Low-temperature relaxation in the one-dimensional Ising model

PHYSICAL REVIEW EVOLUME 53, NUMBER 1JANUARY 1996 Low-temperature relaxation in the one-dimensional Ising model J.Javier Brey and A. Prados Fisica Teorica, Universidad de Sevilla, Apartado de Correos 2065, E$-1080 Sevilla, Spain (Received. 16 August 1995) The decay of the spin-spin time correlation functions in aone-dimensional Ising model with Glauber dynamics is studied. In the low-temperature limit, an asymptotically valid continuous space equation is derived. It is amodified diffusion equation with apurely exponential relaxation term. Its solution leads to an exact Cole-Davison behavior of the spin autocorrelation in the frequency domain, while in the time description aKohlrausch-Williams-Watts (KWW) function followed by an exponential decay is obtained. The exponent of the KWW function, analytically derived, is 1/2, and not 0.63, as has been reported in the literature. PACS number(s): 02.50.Ey, 05.50.+q, 77.22.Ch I. INTRODUCTION Nonexponential relaxation has atracted alot of attention recently, in part motivated by the interest in explaining the origin of the slow relaxation observed in glasses and other complex systems [1]. An important result obtained is that nonexponential relaxation is also shown by systems with an apparent, very simple dynamics. In particular, relaxation in kinetic Ising models has been extensively studied [2,3] and stretched exponential behavior has been found for different choices of the transition rates. Nevertheless, most of the works are numerical, and exact analytical results are scarce. This applies even to the case of one-dimensional systems with Glauber dynamics, which seem to be the simplest possibility. One of the first observations of slow decay in Ising systems was reported by Anderson [4] and later by Bozdemir [5], both in the context of dielectric relaxation in polymers, and using Glauber's model [6]. They studied the frequency behavior of the complex permittivity, which is expressed in terms of the Fourier transform of the spin time autocorrelation funtion. By fitting numerically their results, they conclude that at low temperatures the complex permittivity approaches the Cole-Davidson function with an exponent PGD =1/2. The implications in time of aCole-Davidson distribution in frequencies had been analyzed afew years before by Lindsey and Patterson, also using numerical techniques [7]. According to them, the above distribution is nearly equivalent to astretched exponential or aKohlrausch-Williams-Watts (KWW) function, and they gave the relationship between PcD and the exponent PKvvw. PKw~ =0.63 corresponds to the value PcD =1/2. Consequently, this has been the value used to characterize the relaxation of the spin autocorrelation function in the one-dimensional Ising model with Glauber dynamics [8— 10]. The fact that dielectric relaxation and depolarized light scattering experiments in polymers lead, in general, to values of PKwvv smaller than 0.63 has stimulated the search of other dynamics for the Ising model that give smaller values of PKww [8,9,11]. In some cases, the interactions have been substituted by local constraints on the dynamical events [12]. The objective was to try to make the dynamics of the system more cooperative, at least in the low-temperature limit. In this way, the value PKww =0.5, or very close to it, has been repeatedly obtained. In this work we consider once again the relaxation of the spin time autocorrelation function in the onedimensional Glauber model. There are several reasons for it. First, it is important to derive analytically as many results as possible in order to understand the physical mechanisms that are responsible for the relaxation of the system. For instance, we derive an evolution equation with two easily identifiable terms: one associated with diffusion and the other one with the creation of defects. Similar equations have been introduced as approximations to describe the evolution of some of the models mentioned above. Second, we wanted to verify the value PKww =0.63 and see how the KWW