying again the Beppo-Levy and
6ipi is an integrable function and
theorem to pN, it follows that &-+N_m
p”a.e.
2. Let v ~9 be such that VJ,,~ is absolutely continuous w.r.t. p with density pi and let p = CZE Gipi, then v,I,~is also absolutely continuous with respect to ula, with density P,,~= E”$I,p, (see (1.1) pie Lb). Proof. For any A Ea,, S_,A Ea,_, c a,, for t > 0 and
v,,,(A) = v(S_,A) = s
pi(O) dp(o) S_,A
=
I
S_,A
this proves 2.
Pi(O)
dP(Srm)=
s
Pi(S-rO)
A
dP(m) =
s
Eai(Pi(S-t(m)))
A
dP
5
M. COURBAGE
470
3. Let us denote 1, = Xp”=, 6, and suppose that the sequence A,+,/;l, is decreasing with n. The set 9+ will be the set of all measures v such that: 1) SZ(v,a,)< +cc 2)
foralli=O,
+l,..
.,
(2.15) (2.16)
,i?logp”dp < +cc. s
We introduce now the ;i transformation as the mapping which sends a measure v E 9,+ into v”given by dG = p”d,u. It is clear that /i” is an afine transformation on W which maps a state into a state. To show that 1 is one-to-one let v and v’ be two measures from .!3+ with /Iv = Av’, we have, by using the properties of the conditional expectation E”n,the following relation for the density p”(and similarily for b’): (2.17) and therefore:
J++G) - EYS) = k+,(~n+,- P,)
’
(2.18)
Since c’ = 6, it follows from (2.18) that (2.19)
P”-Pm=P;-PAl,
for any m < n. Taking in (2.19) the limit m -+ -CC and using that p-X = pL * = 1 we have pn = p; for any n and therefore v = v’. We have to show now that Av, evolves under a Markov process. In previous works5,‘0s”), the “equivalence” between an irreversible stochastic process and the unitary group U,:p+(U,p)(o) = p(S_,o) has been constructed for all square integrable distribution functions p(o) via a bounded operator A on L: given by A = 1 sip, ) Pip = E”l(p),
(2.20) p EL; .
We have shown that the family A U,A -I, t 2 0, is a family of bounded densely defined operators which extends to a semi-group of a Markov process with a transition probability given by: (2.21)
p(t, 0, E) = (A -‘U,&,)(w), where CYJI~ is the characteristic
function of the subset E E a.
Remark. In ref. 11, it has been shown that AU,A -’ is a.e. positivity preserving as an operator on Li. This proof can be extended to show that A-‘U_,A, t 3 0 is positivity preserving as an operator on the space of all bounded
IRREVERSIBILITY
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measurable functions B(T, a). For this one has to use the existence of a regular conditional distribution in the Lebesgue measure spaces, that is to say, the existence of ai-measurable function o +P(A ( a,)(o) which is a probability on (r, a) as a set function A +P(A ) ai)( Vo, and equal ai-almost everywhere to E”~(cp,)(o) (see ref. 17). The construction of A, above presented, can be seen more generally if we do not require v”to be absolutely continuous. To do this, consider A defined by (2.20) as a bounded operator on the set B(r, a) of all bounded measurable functions f(w), that is a Banach space with respect to the norm ((f 11 m = sup, Al. It is easy to see from (2.20) that the set function 1(E, o) given by: (2.22)
WY 0) = (&3(o)
is a probability measure on a for all o. This follows from the regularity of the conditional distribution (E”icpJ(o) = P(E ) a,,)(o ) a.e. Then, ;i may be defined as an operator on the space of all o-additive set functions u (r, a) by:
s
s
(@(E) = W&(o) dv(o) = W, m) d/-do). r
(2.23)
r
One verifies that /i defines a contraction operator on the space v(T, a) of all a-additive set functions with the usual norm given by the total variationls). Let us denote by xv = v”.As well known, an arbitrary measure v EV(~, a) defines a functional on B(T, a) by: v(j)= J
r
fdv.
r
Then, by standard arguments, we have, as a result of (2.23) the following relation for v”and f E B(T, a): (W(f)
= q(f)
(2.24)
= v(Af) *
Similarly we have: (v)(f)
= VU-,f
),
feB(r,
4,
(2.25)
from these relations it follows that (;l”ov)(E) = v(U_&&) =
= Av(n
P(t, o, E) dv”.
(2.26)
412
M. COURBAGE
Let us denote by @: the operator (@W(E)
on v(E, B)
P(t, o, E) dv"(o) ,
=
(2.27)
s I-
from (2.26) we have:
Remark. The operator
n defined by (2.23) is one-to-one if /1, as an operator on B(T, a) contains in its range all characteristic functions of a generator of a. This is a direct consequence of (2.24). Such a property is not generally true but depends on the version of the conditional probability which defines /i on B(T, a). However, the restriction of ;i to 9+ does not depend on the version of E”(cp,)(w) and is invertible so that its inverse can map an absolutely continuous measure into a singular one. 4. The X-theorem follows nowfrom thefact that p”,evolves under a Markov process,
then
s
p”Ilog p”,dp d
I
p”l,log p”t,dp s r
for t 6 t’. As a result it follows that sp”, log p”ldp < + cc for Jr p”log p”dp < + GOi.e. V,E~ + if v E 9+. This proves the property 2. Let us finally show that
s
t -+ + co
p”tlog p”,dp -0,
I-
for any v&9+. Let us denote log+ x = (log x)+, then
s
P, log P”,dp G
p”,log+ P”,dp t s
using the relation E”g+iU,p = U,E”p, we have p, =
c 6,EW,pi I
= ulC6iPi-*= I
= c +!J,E”~ -‘pi 1 UtCal+*Pi,
,
t> 0
if
IRREVERSIBILITY
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473
therefore:
On the other hand CiSi+,pi is a.e. decreasing with respect to t for
C6i+,pi= 1+CA,+,(Eai-Eoi-‘)pi\
1,
t+CQ.
Then the application of the Beppo-Levy theorem for the limit t +a~ and the monotonicity of x log+ x yields the result,
3. Applications and examples In the preceding section we have characterized a set of “physical admissible states” for K-systems as the set of all probability measures v having a finite “amount of information” i.e. 1) Q(v,aJ< 2)
+co
foralli,
ijlog@dp<+co. s r
In general the finiteness of Q(v, q,) does not imply the finiteness of G?(v,aJ for all i. However, this is true for B-systems when a, is generated by independent$nite partition i.e. if a, is a-algebra generated by to = V?, B’S’, B= (PI,. . . , Pk) *. To see this, we may write C!(v, V’!_,B’9’) as the expectation with respect to v of the function log v (A,(o))/p (A,(o)), wh ere A,(o) is the cell of V’?,, B’s9 containing 0: s2(v, V o-, B’9) =
dv(w) log s
M”(W)) &4(w)).
(3.1)
r
Similarily we have for Q(v, V !, B’Y): s2(v, V !, B’9) =
dv(w) log s
r
v(W%w)n4(w)) P((B~)(~W,@))
’
* Recall that the product 9 V 9 of two partitions 9 and 9 is the partition that has elements
(3.2)
P, n Q,.
M. COURBAGE
414
Then O(v, V ‘, B'S) - C’(v, V
0, B'9') =
dv(o) log s
r
s
dv log PLWY~)
< -
v((Bg)(o)
1A,(o))
PLUW(~)
)4(w))
(A(w)
r
dv log p(P(o)> < -sup log P(~i).
=
(3.3)
s I-
A similar result holds also for Markov shifts with finite partition. On the other hand, to satisfy condition 2) it is sufficient to insure the square integrability of b,,. This is a result of the following inequality,
r
r
us check
SN =
It
,“y,
$S#i
1+ i
AiEipN AN+,PN
easy to see that,
Ng, (ni -
~N+I)J%'N +
&+I - lN+h"~
(3.6)
IRREVERSIBILITY
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415
As EipN = 0 Vi > N’, EiEj = 6,Ei we get: llWSW(I*=(&+I
-~.+,)211prl(‘+~~~(~i-i.,,,Yl14P,I12.
