Fluctuation theorem, nonlinear response, and the regularity of time reversal
symmetry
Marcello Porta
Citation: Chaos 20, 023111 (2010); doi: 10.1063/1.3396283 View online: https://doi.org/10.1063/1.3396283
View Table of Contents: http://aip.scitation.org/toc/cha/20/2
Fluctuation theorem, nonlinear response, and the regularity
of time reversal symmetry
Marcello Porta
Dipartimento di Fisica, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, 00185 Roma, Italy
共Received 9 October 2009; accepted 26 March 2010; published online 28 May 2010兲
The Gallavotti–Cohen fluctuation theorem共FT兲 implies an infinite set of identities between corre-lation functions that can be seen as a generalization of Green–Kubo formula to the nonlinear regime. As an application, we discuss a perturbative check of the FT relation through these iden-tities for a simple Anosov reversible system; we find that the lack of differentiability of the time reversal operator implies a violation of the Gallavotti–Cohen fluctuation relation. Finally, a brief comparison to Lebowitz–Spohn FT is reported. © 2010 American Institute of Physics.
关doi:10.1063/1.3396283兴
At present time, there is no universally accepted micro-scopic theory of the steady state of nonequilibrium sys-tems; we call nonequilibrium steady state the stationary state of a dissipative system. However, for small external forcings some properties of nonequilibrium steady states can be inferred by considering them as “small perturba-tions” of equilibrium states, which are well described by equilibrium statistical mechanics. As a result, the re-sponse to a small external forcing can be expressed, at least formally, in terms of quantities which are computed at equilibrium, through the so-called linear response
theory. Obviously, the linear response theory loses its
pre-dictive power when the forcings are not small, that is when the assumption of closeness to an equilibrium state is no longer justified; the goal of nonequilibrium statisti-cal mechanics is to provide relations between physistatisti-cal observables in the nonlinear regime that reduce, in par-ticular, to the usual linear response theory in the limit of zero forcings. Analogous to what happens at equilibrium with the ergodic hypothesis, one can attempt a descrip-tion of nonequilibrium steady states by making some as-sumptions on the dynamics and then checking if the the-oretical predictions which are eventually implied by the assumptions are consistent with the experiments; this is what Gallavotti and Cohen proposed with their chaotic
hypothesis (CH). This hypothesis consists of an
appar-ently very strong claim on the dynamics of the particles at the steady state; it assumes that for the purpose of studying macroscopic quantities a system exhibiting cha-otic motions may be considered chacha-otic in a “maximal” sense (which can be made mathematically precise). Ac-cepting this, if the law of motion is reversible under a suitably regular time reversal operation, then it is pos-sible to rigorously prove a nontrivial large deviation law for the rate of entropy production, which is the content of the so-called Gallavotti–Cohen (GC) fluctuation theorem (FT); this law has been successfully checked in many ex-periments, most of which numerical, in the past 15 years. Remarkably, in the limit of zero forcings the GC FT im-plies the Green–Kubo (GK) formula, a well-known result of linear response theory; in this paper we show that with little efforts, the GC FT allows to compute the higher
order corrections to linear response. As an application, we perform a check of the GC fluctuation relation by explicitly evaluating the first correction to linear response in a very simple model, whose dynamics is reversible un-der a continuous but not differentiable time reversal op-eration; we find that the lack of differentiability of the time reversal operator implies a violation of the fluctua-tion relafluctua-tion. However, in the considered case a different large deviation rule is true, which is essentially equivalent to the Lebowitz–Spohn FT.