function emerges from the equations. Our calculations show that the ColeDavidson distribution is asymptotically exact in the lowtemperature limit, with PGD =1/2. Nevertheless, in the time domain one does not obtain asingle KWW function, but an exponential behavior appears in atime window where the relaxation is still relevant. Besides, the exponent of the KWW function describing most of the relaxation is PKww =1/2, showing that the introduction of local constraints does not always increase the degree of cooperativity of the dynamics. Finally, we have previously studied the relaxation of the energy for the same model [10] and found it interesting to compare it with that of adifferent property. The plan of the paper is as follows. The model is reviewed in Sec. II, where ageneral expression for the spin-spin time correlation functions is given. Section III contains the most relevant results. The low-temperature limit is analyzed and acontinuous space equation for the time evolution of correlations is derived. From this equation the Cole-Davidson distribution is easily obtained. Then, relaxation in time domain is considered, leading to the identification of aKWW region at intermediate times. The initial part of the relaxation is the subject of Sec. IV. It is shown that, even though the decay taking 1063-651X/96/53(1)/458(7)/$06. 00 458 1996 The American Physical Society 53 LOW-TEMPERATURE RELAXATION IN THE ONE-. ..459 place in it is negligible at low temperatures, it contains enough information to determine the value of the exponent PKwv(r. Section Vis devoted to some final remarks. A,,(0) =(, ,).,=&~'-'~, (8) This hierarchy is to be solved with the initial condition II. SPIN-SPIN TIME CORRELATION FUNCTIONS We consider an infinite one-dimensional Ising model. The state of the system is specified by the spin vector cr =(o,), where o; =+1is the spin at site i. The energy of the system for aconfiguration 0is where we have introduced J g=tanh +T Upon deriving Eq. (8) we have used that pii (o,tlo', 0) = b.The form of Eq. (7) and the initial conditions given in Eq. (8) imply that A,iis afunction of li — jl. Thus we define functions f„(t)by 'R((r) =— J)(r;o.;+i, with Japositive constant. The evolution ofthe system is described by aMarkov process with Glauber dynamics. Therefore, the conditional probability pi~i((r, tl(r', t') of finding the system in the state oat time t, given it was in the state 0' at time t', t' (t, obeys the master equation f-(t) =A'. (t) =(.+-(t) '(o))" with n=i— j.These functions obey the equations t9 0!p ~f=~f +(f+i+f -i) Ot 2 where nis an integer in the range — oo &n(+oo. The above hierarchy, with the initial condition f(0) =rj~ can be easily solved by standard procedures, for instance, using the eigenfunctions method [13],and the result can be expressed as where R,(r is the configuration obtained from oby flipping the ith spin, i.e.,by changing cr; into — o;, and (s(;(o) is the transition rate for the Hip. The expression of this latter quantity is ~;(o.)=—1—— o'(o' — i+o'+i) 21r 11— rj cos rzg (i )t 7l 1+'g p1Qcosg The one-dimensional Ising model we are dealing with has been used as amodel for dielectric polarization [5]. Under certain simplifying hypotheses, the complex permittivity ewas shown to be related to fo(t) through the relation Here o. is another positive constant defining the time scale of the evolution of the system and pis afunction of the temperature Tof the heat bath given by (k~ is Boltzmann's constant) =1— iu) dte '~' pt, (13) 2J p=tanh QT where ep and eare the limiting lowand high-&equency permittivities, respectively. Prom Eq. (12) we have Our aim is to study the equilibrium time correlation function (cr;(0)o;(t)),~. It is convenient to introduce the set of functions (t) 7d— m(1 — Pcos q)t Dt m1+g2 17f -t =ae'Io(apt). 