(3.7)
By using now the following inequalities <
Ai-AN+l IIEiPi/*
=
Vi < N + 1 ,
1,
lIPill
-
lIPi-1112
<
[(Pill2
9
(3.7) yields:
From this estimate, the L:-convergence test i.e. if
Ent,l
)IPi+111*
i>O
I(
2:
<
may be insured by using the D’Alembert
1
’
or equivalently if
As expected, this result relates the convergence of sN to the choice of the sequence li. For &+,/&+O, i-roe, all measures v such that supi(Ipi+,)(*/ /IpiI/’ < + CO will belong to Q+ (e.g. & = e-“).
Let us summarize this discussion: If (r, B”, a, p) is a K-system, if viaiis absolutely continuous with respect to plai, Vi, with square integrable density pi, i.e., 2 ={ p:dp IIP,41
< +co
and if ;i is constructed as in the proposition 3 of section 2, then v EL@+ if
To get an estimate of the ratio (lpi+, ~~2/IIPi~(2 for a B-system with finite partition we have:
M. COURBAGE
476
where A,,(w) is the cell of V:, Bk9 containing o, n, i > 0. Therefore: v((B~)(o)N,,i(o)) pf+, dp = lim dv nF+cO ~((B~)(~)nA,i(w)) s = lim dv vG%,i(~)) v(W’)(o) n-zc i
dmax’
P((A~,~(~)) P(B~)(~)
/4(o)) 1An,i(~))
1 ,w;> s
pfdp.
We have the following result for any B-system with $nite partition. If vl,0is absolutely continuous w.r.t. v with density p,, such that s pi dp < + co then J p f d,u < + co for all i. If moreover (3.8) Let us consider as an example the baker transformation which is isomorphic to the Bernoulli shift (l/2, l/2). Recall that B acts on the unit square (p . q) in squeezing in the direction of q and stretching in the direction of p and stacking the right half on the left half, more precisely B(P, 4) = (2~9q/2) 0 6~ < l/2 > =(2pl,q/2+ l/2) 1/2dp
< 1.
The a-algebra of the Bore1 sets is generated by the partition 9 = (PO, P,) and P, = {(p, q) 1P 2 l/2}. That is, let where PO= {(P>4) IP < l/2) B”9 = (B”P,,, B”P,) and let V,“+k-’ B’Y the partition of r whose elements are: A 2$J;;, ‘k ,,n+k_, E B”Pj,nB(“+‘)Pi,n . . . ~IB(“+~~‘)P~~. The set of finite unions of these rectangles which are usually called “-cylinders” form an algebra that generates the o-algebra of a Bore1 set of the unit square. A a-additive measure is uniquely defined by its values for the cylinders. The Lebesgue measure is given by: P(A;;..,‘:;:,-,) = l/2k. One has from the definition of these A-set: B’(A;,, .;;$k-,) = A:;;;.:‘l.,+k+,~, . It is easy to see that V ,”B’9 tends, n + - 00, to t,, = V 0, B9 formed by vertical lines. This partition generates %. a, is generated by the partition V’L, BY and clearly V ‘2 B’Y is a trivial partition of r into points which generates a. Cells of &, correspond to contracting fibers. The dilating fibers are cells of V Y, BY’; these are horizontal lines of length 1. The time inversion here is the permutation of p and q. Many non-symmetric measures can be constructed. The simplest class is the class of uniform measures concentrated on a dilating fiber q = qO.It is defined as follows on the “cylinders”:
IRREVERSIBILITY
v(/p--~~;~;~)
=
v
OF KOLMOGOROV
-(~‘-~;,y$)v
DYNAMICAL
SYSTEMS
411
+oq,y,+ )
where m 2 0, n > 1 and v’(A’-;;;;;,$)
= l/2”+’ )
v +(A’::,y,$) = 8i,,uI x . . . x 6,“” ) v + is the Dirac measure concentrated
on q. = X~,u,/2”, Ui= 0 or 1. From the definition, ~1%= plpO,therefore p,, = 1 and then the above proposition shows that v ~9 if {Ai}is given by: ii = l/(1 + a’),
a > 1.
This shows that singular measures uniformly concentrated on the dilating fibers for B-systems are of finite entropy i.e. ologpdp<+cc. s r All the above arguments extend easily to arbitrary Bernoulli and Markov shifts with finite partitions showing that these systems are intrinsically irreversible and intrinsically random systems.
Acknowledgements The author wants to thank I. Prigogine, B. Misra from Brussels University and F. Ledrappier from Paris VI University for fruitful discussions and suggestions.
Note added in proof After communicating our results stated in ref. 7 to Tirapegui he gives, together with Martinez18), another proof of the divergence of the entropy for %- and they studied in two K.U.L. preprints some properties of /T and Q(v,, a,).
Appendix. Martingales and proof of theorem 1 We summarize here some results of the martingales theory (for details see Doob16) and Loeve”) leading to the proof of theorem 1. In what follows E(J) denotes the expectation of the functionf. First of all, recall that the conditional expectation of an integrable function f of the measure space (r, a, CL) with respect to a sub-o-algebra a,, is the (projection) operator which associates to f, an a,,-measurable function E%(J) uniquely (a,-a.e.) defined by:
478
M. COURBAGE
(A.11 for all measurable
sets _!?~a,. In case of countable
. .), .P(f)
c1 = (A,, A,,
1 _po(f)(o)
~
P(4)
=
a, generated
by the partition
is given by:
I
f(o) dp, if-4
~(4) > 0,
(A.21
A,
0, Let x,,(o)
ifwEA,.p(A,)=O.
be a sequence
of integrable
random
variables
in a probability
(r, a, p) and suppose that to each x, corresponds a Bore1 sub-a-algebra that x, is Yn-measurable and PROS,, for n < n’. x, is a martingale
x, is a semi-martingale x, < E”+&+,) As an example,
if
.F, - a.e. .
E”n(x, + ,) = x,,
space P”I, such
(A.3)
if (A.4)
. E”n(f)
is a martingale
whenfis
an integrable
random
variable.
Theorem A. 1) if x, is a semi-martingale
and if 4 is a non-decreasing and convex < + CC for all n then 4(x,,) is a semi-martingale. 2) if x;, is a martingale and C#J a continuous convex function with E(Ic$ (x,)1) < + cc, Vn, then C#J (x,) is a semi-martingale. function
with E&5(x,)()
From the definition it follows that, E(x,) is montone non-decreasing for a semi-martingale, and is constant if and only if x, is a martingale. An important application sequence
of martingales of decreasing
9n the sub-p-algebra measure
on (r,
-4~) =
is the theory
finite partitions generated
of derivatives as defined
.
by CL,, 9, c Y*c
a) and define the function
x,(o)
of measures.
in the section
Let a,, be a
of the paper and
Let v be a probability
by:
v(A,(oN ifp(A,(w)) > 0, ~(An(o)
=o
if&$,(w)) = 0,
(A.9
where A,(w) is the cell of c(, containing w. Then x, is a martingale. x, is the derivative of vlgn restricted to ZJ’,, with respect to ~1~“. From the theorem A, it follows that x,(w) logx,(w) is a semi-martingale and E(x, log x,) =
c p(A,) n
More generally, 9E~9-n+,) tending
log -v(An)
2
IS
n
P G%J
let 9,, be an to a as n +cc
increasing (i.e. UF,,
=
dv log <‘c,(o) .