I. INTRODUCTION
Despite many proposals have been advanced, a general theory of the steady state of dissipative systems is still lack-ing. Nevertheless, it is a remarkable fact that under suitable hypothesis something can be said and physical predictions can be made; for example, CH,1stating that for the purpose of studying macroscopic properties, a system exhibiting cha-otic motions may be regarded as a transitive hyperbolic共that is Anosov兲 one, see Ref.2, implies two remarkable results: the GC FT,3,4holding for reversible systems, and a formula describing the linear response of nonequilibrium systems due to Ruelle.5–7 These results have been and are still widely studied in the physical literature; see Ref.8共where a relation which inspired the FT was empirically discovered兲 and Refs. 9–13, for instance. In this note we focus on the first of these two results; in particular, in Sec. II we show that FT has some nice implications on nonlinear response 共in Ref. 14it has been already pointed out that FT implies the usual linear response theory, that is GK formula and Onsager reciprocity relations兲, while in Sec. III we discuss a check of the FT relation in a simple Anosov reversible system. Interestingly, a simple perturbative calculation shows that the lack of dif-ferentiability of the time reversal operator implies a violation of the GC fluctuation relation. Finally, in Sec. IV we show that in the presence of a nondifferentiable time reversal transformation an identity equivalent to a result proved by Lebowitz and Spohn in Ref. 15is true.
The connection between FT and nonlinear response con-sists of the fact that FT implies identities between correlation
functions of physical observables in a nonequilibrium steady state; this has been pointed out first in Ref. 16, where the authors considered systems whose evolution was stochastic and ruled by a master equation, and independently in Ref.17 in the context of deterministic systems satisfying CH. The results of Ref. 17 are presented in Secs. II and III of this paper.
Before turning to our results, we spend a few words on some of the main features of Anosov systems; we refer the interested reader to Ref. 2 for a modern introduction to the subject. Consider a generic discrete dynamical system xk= Skx0 共this is not a lack of generality, since S can be
thought as the map arising from the Poincaré section of a system evolving in continuous time, see Ref.19兲 and assume that S is Anosov; informally, this means that given a point x the nearby points separate exponentially fast from x in the future and in the past, except when located on a surface Ws共x兲 共stable manifold兲 or Wu共x兲 共unstable manifold兲,
re-spectively, for the future and for the past.
It is a well-known result that Anosov systems admit an invariant measure +, the so-called Sinai–Ruelle–Bowen
共SRB兲 measure;18
in fact, given a sufficiently regular observ-able F共x兲 the following equality holds:
lim T→+⬁ 1 T
兺
j=0 T−1 F共Sjx兲 =冕
+共dx兲F共x兲, 共1.1兲apart from a set of points of zero volume measure. Remark-ably, the SRB measure admits in principle an explicit repre-sentation, similar to the equilibrium Gibbs distribution.
Notice that at equilibrium, that is when the system is stationary and nondissipative, the chaotic hypothesis implies the ergodic hypothesis, in the sense that assuming CH the SRB measure is the Liouville one; but in general, when dis-sipation is present the SRB measure is singular with respect to the volume, that is it is concentrated on a zero volume set. To conclude, the SRB measure verifies a large deviation theorem 共see, for example, Ref.19for a proof of this state-ment for a special choice of F and in the more complex case of Anosov flows兲. In fact, consider the finite time average f =−1兺j=−/2−1/2F共Sjx兲, where F共x兲 is Hölder continuous in x; then, it is possible to prove that there are values f1, f2 such
that if 关a,b兴苸共f1, f2兲, then Prob+共f 苸关a,b兴兲⬃eF共f兲in the
sense that lim
→+⬁
1
log Prob+共f 苸 关a,b兴兲 = maxf
苸关a,b兴F共f兲 共1.2兲
andF共f兲 is analytic and convex in 共f1, f2兲.
II. FLUCTUATION THEOREM AND NONLINEAR RESPONSE
Currently, no universally accepted definition of entropy for a dissipative system has been given. Nevertheless, the rate of entropy production is a well-defined quantity and is proportional to the work per unit time made by the thermo-stats on the system; the proportionality factor is the inverse of the temperature of the thermostats共setting to 1 the Bolt-zmann constant兲. In particular, for a special class of thermo-stats, the Gaussian ones, the entropy production rate
corre-sponds to the phase space contraction, that is to minus the divergence of the equations of motion, see Ref.20; this fact can be taken as a general definition of entropy production if one assumes that the steady state of a large system is not affected by the details of the thermostatting mechanism which ensures the existence of the steady state.