1+92 (14) A' (t) =(*(t) '(0)) =) ) (r;o, 'pi~, (o,tl(r', 0)p.,~((r'), Here Ip is the zeroth-order modified Bessel function. Substitution into Eq. (13) yields where the equilibrium distribution function of the system is 11 E(La)) =Cz 1+r12 [(z~ +o)2 o,2~2]1/2 Bozdemir [5] has also shown numerically that this expression approaches at low temperatures the Cole-Davidson distribution Time derivative of Eq. (5) yields 1 e((d) (1 +zuz~D)PRD ' — A;i(t) =— nA;i(t) +[A, i,(t) +A;+i~(t)]. (7) where 7'CD is aconstant and PcD =1/2. It must be pointed out that Anderson [4] had previously observed 460 J.JAVIER BREY AND A. PRADOS 53 the same behavior in an equivalent model, although in neither of the two references is adetailed analytical derivation of Eq. (16) presented. Lindsey and Patterson have carried out an extensive numerical study about the relation between the two functions fo(t) and e'(tv) and concluded that corresponding approximately to aCole-Davidson function in the &equency domain is aKWW or stretched exponential function for fo(t) [7], i.e., the relaxation spectrum is de6ned as Ag p, =dA (A —AM) g(A). A1 (22) For the long time limit, the asymptotic behavior of Eq. (18) is i/2 — Apt (23) fo(&) =exp[— (&/7Kww)] " They also give an expression, obtained by fitting the numerical data, to obtain PKww &om PCD. In particular, for PCD =1/2 their expression gives PKww 0.63. This latter value has been used often in the literature to compare the relaxation spectrum of Ising models with difFerent dynamics [8,9] and also to claim that different properties ofthe Ising model with agiven dynamics relax with difFerent values of PKww (in the low-temperature liinit) [10]. Here it will be shown that, at low temperatures, the function fo(t) is actually accurately described by a KWW function over atime window that contains most of the relevant part of the relaxation, but the value of the exponent is pKww =1/2. E=1— p(24) and take the limit e((1, which yields g=1—(2e)'i'+ 0(e), AM -n(2e)'~2, (25) (26) i.e.,again an essentially exponential behavior, but in this case with arelaxation frequency Ai. In the hightemperature limit, the relaxation spectrum is very narrow p, (( AM and the relaxation is dominated by the initial exponential Eq. (21). Consider now the low-temperature limit, in which p(T) approaches one from below. To study it we introduce the parameter III. THE LOW-TEMPERATURE LIMIT and A2 fo(t) =dAg(A) e A1 (18) where the relaxation spectrum g(A) is To clearly understand what happens at low temperatures it is convenient to establish first some general features of the function fo(t). From Eq. (12) and by means of asimple change of variable we can write ~1 — 0!C. (27) (28) Therefore, in this limit Ai &( AM and aclear separation of time scales shows up. As aconsequence, there must be an intermediate time region in which the decay is highly nonexponential [3]. We can get more information by considering the average relaxation time (7.)defined as g(A) =- 7r 1+g2 [(A — Ai)(A2 — A)] ~2A (19) and Ai — —n(1 — p) and A2 — —n(1+ p) are the minimum and maximum relaxation rates, respectively. As aconsequence of the inicial condition fo(0) =1, the distribution g(A) is normalized to one. The average frequency is Ag |9 1— g AM =dAAg(A) =—— fo(t) =n,.(20) Bt ~o1+@ The short time behavior of fo(t) is now easily obtained Ag f(t) AM tdA (— A) — (A — A~)t A1 A2 =e"'dAg(A) [1 —(A — AM)t A1 +-, '(A — AM)'t'+ ] — wM e[1+~( 2t2)] i.e.,the initial relaxation is exponential. The width pof It is easily seen that this quantity diverges as efor e— +0. Taking into account that the initial relaxation corresponds to an average relaxation time AM, which goes as e/', we conclude that the initial exponential decay becomes irrelevant in the low-temperature limit. Although it is, of course, possible to study also the behavior of fo(t) at intermediate times using its explicit expression given by Eq. (18), we will follow adifferent approach. There are several reasons for this choice. First, the method we will employ can be useful to study other models for which the exact expression of the relaxation function to be investigated is not known. Second, through our calculations we will obtain an equation for the correlations, valid in the low-temperature limit. Similar equations have been introduced without asolid justiGcation to describe low-temperature relaxation in other models. Adiscussion of such equations in the light of what happens in our case is clarifying. Finally, our approach leads to adirect identification of the mechanisms responsible for the relaxation of the system and to the reason why most of the relevant part of the relaxation is nonexponential. 