siquence generates
of sub-a-algebras (i.e. a, in this case we write
IRREVERSIBILITY
OF KOLMOGOROV
DYNAMICAL
SYSTEMS
479
S6,/“a), and suppose that vlFi is absolutely continuous with respect to plFi for all i; with density pi, then pi is a martingale. Most important results concern the convergence of the martingales and the semi-martingales. Each positive martingale x, converges a.e. to x,. LL-convergence of x, to x, occurs if and only if x, = E*n(x,) or equivalently, if x, is uniformly integrable. The integrability of sup,, x, implies the uniform integrability of x,. By a theorem of Doob (see ref. 16 chap. VII, th. 3.4) a sufficient condition for the integrability of sup x,, x, > 0 is sup E(x, log+ x,) < + 00 . n
(A4
We turn now to the martingale x, defined by (AS) and suppose that G,/“%. this martingale is uniformly integrable x,, converges -Lb and therefore:
lim x,,dp = x,dp n-+0X s s A
If
(A.7)
A
for all A. Consequently,
s
x,dfi=
if A EF~, then for n > m
1 v(B)=v(A). &Qm
A
(A.9
ECA
This, together with (A.7) implies that:
v(A)=
x,dp s A
and therefore vi,,,,is absolutely continuous w.r.t. ~1%. A sufficient condition for the a.e. convergence of semi-martingales x,, is the existence of a finite upper bound of ,?S(lx,l),then the limit of x,, x,, is integrable. For positive semi-martingales, the above condition together with the Fatou lemma imply that: E(x,) < sup, E(x,). If a semi-martingale x, satisfies to sup, E(lx,l) < + cc and moreover to: X”< E”n(x,) .
(A.9)
Then sup, E(x,) < E(x,). If x, is positive, (A.9) implies that lim,,, E(x,) = E(x,). Finally, let x,, n f - 1, be a semi-martingale. Then, lim x, = x_,, n + - 00, exists a.e. and is equal to Ea--(x-J (see pages 328 and 329 of ref. 16). If lim E(x,), n --, - co is finite, then x, converges Lb to x_, and E(x,)+E(x_,) as n + - co. The proof of the theorem I follows now easily.
480
M. COURBAGE
The positivity
of 52(v, a,,): from the convexity
of the function
4(x)
= x log x we
get:
Let
x,(w)
derivative
be the
martingale
of v with respect
(A.5)
associated
with
the
Radon-Nykodim
to p, then
E(x, log x,) = Q(v, c1,). By 2) of the theorem A of this appendix x, log x, is a semi-martingale, E(x, log x,) is non-decreasing and this proves 1). Let us prove
3) of the theorem
Suppose that lim,,, .Y log x into a positive
then
1.
O(v, a,) < + co with S,,Ya, (or cr,/*;“,,) and decompose part x log+ x and a negative part x log- x then
x, log x, = x, log+ x, + x, log- x, )
(A.10)
where:
logxfurx log+ x(resp.
for
0 From
(A. IO)
we
b l(resp.x
< I),
log- x) = x <
l(resp. x 2 1).
have:
E(x, log+ x,) = Q(v, s) - E(x, log- x,) ) (A.11)
G sz(v, c(,) + 1 for x, log- _r, 2 - 1, and therefore: sup E(x, log+ x,) < + co n
The application of the Doob’s theorem (A.6) implies that E(sup, x,) < + CC and xn is uniformly integrable. Then x,, converges LL to an integrable function which will be denoted by p0 with x, = E”n(p,) and, as shown above, ~1,~is absolutely continuous w.r.t. ~1.~ with density pO. It is now clear that 4(x,,) satisfies the condition sup E#$(x,)l) n
< + co
and by the convergence
see
theorem
(A. 11) of semi-martingales
lim 4(x,,) = $(p,,) is integra-
IRREVERSIBILITY
OF KOLMOGOROV
DYNAMICAL
481
SYSTEMS
ble. Now by the Jensen’s inequality we have: 4 (&I) = 4 (WPO)) < EV$ (PO))
(A.12)
which implies that the positive semi-martingale 4(x,) - l/e satisfies (A.9) and the convergence theorem of positive semi-martingales yields
s
lim W (x,)) = pa1%p. dp . This achieves the proof of 3) of theorem 1. To prove 4) let jr pa(w) log pa(o) dp(o) < + co. Then, taking the expectation value on both sides in (A.12) implies also that Q(v, a~) < + co and the same arguments lead to B(v, a,) = j p. log p. dp. Finally to prove 2), suppose that fi(v, a,,) < + co, and let 9 be an arbitrary finite partition L-S < to. Denote by x2(o) the derivative of vlgwith respect to pls (see AS), then by 3) VI,,,is absolutely continuous with density p. and
s s s E-Q,)
dp =
~0
dp
Q
QE9
=
v(Q)
x.do) dp
=
Q
which implies that: (A.13)
E”(L),) = x.2 *
This relation together with the Jensen’s inequality, likely to (A.12) implies d,(&) G EY$ (PO))3 taking the expectation value on both sides yields Q(v, 2) G
~0 log
~0 dp
s
and this achieves the proof of 2).
I) I. Prigogine, From Being to Becoming (Freeman, San Francisco, 1980). 2) S. Goldstein and 0. Penrose, J. Stat. Phys 24 (1981) 325.
482
3) 4) 5) 6) 7) 8) 9) 10) I I) 12) 13) 14) 15) 16) 17) 18)
M. COURBAGE
0. Penrose, Foundations of Statistical Mechanics (Pergamon Press, Oxford, 1970). 1. Prigogine, C. George, F. Henin and L. Rosenfeld, Chem. Scripta 4 (1973) 5. B. Misra, I. Prigogine and M. Courbage, Physica 98A (1979) 1. B. Mirsa and I. Prigogine, in: Proc. Workshop on Long Time Predictions in Dynamical Systems, eds. Szebehely, Horton and Reich1 (Wiley, New York, 1982). M. Courbage and I. Prigogine, Proc. Nat]. Acad. Sci. USA &I (1983) 2412. K. Goodrich. K. Gustafson and B. Misra, Physica lO2A (1980) 379. B. Misra, Proc. Natl. Acad. Sci. USA 75 (1978) 1627. M. Courbage and B. Misra, Physica 104A (1980) 359. S. Goldstein, B. Misra and M. Courbage, J. Stat. Phys. 25 (1981) I1 I. V.A. Rohlin, Amer. Math. Sot. Transl. 71 (1952). P.S. Pinsker, Information and Information Stability of Random Variables and Processes (Holden-Day, San Francisco, 1964). B. Misra and I. Prigogine, Progr. Theoret. Phys. Suppl. 69 (1980) 101. N. Dunford and M. Schwartz, Linear Operators (Academic Press, New York, London, 1961). J.L. Doob, Stochastic Processes (Wiley, New York, 1953). M. Loeve, Probability Theory (Van Nostrand, New York, 1953). S. Martinez and E. Tirapegui, Phys. Lett. 95A (1983) 143.
North-Holland Publishing Co.
INTRINSIC IRREVERSIBILITY OF KOLMOGOROV DYNAMICAL SYSTEMS M. COURBAGE* Laboratoire de Probabilitb,
Tour 56, 4, Place Jussieu, 75230 Paris Cedex 05, France
Received 21 June 1983
We give the mathematical details and various extensions of the results stated in previous work of Courbage and Prigogine. “Intrinsic random systems” are deterministic and conservative dynamical systems for which we can associate two dissipative Markov processes through a one-to-one “change The of representation”, the first leading to equilibrium for t -++cc and the second for r-+-co. microscopic formulation of the second principle of thermodynamics permits to lift the degeneracy by the exclusion of all states that do not approach equilibrium for t + + cc. The set of admitted initial conditions g+ is then characterized by a non-equilibrium entropy functional which is infinite for rejected initial states and takes finite values for admitted initial conditions. Thus, rejected initial states correspond to an infinite amount of information. To realize this selection rule we consider general probability measures on phase space that are not necessarily absolutely continuous and we extend the theory of transition to Markov processes to such measures. Owing to the non-invariance of s+ under the time inversion, the evolution of these states in the new representation can only be given by one of the two possible Markov processes.