In the case of a system evolving in discrete time, which is the case that we want to consider, the entropy production rate is given by 共x兲=−log兩detS共x兲兩, where S is the Jacobian of the time evolution S. Now, assume that the sys-tem is dissipative, i.e., that+⬅兰+共dx兲共x兲⬎0, let p be the
adimensional average over a time of 共x兲, that is p =共+兲−1兺
j=−/2
/2−1共Sjx兲, and call 共p兲 the large deviation
functional of , as defined in Eq. 共1.2兲; assume that CH holds, and that the system is reversible, in the sense that there exists a differentiable isometry I such that IⴰS=S−1ⴰI,
IⴰI=1. Then, as proven by Gallavotti and Cohen, see Ref.3 or Ref.21for a detailed proof from a formal viewpoint, the following result holds.
Fluctuation Theorem: There is pⴱⱖ1 such that for
兩p兩⬍pⴱ,
共− p兲 =共p兲 − p+. 共2.1兲
This result has an interesting corollary. Setting 共q兲dq = Prob +共p苸关q,q+dq兴兲, define 共兲 as 共兲 = lim →+⬁ 1 log
冕
e共q−1兲+共q兲dq. 共2.2兲 Clearly, 共兲 is related to 共p兲 through a Legendre trans-form, that is−共p兲 = max
 共+共p − 1兲 − 共兲兲 共2.3兲
and, moreover,共兲 admits the following expansion:
共兲 =
兺
nⱖ2t1,. . .,t兺
n−1=−⬁ +⬁ 具共 · 兲共St1 ·兲 ¯共Stn−1·兲典 + T n n! ⬅兺
nⱖ2 Cn n n!, 共2.4兲where by具¯典+Twe denote the cumulant with respect to the SRB measure +. It is straightforward to see that the FT
implies an identity for the generating functional 共兲, see Refs. 13and15–17for instance: it follows that as a conse-quence of the relation 共p兲⬃ep+
共−p兲, valid under the
hypothesis of FT,
共兲 = 共− 1 −兲 −+共2+ 1兲. 共2.5兲
Notice that the generating functional of the cumulants is usu-ally defined as 共see Refs. 13, 15, and 16兲 ˜共兲=lim→+⬁ −−1log具e−p+典, and with this definition the relation共2.5兲is replaced by the more familiar ˜共兲=˜共1−兲; but the two definitions are equivalent since 共兲=−+−˜共−兲.
For-mula共2.5兲translates immediately in a relation for+; in fact,
Eq. 共2.5兲evaluated at = 0 becomes
0 =共− 1兲 −+⇒+=
兺
nⱖ2 Cn
共− 1兲n
n! . 共2.6兲
Assuming that the entropy production 共x兲 has the form 共x兲=兺iGiJi
共0兲共x兲+O共G2兲, where 兵G
i其, and 兵Ji
共0兲共x兲其 are,
re-spectively, the forcing parameters and the corresponding cur-rents, it has been shown in Ref. 14that the identity
兺
i,j
GiGjGiGj+
冏
Gគ =0គ=1
2
兺
i,j GiGjGiGjC2兩Gគ =0គ, 共2.7兲which is nothing else than Eq. 共2.6兲 at second order, is equivalent to the GK formula, stating that
Lij⬅Gj具Ji共0兲典+兩Gគ =0គ=
1 2t=−
兺
⬁+⬁
具Ji共0兲共St·兲J共0兲j 共 · 兲典0, 共2.8兲
where by具·典0 we denote the expectation with respect to the
invariant measure at zero forcing共which by CH is the Liou-ville measure兲. Hence, formula共2.6兲can be seen as a gener-alization of GK formula to the nonlinear regime, being an identity for + valid for Gគ ⫽0គ. Moreover, by taking
deriva-tives with respect to in the right-hand side 共rhs兲 and left-hand side 共lhs兲 of Eq.共2.5兲we find that
Cn=
兺
kⱖ0共− 1兲k+nC k+n
k! , nⱖ 2, 共2.9兲
which is a nontrivial identity for the cumulants valid at Gគ ⫽0គ. Finally, these identities can be considerably extended by using a generalized version of FT. Consider a generic observable O odd under time reversal, i.e., such that O共Ix兲=−O共x兲, assume that the system is dissipative and that O+⬅兰+共dx兲O共x兲⫽0, let w be the adimensional average over a time of O共x兲, that is w=共O+兲−1兺j=−/2−1/2O共S
j x兲, and call共p,w兲 the large deviation functional of the joint prob-ability distribution共p,w兲 of p and w; then, under the same hypothesis of FT, the following result holds as a special case of a much more general result in Ref. 22.