53 LOW-TEMPERATURE RELAXATION IN THE ONE-. . . 461 Consider again Eq. (11),but now written in the form — f(t) =o— (f —', f-+i —-', f— i) —— (f +i+ f— i) t" "2" 2" 2 with the initial condition -In/ f„(0)=1—(2~)i/2+ 0(e) (30) f(0) e— II/( and comparing with the expression of f(0) we get Aperturbation expansion of the above hierarchy in powers of e/is singular because f„(t)is bounded by one for all t, while the terms of the expansion of f(0) diverge for ~n~ — +oo. This leads us to scale the variable n, so that the terms of the expansion of f(0) remain finite, even in the limit ~n~ moo. The scaling factor emerges quite naturally for the following reasons. For t=0the time correlation function becomes the equilibrium space correlation function. The correlation length is de6ned by 1+k2 (39) and thus the spin autocorrelation function is fp(s) — =f(0,s) =—dk, e'+" 21 vr 01+k2 (40) Use of this expression in Eq. (13) gives, for the permittivity in the low-temperature limit, ditive way. There is apure relaxation term, describing an exponential relaxation, and adifFusive term. An example of simple processes corresponding to each of the two mechanisms is given in Fig. 1. Notice that the definition of the time scale 8implies that the relevant times correspond to t(o.e) i. Therefore, Eq. (37) does not describe the initial exponential relaxation, which, as we saw, takes place over times of order e/' .Nevertheless, we also know that at very low temperatures initial decay is not relevant, in the sense that the correlation function remains practically constant and equal to its initial value. The solution of Eq. (37) is 1 /lnrj/ (32) 1 e(& (] +t~)1/2 ' where we have introduced In the low-temperature limit (diverges, suggesting that it defines the relevant length scale. Since ~lnrI~ (2e) /, we introduce the scaled length xas In terms of this variable the hierarchy given by Eq. (29) reads 0f(x,t) =— — nf(x,t) —'f(x +(2e) -'/, t) f(x+ (2e)', t) +f(x —(2e)'/ )t) (34) where f(x,t) =f„(t). Now we expand the series and neglect terms of order e.The result is |9 0 f(x,t) =a— ef(x,t) — nef(x, t) Ot BX2 (42) Therefore, we have derived the Cole-Davidson distribution with PCD =1/2, in agreement with previous numerical calculations [4,5]. The characteristic relaxation time is AD =(ne), which is of order unity over the relevant time scale s. The long time limit of Eq. (40) is which coincides with the low-temperature limit of Eq. (23). On the other hand, Eq. (40) does not present an initial exponential decay for the reasons discussed above. The limit 8(& 1corresponds to an intermediate time region in which the "slow" time scale sis small, but the real time tis large. This is precisely the time window that must be analyzed to look for the nonexponential or, dehning also atime scale by 8=Oat, (36) 0Cjl2 Z, f(x s) =— f(* s) +Z,f(* s) (37) The initial condition for this equation is f(x,o) =e-~*~. (38) In this equation the two mechanisms responsible for the relaxation at low temperatures show up clearly in an adFIG. 1. Sketch ofsimple processes corresponding to the two mechanisms of relaxation at low temperatures described by Eq. (37): (a) the difFusive term and (b) the purely exponential relaxation. J.JAVIER BREY AND A. PRADOS 53 relaxation. To do so, we rewrite Eq. (40) as 4 1 f(0,s) =1— du u-'/"e — ", (44) which for 8«1can be approximated by — 2 ln [— ln fpl 4 I ln f(0, s) =— Sduu e"~—s~(45) ~1/2 or, returning to the tvariable, 4 ln nt 10 12 (46) This expression holds for nt «e.By putting together the results we have obtained, we arrive at adescription of the relaxation of the spin autocorrelation function through three well defined time regimes: 'exp[— (2e) ~nt] for nt && 1 f(0,t) =&exp[— (4net/vr) ~]for 1&& nt && e (7rnet) ~exp( — net) for nt ))e (47) This behavior is similar to the one