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Remarks on nonequilibrium correlations in a simple dynamic model
Volume ll0A, number 2
PHYSICS LETTERS
8 July 1985
REMARKS ON NONEQUILIBRIUM CORRELATIONS IN A SIMPLE DYNAMIC MODEL M. C O U R B A G E Laboratoire de Probabilit~s, Universit~ Paris 6, Tour 56, 4 Place Jussieu, 75230 Paris Cedex 05, France
Received 6 February 1985; accepted in revised form 29 April 1985
We study a deterministic conservativedynamic model inspired by the Kac ring model. We show that some initial probability distributions with long range correlation, which go to equilibrium asymptotically,can be transformed, through a transition to new dissipative and stochastic dynamics, into states with damped correlations.
1. Introduction. One of the main problems of the
statistical description of non-equilibrium phenomena is the selection of suitable initial non.stationary ensambles. B0goliubov, in his famous work of 1946 [1 ], postulated for a classical system of N interacting particles that non-stationary ensembles should satisfy a principle of weakening correlations between coordinates of the particles under time evolution. The system being described by the hamiltonian dynamics, the time evolution of the ensemble P t(P l .... P n , q l ..... qn ) is given by the Liouville equation. Such a principle is a constraint for the selection of initial distributions 00(/71 ..... Pn, q l ..... qn). The need for it has come out from discussions on the validity of the Boltzmann equation. As is well known, the hypothesis of the molecular chaos which leads to the Boltzmann equation corresponds to a distribution with no correlations between velocities before collisions, and this contradicts the fact that collisions create correlations even when the initial distribution is uncorrelated. However, the concept of initial correlations plays an important role in the general theory of irreversibility. Such a theory should explain, on the one hand, the existence of a distinction between past and future in the frame of the microscopic dynamics and, on the other hand, the relation between the stochastic processes of physics and the deterministic local description of the dynamics. Such a theory was recently elaborated by the Brussels group (see refs. [ 2 - 4 ] and references therein). As is well known, the irreversible pro0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
cesses of physics are oriented in the sense of increasing disorder in isolated systems. However, the constancy of the usual expression of disorder f Pt log Pt dp dq under the Liouville equation expresses the preservation of order in this dynamic representation. The problem of the transition towards a new dynamics under which this order is not preserved is of central importance. It can be postulated in terms of the transition to a semigroup of a stochastic markovian process with a welldet'med set of initial probability distributions. On account of the time reversal symmetry of the microscopic dynamics, the transition is in general twofold (for t > 0 and t < 0). We therefore express the second law of thermodynamics at the microscopic level as a limitation of the physical admissible states along all possible statistical ensembles: We say that the microscopic system is intrinsically irreversible if the set of initial distributions that go to equilibrium for t ~ +o. is not invariant under time inversion (this operation, I, which corresponds to velocity inversion in classical mechanics, is defined for a flow S t by the relations: I S t I - 1 = S t , 12 = 1). Accordingly, we reject all initial distributions that do not go to equilibrium for t -~ +o0. This entails a selection between the two semi-groups of evolution. Such a scheme has been elaborated in which the transition has been realized by a one-to-one transformation acting on the set of distributions ~,. If/) t evolves under the deterministic dynamics then A r t has a probability density Pt which evolves under a Markov process. Moreover the associated entropy - f Pt log Pt d/a 77
Volume 110A, n u m b e r 2
PHYSICS LETTERS
increases monotonically to a maximum (equilibrium). All rejected states have infinite entropy at t = 0. For quantum systems, a selection rule has been expressed in terms of initial correlations, all initial states with non.transcient precoUisional correlations are rejected and only the ones that are created by collisions are retained (for more details see ref. [4]). Here we will deal with the relation between these two selection rules in a class of models initiated by the Kac ring model [5]. Lately, de Haan has used these models in a study of time oriented initial conditions [6]. More generally, they have been studied for the investigation of the relation between deterministic evolution and stochastic description.
8 July 1985
As in our previous work [3] we select the admissible initial states as the probability distributions u such that
f~n(V)= tim
~
u ( A _i_. . ..... ki~) .....
k ~** i _ n ..... ik=0,1
i_n ..... ik i_n ..... ik X l ° g [ v ( A - n ..... k ) / ~ ( A - n ..... k )] < o o .