0. Introduction More than ever, the problem of the microscopic interpretation of irreversibility remains in the forefront of physics. This is due to the impact of the recognition of the fundamental role played by irreversible processes in widely different fields such as elementary particles, cosmology and self-organization processes in chemistry and biology’). Moreover, it is of fundamental importance to elucidate the connection between the reversible and deterministic laws of physics and the second law of thermodynamics. According to a widely held view, irreversibility is a result of a “coarse-grained” reduction of the (deterministic) dynamics, and, recently, it has been argued that irreversibility appears when the physical observations realise this reduction in a non-symmetric way with respect to time inversion (see e.g. the works of Goldstein and Penrosez13) and Misra and Prigogine14)). Another approach to the microscopic interpretation of irreversibility due to Prigogine and coworkers’) starts from a different viewpoint. Let us first of all recall * Supported in part by Instituts Intemationaux
037%4371/83/0000-0000/$03.00
de Physique et de Chimie Solvay, Brussels.
0 1983 North-Holland
460
M. COURBAGE
that the functional
s
P,(O) logp,(w) d/do)
9
(0.1)
where p, is a Gibbs distribution function of a system of particles in phase space, remains constant in time. This can be interpreted as a preservation of the “order” or the “information” associated to the state P, under the representation of the dynamics by trajectories. Moreover, the constancy of (0.1) can be interpreted in terms of the existence of two opposite processes the first leading to equilibrium distribution of velocities and the second corresponding to the creation of correlations between particles resulting from their interactions. Boltzmann, by neglecting the second process (for physical reasons) was able to derive his #-theorem. The viewpoint of Prigogine et al. proposes to redefine the distribution function P, by a transition to a new representation. This change of representation is interpreted as a redefinition of units in terms of collective modes for highly complex systems involving also a redefinition of the correlations between units. Under this change of representation the time evolution leads, under a probabilistic process, to the damping of the correlations and to equilibrium. The transition to this new representation should be given by a non-unitary transformation /l:p --+ (&I)(O) = p’(o) in such a way that the functional
s
P”,(o)log i%(o) G(o)
(0.2)
describes the information associated to fir and decreases monotonically towards a minimum in equilibrium. The existence and the construction of such a transformation n leading from a deterministic dynamics the Markov process is not always possible. The dynamical systems where such a transition is possible have been studied by Misra et al.‘) and called “intrinsic random systems”. But we are still faced with a degeneracy: intrinsic random systems may be mapped by two distinct Markov processes the first leading to equilibrium for t -+ + co and the second for t + - co. On the macroscopic level we have also a similar situation. For example, to the Fourier equation for t 2 0 leading to the uniform distribution of heat for t -+ + co corresponds an (anti)-Fourrier equation leading to equilibrium for t + - co. As well known, the Fourier equation for t >, 0 cannot admit solution leading to equilibrium for t < 0 and only a distinct equation may describe such an evolution. Therefore, to lift the degeneracy between the two Markov processes we have to discuss the problem of the intrinsic physical distinction (if any) between the two
IRREVERSIBILITY
OF KOLMOGOROV
DYNAMICAL
SYSTEMS
461
possibilities. The selection between the two possibilities is a result of the restriction of the transformation A to a class of admitted initial states which is not symmetric under the time inversion and which is compatible with dynamics as it is propagated in the future. Moreover, in this work we have shown that the entropy acts as a selector of initial conditions, for, the rejected initial conditions correspond to an infinite entropy (0.2) while admitted initial states correspond to a finite entropy. These concepts have been introduced in our paper without proofs. Some of the theorems concerning non-equilibrium entropy and the extension of n have only been stated. For the readers interested in this problem we shall now give the mathematical proofs. But first, to make the presentation self-contained we shall recall the mathematical formulation of the basic concepts previously introduced.
1. Mathematical formulation of the basic concepts An abstract classical dynamical system is a one parameter group of transformations S, acting on o-points of the phase space r and leaving invariant some distinguished measure p on r. This dynamical system can equivalently be represented by a group of unitary operators U, on Li preserving the set of positive functions, U, is related to S, by the Koopman lemma:
P(~)-w,P)(~) = P(StQJ) *
(1.1)
U, can also be defined by (1 .l) as an operator on LL. On the other hand, a stochastic Markov process on the phase space r, preserving p, is given by transition probability P(t, co, E) from o E r to an element E of the a-algebra a of measureable sets in time t. As well known, to any Markov process one can associate a semi-group of operators on Li:
f-W’J-)(a)
=
f WW,
0, do’),
s r
(1.2)
W, is a semi-group of contracting operators on L: called a semi-group of a Markov process. This Markov process defines a group of automorphisms on the space of all probability measures on (r, a) given by: v -+( w’:v)(E)
=
s
P(t, co, E) dv(o) .
E
(1.3)
462
M. COURBAGE
It is easy to see that if v is absolutely continuous w.r.t. p with square integrable density @, then @TV is also absolutely continuous with a density b, given by pl,= VP, where WY
t-cc
for
all probability distributions MEL:, p # 1. In other words, intrinsic random dynamical systems are systems for which the deterministic dynamics for t b 0 can be considered as “equivalent” to a Markov process, for /i can be interpreted as a change of representation. The conceptual relation between physical irreversible processes and the semi-group W: consists in 1) the construction of an H-function (see property vi) which implies that -
1p”,1s p”,dp
(1.5)
J r
increases monotonically to a maximum value, zero, only reached by the function 1 and 2) the non-invariance of W: by the time inversion, that is, W: cannot be extended for t ,< 0 to a semi-group oja Markovprocess (in fact, AU, A -’ cannot be positivity preserving for t < 0, see refs. 5 and 8). This second point means in particular that one should associate by a couple of transformations A and A’ a couple of semi-groups of Markov process WY = AU,A-’ and W;* = A’U,A’-’ for t > 0 and t < 0, respectively, the first approaching the equilibrium when t + + cc monotonically (in the sense of vi) and the second for t + - co. This scheme corresponds to a well known irreversible process like heat conduction etc. where the time inversion t + - t corresponds to the passage from one process to a distinct one. The interesting question of a necessary condition that should be satisfied by a dynamical system in order to be intrinsically random has been discussed by Misra’) who showed that the existence of a Liapounov function (property vi) implies that the system is mixing. As expected, the motion of the system should be highly unstable. In fact it has been shown that all B-systems are intrinsically random5~‘0) and these results have been extended to all K-systems”).
IRREVERSIBILITY
OF KOLMOGOROV
DYNAMICAL
SYSTEMS
463
Nevertheless, the time symmetry breaking of the dynamical group U, into WY and W;* for t 2 0 and t < 0, respectively,
having finite entropy represents the set of admissible states. Now a sysem is intrinsically irreversible if there exists a subset of physically admissible initial states CP tending to equilibrium in future but not in the past. As a consequence, the velocity inverse Vg- of the states % + cannot tend to equilibrium in the future and are therefore exluded by the microscopic formulation of the second principle. It is clear that for mixing systems such a requirement implies that one should go beyond L$ for L:-states tend to equilibrium for both t--r + co and t + - cc. This is also true for all absolutely continuous states. For mixing systems one should consider states which are not absolutely continuous. In the case of unstable hyperbolic systems, it is possible to consider states with broken time symmetry. Examples are measures concentrated on dilating fibers and contracting fibers. In physical models like the Lorentz gas dilating and contracting fibers may have physical interpretation. In the Lorentz gas for example, measures concentrated on dilating fibers correspond to an ensemble of particles which have formed a parallel beam of particles in the distant past before undergoing collisions, while contracting fibers correspond to a set of particles which will be parallel after executing many collisions. The selection rule expresses the unrealizability of experiences in which a set of particles that undergo several collisions will asymptotically emerge with parallel velocity.