Generalized Fluctuation Theorem (GFT): There are
wⴱⱖ1, pⴱⱖ1 such that for 兩p兩⬍pⴱand 兩w兩⬍wⴱ,
共− p,− w兲 =共p,w兲 − p+. 共2.10兲
One can define the generating functional共1,2兲 of the
mixed cumulants of O, in a way analogous to Eq.共2.2兲, 共1,2兲 = lim
→+⬁
1
log
冕
e共q−1兲+1+共t−1兲O+2共q,t兲dqdt. 共2.11兲 Again, 共1,2兲 is related to 共p,w兲 through a Legendretransform, and moreover, it can be expressed as 共1,2兲 =
兺
kⱖ2m,n兺
ⱖ0, m+n=k 1 n 2 m n!m!Cn,m, 共2.12兲 where Cn,m=1 n 2 m共 1,2兲兩គ=0គ, that is Cn,m is given by asum over times of mixed cumulants of 共Stix兲, O共Stjx兲,
0ⱕiⱕn, and 0ⱕ jⱕm. Then, in full analogy with what has already been discussed, it is easy to show that GFT translates
into an identity for the generating functional, namely, 共1,2兲 = 共− 1 −1,−2兲 − 共21+ 1兲+− 22O+,
共2.13兲 which implies the following relations:
O+=
兺
kⱖ2 共− 1兲k 2k−1 l=0,2l兺
ⱕk−1 Ck−共2l+1兲,2l+1 共2l + 1兲!共k − 共2l + 1兲兲!, 共2.14兲 Cl,n−l=兺
kⱖ0 共− 1兲n+k k! Ck+l,n−l, nⱖ 2, l ⱕ n. 共2.15兲Equation共2.14兲is obtained setting1=2= − 1
2 in Eq.共2.13兲,
while Eq.共2.15兲can be proved differentiating with respect to 1,2the rhs and the lhs of Eq.共2.13兲. It is interesting to see
what happens to formula共2.14兲in the linear regime. Assum-ing that 共x兲=兺iGiJi
共0兲共x兲+O共G2兲 we can rewrite Eq. 共2.14兲
as O+=
兺
kⱖ2,k even 1 共k − 1兲! 共− 1兲k 2k−1 C1,k−1 +兺
kⱖ3,k odd 1 k! 共− 1兲k 2k−1 C0,k+ O共G 2兲 =1 2C1,1+kⱖ4,k even兺
1 共k − 1兲! 共− 1兲k 2k−1 C1,k−1 +兺
kⱖ3,k odd 1 k! 共− 1兲k 2k−1 C0,k =1 2C1,1+kⱖ3,k odd兺
1 k! 共− 1兲k 2k−1冉
C0,k− 1 2C1,k冊
+ O共G 2兲, 共2.16兲 and from Eq. 共2.15兲 we find that for k odd C0,k−1 2C1,k = O共G2兲. Hence, O+= 1 2C1,1+ O共G 2兲 =1 2t=−
兺
⬁ ⬁ 具共St x兲O共x兲典++ O共G2兲, 共2.17兲 which gives GiO+兩Gគ =0គ= 1 2t=−兺
⬁ ⬁ 具Ji共0兲共S t x兲O共x兲典0. 共2.18兲Formula 共2.18兲 describes the linear response of a generic observable odd under time reversal; this result can be seen as a special case of the remarkable linear response formula ob-tained by Ruelle,5,6valid in a much more general context.