This condition is equivalent to the selection of all distributions u that, restricted to each sub-o-algebra a~n, where a~n is generated by the partition At n , . . . , i k .... ----n..... k .... , are
absolutely c o n t i n u o u s w i t h respect
to/~ and with densities Pi = dv/dla I ~ i satisfying 2. Ergodic properties o f the deterministic model.
Consider two species of particles distributed on Z in the following way: at each site i E Z, there exists one particle from each kind. The state of the first kind (called ball) r/i takes two values: + 1 (white ball) and - 1 (black ball). The states e i of the second kind take also two values: - 1 (marker state) and +1 (nonmarker). A microscopic state of the system is any sequence x = {r/i, e i} and the phase space F is [K × K] z, K = {-1, +1}. At regular time intervals the dynamic transformation changes a state {~i, ei) as follows: a ball is shifted from the site i to i - 1 with change of color if the second-kind particle at i is in a marker state, as to this last particle, it is shifted simply to the left too. Then we have: S:
~2n(V) = f P i log Pi d/a < oo.
3. Admissible initial correlations. Any measure u on a~ is uniquely defined by its finite dimensional joint distributions
, . (r/n, en) .... en+k_l) . V(/ln .... n+k-1 ) = POln, rln-1, ..., en, en-1,...).
(4) The correlation between the random variables ~i, ei can be investigated in writing (4) in terms of r/i, ei; by using elementary arguments. In fact, any distribution /9 can be written as
x = {Hi, ei) ~ S x = {*li(Sx), ei(Sx)} ,
n+k-1
P(rln,..., en) = 1122k +
71i(Sx) = ei+l(X ) */i+l(X),
ei(Sx) = ei+1 (x ) .
(1)
We take as an equilibrium state/a the distribution in which all '1i, ej are independent with probability (~-, ~). 1 /a is uniquely defined by its value on the "cylindric" sets: A(Ul,Ox) ..... (uk, oD n ..... n+k-1 ={xEFl~n=Ul,
e n = ° l ....
(3)
It is now easy to see that this system is the Bernoulli 1 1 system (~, ~, 4, ~), leading to mixing (for these properties, see ref. [7]). 78
Ci(n, k)~i
n+k-I
+ ~
d ( n , klej + ...
j=n
+
D
(5)
}' (2)
, , (ul, ol) ..... (uk, yD, = 1/22k )
i=n
il ..... 11
which generate the o-algebra of measurable sets• Then lat.'~n ..... n+k -1)
(3')
F
In this sum, r ranges from 0 (in the case of product of e's only) to k, and similarly for l. We denote C!l ..... li
-c[. By straightforward computation we get
tl,...,i n
P(rli, rli) - POli)P(rlj) = ( l /2 2 k - 2)( Ci,j - CiCi)~fl j ,
that is, ~i and 77i are independent if and only if Ci, i(n , k) = Ci(n , k ) C j ( n , k). This extends to any col-
Volume 110A, number 2
PHYSICS LETTERS
lection of variables (~il, ""' *lit, eh .... , e/t). Now, "collisions" between "balls" and "markers" create correlation between them if they were not correlated. In fact, the "Liouville equation" for v is vt(A ) = v(S'-tA) forA E sff and this implies: , . - t .(nn,en) ......... pt(~ln, 71n_l, ..., en, en_l, ...) = v(b A n ..... n+k-1 )" (6)
Then we have from (1), (4), (5) and (6):
=
+ ~-l Cil+l,i2+leilei2rlil~i2 + .... ii < i2 From this formula we may compute the coeffiI I" l" cients Ci, C7 and Cil o f v 1 = vS -1 and we get:
C; ] = 0 '
u t'° ) = . ( n . = uv t, en = v ) .
(7)
A physical change o f representation should tramform the states in such a way that the correlations far away on the left be damped. The A transformation which realizes the transition to this representation can be constructed as in ref. [3]. Therefore, we compute the expression of a single initial correlation between */n and en in the new representation and fred
n+k-1 n+k-1 + ~ c/+le/+ ~ t'd"+1 ~i+1 ei~Tiei j =r i, ]=n
Ci, = C/+I '
fibers "have infinite entropy (3)" and are rejected (for details see ref. [3]). Although states that are concentrated on dilating fibers with infinitely long range correlations are admitted, they do not represent physical states. In fact, the deterministic evolution shifts the correlations between e and ~ without damping according to the relation
vtO?n_ t = u, en_ t = v) --- v(S_tA(nU_'~))
Pl (r~,, *ln-1 .... , en, en-I .... ) n+k-1 = 1)22k + ~ Ci+leff/i i--.