Whenever the arrow of time is given by %+, the second principle become a selection principle which eliminates non-admissible initial states. The entropy
M. COURBAGE
464
functional plays the role of selector of initial states displaying the amount of information necessary for their preparation, and the admissible states are all states with$nite entropy. This set, that will be denoted 9+, has a time broken symmetry as it contains the V+ set. The initial conditions selected by the second law correspond to a finite amount of information while the initial conditions rejected correspond to an infinite amount of information. In this note, we show that such an entropy can be constructed for unstable systems of K-type, where the class GP is the set of probability measures uniformly concentrated on a dilating fiber. This entropy is a Liapounov function over the set of states. Whenever the set of initial states 9?+ is selected we show that there exists an afine transformation on this set under which the reversible deterministic evolution of these states is transformed into an irreversible Markovian evolution. This transformation extends the operator constructed in refs. 5, 10 and 11 for L:-states. The non-invariance of .9+ under the time inversion lifts the degeneracy between @‘: for t 2 0 and tit:* for t < 0 as the set 9- on which act ,4 ’ and @I: contains the set V- which has been rejected by the selection rule. Consequently, the K-systems are intrinsically random and intrinsically irreversible when the invariant measure p represents equilibrium state.
2. Intrinsic irreversibility of Kolmogorov dynamic systems First we recall the definition of intrinsic irreversibility of dynamical systems introduced in ref. 7. An abstract dynamical system is a Lebesgue space12) (r, a, ,u), where p is a normalized positive measure on the Bore1 o-algebra a of measurable subsets of f, on which acts a group of measure preserving transformations S,, f E R. For simplicity, we suppose that r is compact. The system is reversible if it has a time velocity inversion, I, introduced in section 1, also called a time inversion transformation. This transformation defines the operator on the space of measures:
v-+vv=v’:v’(E)=v(ZE).
(2.1)
All dynamical systems considered below will be of this type. The states of the system are all probability measures on (r, a) and we denote this set by Y. S, induces on Y a dynamical automorphisms group a, = v E~+U,V E Y (ct,V)(E) = v,(E) = V(S_,E) .
(2.2)
Let us define the approach to equilibrium of a state v. We say that v, approaches
IRREVERSIBILITY
OF KOLMOGOROV DYNAMICAL SYSTEMS
the equilibrium as t + + co if v,+p in the W*-topology
ji
dvt-
465
i.e. if
jt?-cb
(2.3)
r
r
for all continuous functions J More precisely, a reversible dynamic system is called intrinsically irreversible and intrinsically random if: 1) There exists a set of initial states W CY that is affine and not invariant by the time inversion V, i.e. .9+ contains a subset V+ of probability measures, such that its time inverse W cannot tend to p as t--f + co, i.e. Vv E YV+, vr+p, t-++CO.
2) c@+ C 9+, Vt. 3) VVE~+, v,+p, ast-++oo. 4) There exists an affine transformation /1”on W that realizes an “equivalence” between 01,and some irreversible Markov process in the following sense: i) /i maps a state v into a state, denoted G, and /ip = p. ii) /1”is one-to-one. iii) Aa,v = FP:fTv for any v E9+, where @‘: is a semi-group of automorphisms of a Markov process associated to a transition probability P(t, co, E) i.e.
dv”(o)P(t, CO,E) .
(I@‘:+)(E) =
(2.4)
s
r
iv) v”is absolutely continuous w.r.t. p with a density fiOsuch that
(2.5)
v) xv, is absolutely continuous with a density p”,verifying an &‘-theorem that is,
s
Q(v,)= Ptlogp”,dp
(2.6)
r
decreases monotonically
and tends to zero (= G(p)) as t + + CO.
A set of initial states Qt satisfying the above conditions will be called a set of admissible initial states.
466
M. COURBAGE
Clearly, the property 1) asserts that the class %+ allows to display the arrow of time. This property is not a consequence of the dynamics, it represents a supplementary property of the physical system. Evidently G!Z+does not determine the class of states 9+ physically admissible, this class of states should be characterized by the “amount of information” relatively to equilibrium and necessary for their realization. Such a functionnal of states should satisfy to the conditions: 1) to be finite for %+ but infinite for V, 2) to tend monotonically to zero when the state tend to equilibrium. We present here the construction of such /i’ for K-systems. We shall show that this construction coincides with the operator II previously constructed for densities in L:. We characterize the set 59’ and we define 1 for all states in LB+. We recall that the “canonical” method to construct the operator /1 on Lz permits to associate with the unitary group U, (1.1) two distinct Markov processes for t > 0 and t < 0. We will not recall this method which introduces II as a function of an internal time operator5). All proofs of this property for K-systems can be found in ref. 11. Here we present directly 1 and show that all states of 9!+ transformed by /1”,evolve under a Markov process. This shows, when the dilating fibers are chosen as a class %?+, that a K-system is intrinsically random and intrinsically irreversible. We recall first of all, that a K-system is a dynamic system having a distinguished sub-a-algebra a,, such that (1) S,a, = a,ca,, if t
IRREVERSIBILITY
OF KOLMOGOROV
DYNAMICAL
SYSTEMS
467
(2.7)
We recall that a sequence of finite partitions is said decreasing if 9,+, is a refinement of Y’,(Y,,+, > 9,,). It is well known’*) that if (r, a,,, p) is a Lebesque (separable) measure space, there exists a decreasing sequence of finite partitions 9, which tend to the partition to that generates q. We have the following theorem: Theorem 1 (with the above notations)
1) sZ(v, 9,) is a positive and non-decreasing Q(v, a,) as n+co:
function of n, which tends to
Q(v, a,) = lim Q(v, Y’,) .
(2.8)
n-cc
2) WV, a01= ;yg WV, 2) .
(2.9)
).
3) If sZ(v, a,) < + co, then the measure vlOO restriction of v to a, is absolutely continuous with respect to p. In this case:
WV, a,> =
s
p.log p. +
(2.10)
,
r
where p. is the Radon-Nykodim density of vlaOwith respect to p. 4) Conversely, if vl%is absolutely continuous w.r.t. p with density p. and if { p. log p. dp < + 00 then Q(v, eo) = jr p. log PO GL. This theorem follows straightforwardly from the Martingales theory. We recall in the appendix some basic results of this theory and the proof of theorem l*. In the case of distributions vp with density p satisfying l p2 dp < + co Q(v,, ao) is identical to the entropy introduced in the work of Misra and Prigogine14)
WV,, a,> =
Pop
(2.11)
log Pop dm ,
where POis the projection onto Li(a,)c * We have learned from F. Ledrappier by Perez, see ref. 13.
Lt. This results from the above theorem.
that this theorem
has been proved
by Dobrushin
and later
M. COLJRBAGE
468
A functionnal such as (2.11) has been also considered by Goldstein and Penrose’) as a model of entropy for absolutely continuous measures v,. Theorem 2. Let v be an initial probability
1) a(~,. a,) decreases monotonically t-*+a, 2) VI--+/&t++cQ.
measure such that sZ(v, a,J < + cc, then: with t and tends to Q(p, a,,) = 0, when
1) From the theorem I, ~1,~is absolutely continuous with density po. For t > 0, a_, c a, and VI,_,is absolutely continuous with density E” i(p,J. (Definition and properties of the conditional expectation E”f are given in the appendix.) E”~~(p,) is a martingale which converges Lh to 1 and E” ‘(p,,) log E”~ r(pJ is a semi-martingale which converges LL to 0. Now, from (2.7), and 4) of theorem 1 we get:
Proof.
Q(v,, P’,) = Q(v, S-,.9’,)
1
(2.12)
lim 52(v, S-,9,> = Q(v, S-,a,), n>7 and therefore Q(v,, a”) = WV, S-,a,) = j E”ml(p,)log E”m(p,)dp \
0.
2) let A ~a,. For I sufficiently great S-J v,(A) = v(S,A) =
t-cc
.
E a,, then
s
~(1dp
.S-,A
(2.13)
here qE is the characteristic
function of a set E.
Any mixing system, S,, has the following property:
s
.f(~k(S,o) d/do) -
r
gh. r
(2.14)
I
if f6 Lh and g is a bounded measurable function Putting in (2.14) g = qA and f = p,, we get: v,(A)-
cl(A),
r-r+%’
VA Ea,.
The same property extends easily for any A E a, by c/3 approximation
argument.