found in Ref. [10] for the linear relaxation of the energy following an homogeneous temperature perturbation, also for the Ising model with Glauber dynamics. The only relevant difference is that the prefactor of the exponential in the long time regime goes, in the case ofthe energy, as t/.Moreover, Shore and Zwanzig [14] have obtained the same three temporal regimes in aone-dimensional lattice model. Thus we have shown that at intermediates times fo(t) is described by astretched exponential function with PKww =1/2. Note that this is an exact result, valid in the asymptotic limit of low temperatures. An example is given in Fig. 2, where fo(t) is plotted for e=10 The solid line is the KWW function given by Eq. (46). We can estimate roughly the time interval in which the KWW function gives an accurate description of fo(t) by considering that it is bounded by its intersections t-1 and t2 with the initial and the final exponentials. Prom Eq. (47) one gets ti — —2/nor and t2 0.56/ne. The correlation function at these two times takes the values f(O, ti) =exp[— 2~e~/m] and f(O, t2) 0.42, respectively. If we consider, for instance, atemperature corresponding to e=10,it is f(0,ti) 0.99, showing that the initial exponential decay becomes negligible in the low-temperature limit. On the other hand, the value of f(0,t2) does not depend on the teinperature and there is no reason to expect the KWW function to give an accurate description of the relaxation for values of f(0,t) smaller than 0.4. Between this value and the asymptotic exponential decay at long times the apparent value of PKww increases &om 0.5to 1. Of course, one can force the complete relaxation to be fitted by aunique KWW function, then getting avalue ofPKipvw in the above interval. This is the way in which Lindsey and Petterson obtained numerically the value PKww 0.63. Although it is clear that this value gives some information about the FIG. 2. Plot of the equilibrium spin autocorrelation function for atemperature value corresponding to e=10 .The diamonds are the numerical integration of Eq. (18) and the solid line is the KWW function of Eq. (46). Note that the logarithm scale amplifies the discrepancies, especially for short times where the difference between the exact solution and the KWW function is in fact negligible, as discussed in the text. average relaxation of f(0,t), one cannot conclude from it that the relaxation is less cooperative (the spectrum narrower) in this model than in other systems for which afitting of the data gives PKivw 0.5. IV. THE INITIAL PART OF THE RELAXATION q„(t) =f„(t). |9 (48) They obey the same hierarchy as f,i.e., — q- =— n(q- ——,q--i ——,q+i) —— (q-— i+q-+i), (49) but the initial condition is different. Asimple calculation gives q„(0) =[— n(2e)'~ +O(ne)] 8„o, (50) where bis the Kronecker function. Since the terms in the expansion of q(0) are bounded for all n, we can now obtain aseries expansion solution of the hierarchy given Although the results in the preceding section lead to a complete description of the relaxation of the equilibrium spin autocorrelation function at low temperatures, it is interesting to consider an alternative analysis, which is valid in the short time regime. In fact, the procedure we will follow here is also appropriate for those models in which the final exponential relaxation either does not exist or is irrelevant. This seems to be the case, for instance, if alocal constraint allowing only Hips conserving the energy is introduced in the model we are considering [8,9,11]. As discussed below Eqs. (29) and (30), they are not useful, in their present form, to carry out an expansion in powers of e/.However, the difficulty is easily surpassed by going to the hierarchy for the time derivatives 53 LOW-TEMPERATURE RELAXATION IN THE ONE-. ..463 by Eq. (49). We write q„(t) =— n(2e) it, '„(t)+O(ne). 