Ci: = ":,i+1 i+1,
8 July 1985
iq=]'
= u, e . = v) = (AO(A
# '°)
= •_nv(A (u'v)) + (1 - ),n)la(A(nu' v)).
(8)
Similar computations yield for neighbouring sites
=(7/+ 1, i = j . (Av)(an..+l )
Thus, many initially uncorrelated distributions will be correlated after one collision (i.e. C[C q = C/). Let us now consider some initial correlations compatible with the selection rule: (A) The simplest case is given by "local" correlations among (rli, el), i, / E [ - n , +n], the distribution being, outside, equal to/a or absolutely continuous with respect to #. These are simply shifted on the left without damping. (B) The states that are absolutely continuous only when restricted on a~n, Vn ~ Z, correspond to states concentrated on "dilating fibers" and that approach equilibrium for t -~ +oo but not for t ~ _o0 [3]. Here, these are probability distributions concentrated on a given configuration 07i, ei) fixed on s i t e s / < - n and are "local" or "quasi local" on sites i > - n . This is a special case of long range correlations that escape to infinity under the deterministic evolution. The oppo. site distribution concentrated on sites i > n, and equivalent to/a for i ~ n is an example of long range correlation that is amplified by the dynamics for t +oo. These distributions concentrated on "contracting"
=
+
+ (1 - X_ (,+l))O(An~n+l).
- X_n)V(A;,+I)
(9)
More generally (Av)(A n ..... n+k-1) will assign dif. ferent damping coefficients to different correlations. We see from (8) that .~ assigns a coefficient )'-n (decreasing to zero for n --> _oo) to the correlation v(A(U, 0), this is the damping or "old" correlations. From (9) we see that .~ accounts for correlations between many particles. We must however notice that the model presented here is too simple to have "realistic" signification, for, the same particles ei and r/i will collide again and again. In more realistic systems correlations between many particles correspond to collisions with new particles. We shall study elsewhere models where particles may collide with other new particles. We now conclude that: contrary to the deterministic dynamics which simply shifts the correlations, the associated stochastic dynamics will damp them according to the relation (following from (8)) 79
Volume 110A, number 2
PHYSICS LETTERS
"~ (u,v) (Avt)(An- t ) = ~t_n+tl)(A(nUVt'°) ) + (1 -- )k_n+t)ld(A(nU'°)). In this model, some admissible long range correlations with constant amplitude tend to equilibrium as t ~ +oo under the deterministic evolution. Nevertheless, only in the new representation they satisfy the principle o f weakening correlations at distant sites and are damped under new time evolution. We acknowledge the Financial support o f Solvay Institute o f the Universit6 Libre de Bruxelles where a part o f this work has been done. We are grateful to Professors C. George, M. de Haan, B. Misra and I. Prigogine for many stimulating discussions.
80
8 July 1985
References [ 1 ] N.N. Bogoliubov, Problems of a dynamical theory in statistical physics, in: Studies in statistical mechanics, Vol. 1, eds. J, Boe~ and G.E. Uldenbeck (North-Holland, Amsterdam, 1962). [2] B. Mirsa and I. I~igogine, Time, probability and dynamics, Proc. Workshop on Long time prediction in nonlinear conservative dynamical systems (Austin, 1981), eds. C.W. Horton, L.E. Reiehl and V. Szebely (Wiley-lnterscience, New York, 1982). [3] M. Courbage and I. Prigogine, Proc. Nat. Acad. Sci. USA 80 (1983) 2412; M. Courhage, Physica 122A (1983) 459. [4] I. Prigogine and C. George, Proe. Nat. Acad. Sci. USA 80 (1983). [5] M, Kac, Probability and related topics in physical sciences (Inte~science, New York, 1959). [6] M. de Haan, Physica 122A (1983) 345. [7] I.P. Cornfeld, S.V. Fomin and Ya.G. Sinai, Ergodic theory (Springer, Berlin, 1982).