IRREVERSIBILITY
OF KOLMOGOROV
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469
SYSTEMS
Although Q(v, a,J can be a selector of initial conditions it cannot represent the total information of the state v relatively to I(. It corresponds only to the information of vlaOrelatively to pl%, and, for all states such that v(,,~= pl(.,, Q(v, a,,) = 0. However, such states are not in equilibrium and their information with respect to finer sub-c-algebras can be non-vanishing. Then we shall take into account the whole information with respect to all ai, sZ(v, a,). Such an entropy is the result of a transformation /1 of v into 5 given on a and not contracted to a sub-g-algebra. Under this “change of representation” a, is transformed into a Markov process with an &‘-theorem associated to states v having finite relative information Q(v, ai) with respect to all ai. We want now to define the transformation 1 and its domain 9+. This will be presented as a result of four propositions. 1. Let 6i be a sequence of positive numbers summing to 1: X:_+zai = 1 and let v be a state such that vi,,; is absolutely continuous w.r.t. u/08 with density pi then C 6,p, converges, a.e. and L:, to an integrable function p”such that jr p”dp = 1. Proof. By the Beppo-Levy
theorem jSN= P,
Applying again the Beppo-Levy and
6ipi is an integrable function and
theorem to pN, it follows that &-+N_m
p”a.e.
2. Let v ~9 be such that VJ,,~ is absolutely continuous w.r.t. p with density pi and let p = CZE Gipi, then v,I,~is also absolutely continuous with respect to ula, with density P,,~= E”$I,p, (see (1.1) pie Lb). Proof. For any A Ea,, S_,A Ea,_, c a,, for t > 0 and
v,,,(A) = v(S_,A) = s
pi(O) dp(o) S_,A
=
I
S_,A
this proves 2.
Pi(O)
dP(Srm)=
s
Pi(S-rO)
A
dP(m) =
s
Eai(Pi(S-t(m)))
A
dP
5
M. COURBAGE
470
3. Let us denote 1, = Xp”=, 6, and suppose that the sequence A,+,/;l, is decreasing with n. The set 9+ will be the set of all measures v such that: 1) SZ(v,a,)< +cc 2)
foralli=O,
+l,..
.,
(2.15) (2.16)
,i?logp”dp < +cc. s
We introduce now the ;i transformation as the mapping which sends a measure v E 9,+ into v”given by dG = p”d,u. It is clear that /i” is an afine transformation on W which maps a state into a state. To show that 1 is one-to-one let v and v’ be two measures from .!3+ with /Iv = Av’, we have, by using the properties of the conditional expectation E”n,the following relation for the density p”(and similarily for b’): (2.17) and therefore:
J++G) - EYS) = k+,(~n+,- P,)
’
(2.18)
Since c’ = 6, it follows from (2.18) that (2.19)
P”-Pm=P;-PAl,
for any m < n. Taking in (2.19) the limit m -+ -CC and using that p-X = pL * = 1 we have pn = p; for any n and therefore v = v’. We have to show now that Av, evolves under a Markov process. In previous works5,‘0s”), the “equivalence” between an irreversible stochastic process and the unitary group U,:p+(U,p)(o) = p(S_,o) has been constructed for all square integrable distribution functions p(o) via a bounded operator A on L: given by A = 1 sip, ) Pip = E”l(p),
(2.20) p EL; .
We have shown that the family A U,A -I, t 2 0, is a family of bounded densely defined operators which extends to a semi-group of a Markov process with a transition probability given by: (2.21)
p(t, 0, E) = (A -‘U,&,)(w), where CYJI~ is the characteristic
function of the subset E E a.
Remark. In ref. 11, it has been shown that AU,A -’ is a.e. positivity preserving as an operator on Li. This proof can be extended to show that A-‘U_,A, t 3 0 is positivity preserving as an operator on the space of all bounded
IRREVERSIBILITY
OF KOLMOGOROV
DYNAMICAL
SYSTEMS
471
measurable functions B(T, a). For this one has to use the existence of a regular conditional distribution in the Lebesgue measure spaces, that is to say, the existence of ai-measurable function o +P(A ( a,)(o) which is a probability on (r, a) as a set function A +P(A ) ai)( Vo, and equal ai-almost everywhere to E”~(cp,)(o) (see ref. 17). The construction of A, above presented, can be seen more generally if we do not require v”to be absolutely continuous. To do this, consider A defined by (2.20) as a bounded operator on the set B(r, a) of all bounded measurable functions f(w), that is a Banach space with respect to the norm ((f 11 m = sup, Al. It is easy to see from (2.20) that the set function 1(E, o) given by: (2.22)
WY 0) = (&3(o)
is a probability measure on a for all o. This follows from the regularity of the conditional distribution (E”icpJ(o) = P(E ) a,,)(o ) a.e. Then, ;i may be defined as an operator on the space of all o-additive set functions u (r, a) by:
s
s
(@(E) = W&(o) dv(o) = W, m) d/-do). r
(2.23)
r
One verifies that /i defines a contraction operator on the space v(T, a) of all a-additive set functions with the usual norm given by the total variationls). Let us denote by xv = v”.As well known, an arbitrary measure v EV(~, a) defines a functional on B(T, a) by: v(j)= J
r
fdv.
r
Then, by standard arguments, we have, as a result of (2.23) the following relation for v”and f E B(T, a): (W(f)
= q(f)
(2.24)
= v(Af) *
Similarly we have: (v)(f)
= VU-,f
),
feB(r,
4,
(2.25)
from these relations it follows that (;l”ov)(E) = v(U_&&) =
= Av(n
P(t, o, E) dv”.
(2.26)
412
M. COURBAGE
Let us denote by @: the operator (@W(E)
on v(E, B)
P(t, o, E) dv"(o) ,
=
(2.27)
s I-
from (2.26) we have:
Remark. The operator
n defined by (2.23) is one-to-one if /1, as an operator on B(T, a) contains in its range all characteristic functions of a generator of a. This is a direct consequence of (2.24). Such a property is not generally true but depends on the version of the conditional probability which defines /i on B(T, a). However, the restriction of ;i to 9+ does not depend on the version of E”(cp,)(w) and is invertible so that its inverse can map an absolutely continuous measure into a singular one. 4. The X-theorem follows nowfrom thefact that p”,evolves under a Markov process,
then
s
p”Ilog p”,dp d
I
p”l,log p”t,dp s r
for t 6 t’. As a result it follows that sp”, log p”ldp < + cc for Jr p”log p”dp < + GOi.e. V,E~ + if v E 9+. This proves the property 2. Let us finally show that
s
t -+ + co
p”tlog p”,dp -0,
I-
for any v&9+. Let us denote log+ x = (log x)+, then
s
P, log P”,dp G
p”,log+ P”,dp t s
using the relation E”g+iU,p = U,E”p, we have p, =
c 6,EW,pi I
= ulC6iPi-*= I
= c +!J,E”~ -‘pi 1 UtCal+*Pi,
,
t> 0
if
IRREVERSIBILITY
OF KOLMOGOROV
DYNAMICAL
SYSTEMS
473
therefore:
On the other hand CiSi+,pi is a.e. decreasing with respect to t for
C6i+,pi= 1+CA,+,(Eai-Eoi-‘)pi\
1,
t+CQ.
Then the application of the Beppo-Levy theorem for the limit t +a~ and the monotonicity of x log+ x yields the result,
3. Applications and examples In the preceding section we have characterized a set of “physical admissible states” for K-systems as the set of all probability measures v having a finite “amount of information” i.e. 1) Q(v,aJ< 2)
+co
foralli,
ijlog@dp<+co. s r
In general the finiteness of Q(v, q,) does not imply the finiteness of G?(v,aJ for all i. However, this is true for B-systems when a, is generated by independent$nite partition i.e. if a, is a-algebra generated by to = V?, B’S’, B= (PI,. . . , Pk) *. To see this, we may write C!(v, V’!_,B’9’) as the expectation with respect to v of the function log v (A,(o))/p (A,(o)), wh ere A,(o) is the cell of V’?,, B’s9 containing 0: s2(v, V o-, B’9) =
dv(w) log s
M”(W)) &4(w)).