III. CHECK OF FLUCTUATION THEOREM FOR A SIMPLE ANOSOV SYSTEM
In this section we will perform a check of the fluctuation relation共2.1兲for a simple Anosov model, the perturbed Ar-nold cat map, starting from the identities 共2.6兲and共2.9兲; as we are going to see, the lack of differentiability of the time reversal operator implies a violation of Eq. 共2.1兲. Consider
the following discrete evolution on the bidimensional torus T2: xគk= Skxគ0 mod 2, 共3.1兲 where Sxគ = Sxគ + fគ共xគ兲, S ⬅
冉
1 1 1 2冊
, គ共xគ兲 =f冉
f1共xគ兲 f2共xគ兲冊
, 共3.2兲 and f1共xគ兲, f2共xគ兲 are trigonometric polynomials. The map S isthe so-called Arnod cat map, which is the simplest example of Anosov map: in fact, the eigenvalues+,−of S are such
that+⬎1, −⬍1. Moreover, the map S is reversible, that is
there exists I such that IⴰS=S−1ⴰI, IⴰI=1, where
I =
冉
− 1 0− 1 1
冊
. 共3.3兲Notice that since S is conservative 共det S=1兲 += 0, which
makes meaningless the fluctuation relation 共2.1兲if=0 共the adimensional quantity p is not defined兲; but one can derive the analogous of Eq. 共2.1兲 for the dimensional quantity p
⬘
=+p, and this relation becomes trivial if the evolution isconservative because in this case p
⬘
= 0.Consider now ⫽0. By the structural stability of Anosov systems, the evolution generated by S is still Anosov provided is chosen small enough. In fact, there exists 0⬎0 such that for ⬍0 a conjugation H, defined by the identity SⴰH= HⴰS, can be explicitly constructed through a convergent power series in, and it turns out that H共xគ兲 is Hölder continuous in xគ.2 In an analogous way, the SRB measure can be explicitly constructed, and it follows that the expectations of Hölder continuous functions exist and are analytic in .2 In particular it follows that, generi-cally,
+⬅ 具− log兩detS兩典+⬎ 0, 共3.4兲
which means that the system is dissipative, so that the invari-ant measure is singular with respect to the volume; hence, the check of the fluctuation relation共2.1兲is nontrivial in this case. Notice that the proof of FT 共Ref.21兲 requires that the evolution is reversible, and in particular that the time rever-sal operator is differentiable; in our specific case the exis-tence of H implies that I= HⴰIⴰH−1 verifies IⴰS= S−1 ⴰI, but due to the mild regularity properties of H, I共xគ兲 is likely to be not differentiable. Here, the time reversal opera-tor Iassociated to the map S is perturbatively constructed as a convergent power series in starting from I; time rever-sal symmetry is not destroyed by small perturbations. How-ever, it is interesting to notice that there exist classes of dis-sipative Anosov flows for which time reversal symmetry holds for any value of the forcings; see the “Gaussian ther-mostat case” of Ref.23for instance, where the time reversal operation is simply the inversion of the velocities. The con-tinuous time dynamics of the Anosov flows can be reduced to discrete ones by repeating the analysis of Ref. 19.