PHYSICS LETTERS
8 July 1985
REMARKS ON NONEQUILIBRIUM CORRELATIONS IN A SIMPLE DYNAMIC MODEL M. C O U R B A G E Laboratoire de Probabilit~s, Universit~ Paris 6, Tour 56, 4 Place Jussieu, 75230 Paris Cedex 05, France
Received 6 February 1985; accepted in revised form 29 April 1985
We study a deterministic conservativedynamic model inspired by the Kac ring model. We show that some initial probability distributions with long range correlation, which go to equilibrium asymptotically,can be transformed, through a transition to new dissipative and stochastic dynamics, into states with damped correlations.
1. Introduction. One of the main problems of the
statistical description of non-equilibrium phenomena is the selection of suitable initial non.stationary ensambles. B0goliubov, in his famous work of 1946 [1 ], postulated for a classical system of N interacting particles that non-stationary ensembles should satisfy a principle of weakening correlations between coordinates of the particles under time evolution. The system being described by the hamiltonian dynamics, the time evolution of the ensemble P t(P l .... P n , q l ..... qn ) is given by the Liouville equation. Such a principle is a constraint for the selection of initial distributions 00(/71 ..... Pn, q l ..... qn). The need for it has come out from discussions on the validity of the Boltzmann equation. As is well known, the hypothesis of the molecular chaos which leads to the Boltzmann equation corresponds to a distribution with no correlations between velocities before collisions, and this contradicts the fact that collisions create correlations even when the initial distribution is uncorrelated. However, the concept of initial correlations plays an important role in the general theory of irreversibility. Such a theory should explain, on the one hand, the existence of a distinction between past and future in the frame of the microscopic dynamics and, on the other hand, the relation between the stochastic processes of physics and the deterministic local description of the dynamics. Such a theory was recently elaborated by the Brussels group (see refs. [ 2 - 4 ] and references therein). As is well known, the irreversible pro0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
cesses of physics are oriented in the sense of increasing disorder in isolated systems. However, the constancy of the usual expression of disorder f Pt log Pt dp dq under the Liouville equation expresses the preservation of order in this dynamic representation. The problem of the transition towards a new dynamics under which this order is not preserved is of central importance. It can be postulated in terms of the transition to a semigroup of a stochastic markovian process with a welldet'med set of initial probability distributions. On account of the time reversal symmetry of the microscopic dynamics, the transition is in general twofold (for t > 0 and t < 0). We therefore express the second law of thermodynamics at the microscopic level as a limitation of the physical admissible states along all possible statistical ensembles: We say that the microscopic system is intrinsically irreversible if the set of initial distributions that go to equilibrium for t ~ +o. is not invariant under time inversion (this operation, I, which corresponds to velocity inversion in classical mechanics, is defined for a flow S t by the relations: I S t I - 1 = S t , 12 = 1). Accordingly, we reject all initial distributions that do not go to equilibrium for t -~ +o0. This entails a selection between the two semi-groups of evolution. Such a scheme has been elaborated in which the transition has been realized by a one-to-one transformation acting on the set of distributions ~,. If/) t evolves under the deterministic dynamics then A r t has a probability density Pt which evolves under a Markov process. Moreover the associated entropy - f Pt log Pt d/a 77
Volume 110A, n u m b e r 2
PHYSICS LETTERS
increases monotonically to a maximum (equilibrium). All rejected states have infinite entropy at t = 0. For quantum systems, a selection rule has been expressed in terms of initial correlations, all initial states with non.transcient precoUisional correlations are rejected and only the ones that are created by collisions are retained (for more details see ref. [4]). Here we will deal with the relation between these two selection rules in a class of models initiated by the Kac ring model [5]. Lately, de Haan has used these models in a study of time oriented initial conditions [6]. More generally, they have been studied for the investigation of the relation between deterministic evolution and stochastic description.
8 July 1985
As in our previous work [3] we select the admissible initial states as the probability distributions u such that
f~n(V)= tim
~
u ( A _i_. . ..... ki~) .....
k ~** i _ n ..... ik=0,1
i_n ..... ik i_n ..... ik X l ° g [ v ( A - n ..... k ) / ~ ( A - n ..... k )] < o o .