(3.1)
r
Similarily we have for Q(v, V !, B’Y): s2(v, V !, B’9) =
dv(w) log s
r
v(W%w)n4(w)) P((B~)(~W,@))
’
* Recall that the product 9 V 9 of two partitions 9 and 9 is the partition that has elements
(3.2)
P, n Q,.
M. COURBAGE
414
Then O(v, V ‘, B'S) - C’(v, V
0, B'9') =
dv(o) log s
r
s
dv log PLWY~)
< -
v((Bg)(o)
1A,(o))
PLUW(~)
)4(w))
(A(w)
r
dv log p(P(o)> < -sup log P(~i).
=
(3.3)
s I-
A similar result holds also for Markov shifts with finite partition. On the other hand, to satisfy condition 2) it is sufficient to insure the square integrability of b,,. This is a result of the following inequality,
r
r
us check
SN =
It
,“y,
$S#i
1+ i
AiEipN AN+,PN
easy to see that,
Ng, (ni -
~N+I)J%'N +
&+I - lN+h"~
(3.6)
IRREVERSIBILITY
OF KOLMOGOROV
DYNAMICAL
SYSTEMS
415
As EipN = 0 Vi > N’, EiEj = 6,Ei we get: llWSW(I*=(&+I
-~.+,)211prl(‘+~~~(~i-i.,,,Yl14P,I12.
(3.7)
By using now the following inequalities <
Ai-AN+l IIEiPi/*
=
Vi < N + 1 ,
1,
lIPill
-
lIPi-1112
<
[(Pill2
9
(3.7) yields:
From this estimate, the L:-convergence test i.e. if
Ent,l
)IPi+111*
i>O
I(
2:
<
may be insured by using the D’Alembert
1
’
or equivalently if
As expected, this result relates the convergence of sN to the choice of the sequence li. For &+,/&+O, i-roe, all measures v such that supi(Ipi+,)(*/ /IpiI/’ < + CO will belong to Q+ (e.g. & = e-“).
Let us summarize this discussion: If (r, B”, a, p) is a K-system, if viaiis absolutely continuous with respect to plai, Vi, with square integrable density pi, i.e., 2 ={ p:dp IIP,41
< +co
and if ;i is constructed as in the proposition 3 of section 2, then v EL@+ if
To get an estimate of the ratio (lpi+, ~~2/IIPi~(2 for a B-system with finite partition we have:
M. COURBAGE
476
where A,,(w) is the cell of V:, Bk9 containing o, n, i > 0. Therefore: v((B~)(o)N,,i(o)) pf+, dp = lim dv nF+cO ~((B~)(~)nA,i(w)) s = lim dv vG%,i(~)) v(W’)(o) n-zc i
dmax’
P((A~,~(~)) P(B~)(~)
/4(o)) 1An,i(~))
1 ,w;> s
pfdp.
We have the following result for any B-system with $nite partition. If vl,0is absolutely continuous w.r.t. v with density p,, such that s pi dp < + co then J p f d,u < + co for all i. If moreover (3.8) Let us consider as an example the baker transformation which is isomorphic to the Bernoulli shift (l/2, l/2). Recall that B acts on the unit square (p . q) in squeezing in the direction of q and stretching in the direction of p and stacking the right half on the left half, more precisely B(P, 4) = (2~9q/2) 0 6~ < l/2 > =(2pl,q/2+ l/2) 1/2dp
< 1.
The a-algebra of the Bore1 sets is generated by the partition 9 = (PO, P,) and P, = {(p, q) 1P 2 l/2}. That is, let where PO= {(P>4) IP < l/2) B”9 = (B”P,,, B”P,) and let V,“+k-’ B’Y the partition of r whose elements are: A 2$J;;, ‘k ,,n+k_, E B”Pj,nB(“+‘)Pi,n . . . ~IB(“+~~‘)P~~. The set of finite unions of these rectangles which are usually called “-cylinders” form an algebra that generates the o-algebra of a Bore1 set of the unit square. A a-additive measure is uniquely defined by its values for the cylinders. The Lebesgue measure is given by: P(A;;..,‘:;:,-,) = l/2k. One has from the definition of these A-set: B’(A;,, .;;$k-,) = A:;;;.:‘l.,+k+,~, . It is easy to see that V ,”B’9 tends, n + - 00, to t,, = V 0, B9 formed by vertical lines. This partition generates %. a, is generated by the partition V’L, BY and clearly V ‘2 B’Y is a trivial partition of r into points which generates a. Cells of &, correspond to contracting fibers. The dilating fibers are cells of V Y, BY’; these are horizontal lines of length 1. The time inversion here is the permutation of p and q. Many non-symmetric measures can be constructed. The simplest class is the class of uniform measures concentrated on a dilating fiber q = qO.It is defined as follows on the “cylinders”:
IRREVERSIBILITY
v(/p--~~;~;~)
=
v
OF KOLMOGOROV
-(~‘-~;,y$)v
DYNAMICAL
SYSTEMS
411
+oq,y,+ )
where m 2 0, n > 1 and v’(A’-;;;;;,$)
= l/2”+’ )
v +(A’::,y,$) = 8i,,uI x . . . x 6,“” ) v + is the Dirac measure concentrated
on q. = X~,u,/2”, Ui= 0 or 1. From the definition, ~1%= plpO,therefore p,, = 1 and then the above proposition shows that v ~9 if {Ai}is given by: ii = l/(1 + a’),
a > 1.
This shows that singular measures uniformly concentrated on the dilating fibers for B-systems are of finite entropy i.e. ologpdp<+cc. s r All the above arguments extend easily to arbitrary Bernoulli and Markov shifts with finite partitions showing that these systems are intrinsically irreversible and intrinsically random systems.
Acknowledgements The author wants to thank I. Prigogine, B. Misra from Brussels University and F. Ledrappier from Paris VI University for fruitful discussions and suggestions.
Note added in proof After communicating our results stated in ref. 7 to Tirapegui he gives, together with Martinez18), another proof of the divergence of the entropy for %- and they studied in two K.U.L. preprints some properties of /T and Q(v,, a,).
Appendix. Martingales and proof of theorem 1 We summarize here some results of the martingales theory (for details see Doob16) and Loeve”) leading to the proof of theorem 1. In what follows E(J) denotes the expectation of the functionf. First of all, recall that the conditional expectation of an integrable function f of the measure space (r, a, CL) with respect to a sub-o-algebra a,, is the (projection) operator which associates to f, an a,,-measurable function E%(J) uniquely (a,-a.e.) defined by:
478
M. COURBAGE
(A.11 for all measurable
sets _!?~a,. In case of countable
. .), .P(f)
c1 = (A,, A,,
1 _po(f)(o)
~
P(4)
=
a, generated
by the partition
is given by:
I
f(o) dp, if-4
~(4) > 0,
(A.21
A,
0, Let x,,(o)
ifwEA,.p(A,)=O.
be a sequence
of integrable
random
variables
in a probability
(r, a, p) and suppose that to each x, corresponds a Bore1 sub-a-algebra that x, is Yn-measurable and PROS,, for n < n’. x, is a martingale
x, is a semi-martingale x, < E”+&+,) As an example,
if
.F, - a.e. .
E”n(x, + ,) = x,,
space P”I, such
(A.3)
if (A.4)
. E”n(f)
is a martingale
whenfis
an integrable
random
variable.