One can ask what happens to the fluctuation relation in the simple case we are considering; to understand this, we
make the choice f1共xគ兲=sin共x1兲+sin共2x1兲, f2共xគ兲=0. An ex-plicit computation shows that the linear response is still valid, as expected, and that
C3= − 123+ O共4兲, 共3.5兲
i.e., formula共2.9兲with n = 3 is false at the lowest nontrivial order in perturbation theory, since it tells that C3= C4/2 + O共5兲 and C
4= O共4兲. This is enough to say that Eq.共2.1兲is
violated; in fact, from Eq. 共2.3兲 it follows that17 共using the linear response relation +=共1/2兲C2+ O共3兲兲:
共p兲 = −共p − 1兲2 4
关
++ O共 3兲兴
+共p − 1兲 3 48 关C3+ O共 4兲兴 + O共共p − 1兲44兲, 共3.6兲 that is 共p兲 −共− p兲 = p关
++ O共3兲兴
+ p3冋
C3 24+ O共 4兲册
+ O共p55兲, 共3.7兲and Eq. 共3.5兲implies that the difference 共p兲−共−p兲 is not linear in p关the coefficient of p3in Eq.共3.7兲is nonzero兴. This
result also shows that as expected, I cannot be differen-tiable: a check that would be not so easy without using the FT.
IV. CONCLUSIONS
The GC FT implies nontrivial identities between corre-lation functions valid at nonzero forcing Gគ which reduce to the usual linear response in the limit Gគ →0គ. Through a check of these identities in a simple Anosov system, we have shown that it is essential that the time reversal transformation be共continuously兲 differentiable for the fluctuation relation to hold. Indeed, we considered a model that admits a time re-versal transformation which is only Hölder continuous, and we have found that the identities implied by the fluctuation relation are not true at the third order in the forcing 共while the linear response, corresponding to the second order, is still valid, as expected兲; this is enough to prove that the fluctua-tion relafluctua-tion cannot hold in the case we considered. Notice, however, that in physical applications the time reversal trans-formation is usually regular; for instance, it can correspond to the inversion of velocities共or to more subtle permutations of coordinates, see Ref.24兲.
To conclude, it is interesting to note that in the presence of a nondifferentiable time reversal operator I a different FT holds, the Lebowitz–Spohn one, see Ref.15; strictly speak-ing this theorem has been proved in the context of general Markov processes, but it can be understood also in the case of deterministic chaotic dynamics. This result applies in par-ticular to systems that are 共suitably small兲 perturbations of reversible Anosov ones, since, as it has been pointed out in Sec. III, by the structural stability of Anosov dynamics the time reversal symmetry is not destroyed by the perturbation 共although it will be in general only Hölder continuous in x兲. The fact that a fluctuation relation still holds is a natural consequence of the Gibbsian nature of the invariant measure
describing the steady state; this has been pointed out in Ref. 25where a FT for the one-dimensional Ising model in an external field was derived, and then systematized in Ref. 26, where large deviation rules for Gibbs states involving transformations different from time reversal were discussed. Following step by step the proof of GC FT,21one can see that the large deviation functional˜共p˜兲 of the dimensionless quantity ˜ =p 共˜+兲−1兺
j=−/2
/2−1˜共Sj
x兲, where ˜共x兲=−u共x兲
+u共Ix兲 and u共x兲 is the sum of the positive Lyapunov
expo-nents corresponding to the local unstable manifold Wu共x兲 of S, verifies
˜共− p˜兲 =˜共p˜兲 − p˜ ˜+ 共4.1兲
共of course under the physical restriction on p˜ to vary within the analyticity interval of˜, see Ref. 21兲, which reduces to Eq. 共2.3兲 if I is differentiable, since in this case u共Ix兲=−s共x兲. Formula共4.1兲is equivalent to the Lebowitz–
Spohn FT; in fact,˜共x兲 is proportional to the logarithm of the ratio of the SRB probabilities of the trajectories x−/2, x−/2+1, . . . , x/2−1and of its time reversed, which is pre-cisely the “action functional” introduced in Ref.15.
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