This condition is equivalent to the selection of all distributions u that, restricted to each sub-o-algebra a~n, where a~n is generated by the partition At n , . . . , i k .... ----n..... k .... , are
absolutely c o n t i n u o u s w i t h respect
to/~ and with densities Pi = dv/dla I ~ i satisfying 2. Ergodic properties o f the deterministic model.
Consider two species of particles distributed on Z in the following way: at each site i E Z, there exists one particle from each kind. The state of the first kind (called ball) r/i takes two values: + 1 (white ball) and - 1 (black ball). The states e i of the second kind take also two values: - 1 (marker state) and +1 (nonmarker). A microscopic state of the system is any sequence x = {r/i, e i} and the phase space F is [K × K] z, K = {-1, +1}. At regular time intervals the dynamic transformation changes a state {~i, ei) as follows: a ball is shifted from the site i to i - 1 with change of color if the second-kind particle at i is in a marker state, as to this last particle, it is shifted simply to the left too. Then we have: S:
~2n(V) = f P i log Pi d/a < oo.
3. Admissible initial correlations. Any measure u on a~ is uniquely defined by its finite dimensional joint distributions
, . (r/n, en) .... en+k_l) . V(/ln .... n+k-1 ) = POln, rln-1, ..., en, en-1,...).
(4) The correlation between the random variables ~i, ei can be investigated in writing (4) in terms of r/i, ei; by using elementary arguments. In fact, any distribution /9 can be written as
x = {Hi, ei) ~ S x = {*li(Sx), ei(Sx)} ,
n+k-1
P(rln,..., en) = 1122k +
71i(Sx) = ei+l(X ) */i+l(X),
ei(Sx) = ei+1 (x ) .
(1)
We take as an equilibrium state/a the distribution in which all '1i, ej are independent with probability (~-, ~). 1 /a is uniquely defined by its value on the "cylindric" sets: A(Ul,Ox) ..... (uk, oD n ..... n+k-1 ={xEFl~n=Ul,
e n = ° l ....
(3)
It is now easy to see that this system is the Bernoulli 1 1 system (~, ~, 4, ~), leading to mixing (for these properties, see ref. [7]). 78
Ci(n, k)~i
n+k-I
+ ~
d ( n , klej + ...
j=n
+
D
(5)
}' (2)
, , (ul, ol) ..... (uk, yD, = 1/22k )
i=n
il ..... 11
which generate the o-algebra of measurable sets• Then lat.'~n ..... n+k -1)
(3')
F
In this sum, r ranges from 0 (in the case of product of e's only) to k, and similarly for l. We denote C!l ..... li
-c[. By straightforward computation we get
tl,...,i n
P(rli, rli) - POli)P(rlj) = ( l /2 2 k - 2)( Ci,j - CiCi)~fl j ,
that is, ~i and 77i are independent if and only if Ci, i(n , k) = Ci(n , k ) C j ( n , k). This extends to any col-
Volume 110A, number 2
PHYSICS LETTERS
lection of variables (~il, ""' *lit, eh .... , e/t). Now, "collisions" between "balls" and "markers" create correlation between them if they were not correlated. In fact, the "Liouville equation" for v is vt(A ) = v(S'-tA) forA E sff and this implies: , . - t .(nn,en) ......... pt(~ln, 71n_l, ..., en, en_l, ...) = v(b A n ..... n+k-1 )" (6)
Then we have from (1), (4), (5) and (6):
=
+ ~-l Cil+l,i2+leilei2rlil~i2 + .... ii < i2 From this formula we may compute the coeffiI I" l" cients Ci, C7 and Cil o f v 1 = vS -1 and we get:
C; ] = 0 '
u t'° ) = . ( n . = uv t, en = v ) .
(7)
A physical change o f representation should tramform the states in such a way that the correlations far away on the left be damped. The A transformation which realizes the transition to this representation can be constructed as in ref. [3]. Therefore, we compute the expression of a single initial correlation between */n and en in the new representation and fred
n+k-1 n+k-1 + ~ c/+le/+ ~ t'd"+1 ~i+1 ei~Tiei j =r i, ]=n
Ci, = C/+I '
fibers "have infinite entropy (3)" and are rejected (for details see ref. [3]). Although states that are concentrated on dilating fibers with infinitely long range correlations are admitted, they do not represent physical states. In fact, the deterministic evolution shifts the correlations between e and ~ without damping according to the relation
vtO?n_ t = u, en_ t = v) --- v(S_tA(nU_'~))
Pl (r~,, *ln-1 .... , en, en-I .... ) n+k-1 = 1)22k + ~ Ci+leff/i i--.