Theorem A. 1) if x, is a semi-martingale
and if 4 is a non-decreasing and convex < + CC for all n then 4(x,,) is a semi-martingale. 2) if x;, is a martingale and C#J a continuous convex function with E(Ic$ (x,)1) < + cc, Vn, then C#J (x,) is a semi-martingale. function
with E&5(x,)()
From the definition it follows that, E(x,) is montone non-decreasing for a semi-martingale, and is constant if and only if x, is a martingale. An important application sequence
of martingales of decreasing
9n the sub-p-algebra measure
on (r,
-4~) =
is the theory
finite partitions generated
of derivatives as defined
.
by CL,, 9, c Y*c
a) and define the function
x,(o)
of measures.
in the section
Let a,, be a
of the paper and
Let v be a probability
by:
v(A,(oN ifp(A,(w)) > 0, ~(An(o)
=o
if&$,(w)) = 0,
(A.9
where A,(w) is the cell of c(, containing w. Then x, is a martingale. x, is the derivative of vlgn restricted to ZJ’,, with respect to ~1~“. From the theorem A, it follows that x,(w) logx,(w) is a semi-martingale and E(x, log x,) =
c p(A,) n
More generally, 9E~9-n+,) tending
log -v(An)
2
IS
n
P G%J
let 9,, be an to a as n +cc
increasing (i.e. UF,,
=
dv log <‘c,(o) .
siquence generates
of sub-a-algebras (i.e. a, in this case we write
IRREVERSIBILITY
OF KOLMOGOROV
DYNAMICAL
SYSTEMS
479
S6,/“a), and suppose that vlFi is absolutely continuous with respect to plFi for all i; with density pi, then pi is a martingale. Most important results concern the convergence of the martingales and the semi-martingales. Each positive martingale x, converges a.e. to x,. LL-convergence of x, to x, occurs if and only if x, = E*n(x,) or equivalently, if x, is uniformly integrable. The integrability of sup,, x, implies the uniform integrability of x,. By a theorem of Doob (see ref. 16 chap. VII, th. 3.4) a sufficient condition for the integrability of sup x,, x, > 0 is sup E(x, log+ x,) < + 00 . n
(A4
We turn now to the martingale x, defined by (AS) and suppose that G,/“%. this martingale is uniformly integrable x,, converges -Lb and therefore:
lim x,,dp = x,dp n-+0X s s A
If
(A.7)
A
for all A. Consequently,
s
x,dfi=
if A EF~, then for n > m
1 v(B)=v(A). &Qm
A
(A.9
ECA
This, together with (A.7) implies that:
v(A)=
x,dp s A
and therefore vi,,,,is absolutely continuous w.r.t. ~1%. A sufficient condition for the a.e. convergence of semi-martingales x,, is the existence of a finite upper bound of ,?S(lx,l),then the limit of x,, x,, is integrable. For positive semi-martingales, the above condition together with the Fatou lemma imply that: E(x,) < sup, E(x,). If a semi-martingale x, satisfies to sup, E(lx,l) < + cc and moreover to: X”< E”n(x,) .
(A.9)
Then sup, E(x,) < E(x,). If x, is positive, (A.9) implies that lim,,, E(x,) = E(x,). Finally, let x,, n f - 1, be a semi-martingale. Then, lim x, = x_,, n + - 00, exists a.e. and is equal to Ea--(x-J (see pages 328 and 329 of ref. 16). If lim E(x,), n --, - co is finite, then x, converges Lb to x_, and E(x,)+E(x_,) as n + - co. The proof of the theorem I follows now easily.
480
M. COURBAGE
The positivity
of 52(v, a,,): from the convexity
of the function
4(x)
= x log x we
get:
Let
x,(w)
derivative
be the
martingale
of v with respect
(A.5)
associated
with
the
Radon-Nykodim
to p, then
E(x, log x,) = Q(v, c1,). By 2) of the theorem A of this appendix x, log x, is a semi-martingale, E(x, log x,) is non-decreasing and this proves 1). Let us prove
3) of the theorem
Suppose that lim,,, .Y log x into a positive
then
1.
O(v, a,) < + co with S,,Ya, (or cr,/*;“,,) and decompose part x log+ x and a negative part x log- x then
x, log x, = x, log+ x, + x, log- x, )
(A.10)
where:
logxfurx log+ x(resp.
for
0 From
(A. IO)
we
b l(resp.x
< I),
log- x) = x <
l(resp. x 2 1).
have:
E(x, log+ x,) = Q(v, s) - E(x, log- x,) ) (A.11)
G sz(v, c(,) + 1 for x, log- _r, 2 - 1, and therefore: sup E(x, log+ x,) < + co n
The application of the Doob’s theorem (A.6) implies that E(sup, x,) < + CC and xn is uniformly integrable. Then x,, converges LL to an integrable function which will be denoted by p0 with x, = E”n(p,) and, as shown above, ~1,~is absolutely continuous w.r.t. ~1.~ with density pO. It is now clear that 4(x,,) satisfies the condition sup E#$(x,)l) n
< + co
and by the convergence
see
theorem
(A. 11) of semi-martingales
lim 4(x,,) = $(p,,) is integra-
IRREVERSIBILITY
OF KOLMOGOROV
DYNAMICAL
481
SYSTEMS
ble. Now by the Jensen’s inequality we have: 4 (&I) = 4 (WPO)) < EV$ (PO))
(A.12)
which implies that the positive semi-martingale 4(x,) - l/e satisfies (A.9) and the convergence theorem of positive semi-martingales yields
s
lim W (x,)) = pa1%p. dp . This achieves the proof of 3) of theorem 1. To prove 4) let jr pa(w) log pa(o) dp(o) < + co. Then, taking the expectation value on both sides in (A.12) implies also that Q(v, a~) < + co and the same arguments lead to B(v, a,) = j p. log p. dp. Finally to prove 2), suppose that fi(v, a,,) < + co, and let 9 be an arbitrary finite partition L-S < to. Denote by x2(o) the derivative of vlgwith respect to pls (see AS), then by 3) VI,,,is absolutely continuous with density p. and
s s s E-Q,)
dp =
~0
dp
Q
QE9
=
v(Q)
x.do) dp
=
Q
which implies that: (A.13)
E”(L),) = x.2 *
This relation together with the Jensen’s inequality, likely to (A.12) implies d,(&) G EY$ (PO))3 taking the expectation value on both sides yields Q(v, 2) G
~0 log
~0 dp
s
and this achieves the proof of 2).
I) I. Prigogine, From Being to Becoming (Freeman, San Francisco, 1980). 2) S. Goldstein and 0. Penrose, J. Stat. Phys 24 (1981) 325.
482
3) 4) 5) 6) 7) 8) 9) 10) I I) 12) 13) 14) 15) 16) 17) 18)
M. COURBAGE
0. Penrose, Foundations of Statistical Mechanics (Pergamon Press, Oxford, 1970). 1. Prigogine, C. George, F. Henin and L. Rosenfeld, Chem. Scripta 4 (1973) 5. B. Misra, I. Prigogine and M. Courbage, Physica 98A (1979) 1. B. Mirsa and I. Prigogine, in: Proc. Workshop on Long Time Predictions in Dynamical Systems, eds. Szebehely, Horton and Reich1 (Wiley, New York, 1982). M. Courbage and I. Prigogine, Proc. Nat]. Acad. Sci. USA &I (1983) 2412. K. Goodrich. K. Gustafson and B. Misra, Physica lO2A (1980) 379. B. Misra, Proc. Natl. Acad. Sci. USA 75 (1978) 1627. M. Courbage and B. Misra, Physica 104A (1980) 359. S. Goldstein, B. Misra and M. Courbage, J. Stat. Phys. 25 (1981) I1 I. V.A. Rohlin, Amer. Math. Sot. Transl. 71 (1952). P.S. Pinsker, Information and Information Stability of Random Variables and Processes (Holden-Day, San Francisco, 1964). B. Misra and I. Prigogine, Progr. Theoret. Phys. Suppl. 69 (1980) 101. N. Dunford and M. Schwartz, Linear Operators (Academic Press, New York, London, 1961). J.L. Doob, Stochastic Processes (Wiley, New York, 1953). M. Loeve, Probability Theory (Van Nostrand, New York, 1953). S. Martinez and E. Tirapegui, Phys. Lett. 95A (1983) 143.