Ci: = ":,i+1 i+1,
8 July 1985
iq=]'
= u, e . = v) = (AO(A
# '°)
= •_nv(A (u'v)) + (1 - ),n)la(A(nu' v)).
(8)
Similar computations yield for neighbouring sites
=(7/+ 1, i = j . (Av)(an..+l )
Thus, many initially uncorrelated distributions will be correlated after one collision (i.e. C[C q = C/). Let us now consider some initial correlations compatible with the selection rule: (A) The simplest case is given by "local" correlations among (rli, el), i, / E [ - n , +n], the distribution being, outside, equal to/a or absolutely continuous with respect to #. These are simply shifted on the left without damping. (B) The states that are absolutely continuous only when restricted on a~n, Vn ~ Z, correspond to states concentrated on "dilating fibers" and that approach equilibrium for t -~ +oo but not for t ~ _o0 [3]. Here, these are probability distributions concentrated on a given configuration 07i, ei) fixed on s i t e s / < - n and are "local" or "quasi local" on sites i > - n . This is a special case of long range correlations that escape to infinity under the deterministic evolution. The oppo. site distribution concentrated on sites i > n, and equivalent to/a for i ~ n is an example of long range correlation that is amplified by the dynamics for t +oo. These distributions concentrated on "contracting"
=
+
+ (1 - X_ (,+l))O(An~n+l).
- X_n)V(A;,+I)
(9)
More generally (Av)(A n ..... n+k-1) will assign dif. ferent damping coefficients to different correlations. We see from (8) that .~ assigns a coefficient )'-n (decreasing to zero for n --> _oo) to the correlation v(A(U, 0), this is the damping or "old" correlations. From (9) we see that .~ accounts for correlations between many particles. We must however notice that the model presented here is too simple to have "realistic" signification, for, the same particles ei and r/i will collide again and again. In more realistic systems correlations between many particles correspond to collisions with new particles. We shall study elsewhere models where particles may collide with other new particles. We now conclude that: contrary to the deterministic dynamics which simply shifts the correlations, the associated stochastic dynamics will damp them according to the relation (following from (8)) 79
Volume 110A, number 2
PHYSICS LETTERS
"~ (u,v) (Avt)(An- t ) = ~t_n+tl)(A(nUVt'°) ) + (1 -- )k_n+t)ld(A(nU'°)). In this model, some admissible long range correlations with constant amplitude tend to equilibrium as t ~ +oo under the deterministic evolution. Nevertheless, only in the new representation they satisfy the principle o f weakening correlations at distant sites and are damped under new time evolution. We acknowledge the Financial support o f Solvay Institute o f the Universit6 Libre de Bruxelles where a part o f this work has been done. We are grateful to Professors C. George, M. de Haan, B. Misra and I. Prigogine for many stimulating discussions.
80
8 July 1985
References [ 1 ] N.N. Bogoliubov, Problems of a dynamical theory in statistical physics, in: Studies in statistical mechanics, Vol. 1, eds. J, Boe~ and G.E. Uldenbeck (North-Holland, Amsterdam, 1962). [2] B. Mirsa and I. I~igogine, Time, probability and dynamics, Proc. Workshop on Long time prediction in nonlinear conservative dynamical systems (Austin, 1981), eds. C.W. Horton, L.E. Reiehl and V. Szebely (Wiley-lnterscience, New York, 1982). [3] M. Courbage and I. Prigogine, Proc. Nat. Acad. Sci. USA 80 (1983) 2412; M. Courhage, Physica 122A (1983) 459. [4] I. Prigogine and C. George, Proe. Nat. Acad. Sci. USA 80 (1983). [5] M, Kac, Probability and related topics in physical sciences (Inte~science, New York, 1959). [6] M. de Haan, Physica 122A (1983) 345. [7] I.P. Cornfeld, S.V. Fomin and Ya.G. Sinai, Ergodic theory (Springer, Berlin, 1982).