0.9— ~o~*%+ og + Substitution into Eq. (49) leads to ~C=— n(( —2C -i —2C +i) Ot to be solved with the initial condition (52) 0.5— + o+ + oo+ +0+0 o0+0 +o~ o++ o („(0)=h„p. (5S) — 4— 20246810 12 14 ln o.t 1 g(t) =—dq cos nq e 0(54) and introduction of this into Eq. (51) yields q„(t) =— n(2e) e'I„(nt) +O(ne), (55) Thus gobeys asymmetric random walk master equation with the particle located initially at the origin. The solution of the equation is [13] FIG. 3. Plot of the time-dependent parameter P,s, defined in Eq. (58), for three different values of the temperature, namely, e=10 (diamonds), 10 (crosses), and 10 (circles). For both very short and very long times, P,z-+ 1, while as the temperature decreases an intermediate time window develops with P,z=1/2. The decay from its short time value to its intermediate one is independent of the temperature, as discussed in the text. where I„(t) is the nth-order modified Bessel function. From this result we have t (t) =f,(o) +dt'q, (t') 0 =1—(2e) ~nt e'[Ip(nt) +Ii(at)] +O(net) (56) ln fp(t) =— (2e)'~ nt e'[Ip(nt) +Ii(at)]. (57) This equation can be shown to be equivalent to the one reported by Budmir and Skinner for the constant energy Ising model [15]. If we now consider here nt «1, we recover once again the initial exponential relaxation of Eqs. (21) and (26), while for e)& at )& 1, the KWW decay of Eq. (46) is found. Although Eq. (57) reproduces both the initial exponential decay and the KWW intermediate function, it is appropriate to underline that it is not able to describe most of the relevant part of the relaxation, which takes place on atime scale for which that expression is not valid. Of course, amodification of the dynamics of the system can lead to achange of the relevant time scale in such away that the anal exponential decay is negligible. Then the relaxation is accurately given by Eq. (57). This may be the case in Refs. [9,11]. The analysis in this section has another interesting consequence. Let us de6ne an efFective time-dependent KWW exponent by t9 P,g(t) =ln [— ln fp(t)] .(58) Neglecting O(net) in this expression is not correct for all times. In particular, the result does not verify limy~ fp(t) =0. Only in the limit net && 1can we approximate Use of Eq. (57) yields p,~(t) =(nt e'[Ip(at) +Ii(nt)] j.(59) ~(t) =1—~(t) (6o) where vis aparameter going to zero as T— +0. It follows that ln [— ln P(t)]ln v+ln ((t) (61) and P,~will be independent of v. V. CONCLUSION We have studied in detail the time evolution of the equilibrium spin-spin time correlation functions in the one-dimensional Ising model with Glauber dynamics. In the low-temperature limit, we have derived acontinuous space equation for the evolution of these correlation The important point is that this expression, valid for short times and in the low-temperature limit, does not depend on e. On the other hand, it correctly predicts the decay of P,@from its initial value P,s=1to the intermediate value P,s=1/2. This result is confirmed by the curves plotted in Fig. 3. Note that the time window described by the KWW function, which corresponds to the horizontal part of the curves, increases as the temperature decreases. Asiinilar behavior of p,tr has been found in other models [16] and for difFerent relaxation properties [10,17]. The physical origin of the general property can be roughly explained in the following way. Let P(t) be arelaxation property verifying P(0) =1. As the temperature goes to zero, one expects the system to be frozen and therefore in the low-temperature limit and for short times we can write 464 J.JAVIER BREY AND A. PRADOS 53 functions and identified &om it the main mechanisms of relaxation at low temperatures. For the equilibrium spin autocorrelation function, an exact Cole-Davidson behavior with PCD =1/2 is obtained in the frequency picture, while in the time domain aKWW function, with an exponent PKww =1/2, followed by an exponential is found. The value PKww =0.63 reported previously in the literature could then be considered as giving some information about the average shape of the relaxation curve of the spin time autocorrelation function, but it cannot be concluded that the relaxation in Glauber's model is less cooperative than in other systems with locally constrained dynamics. For the initial part of the relaxation, before the KWW behavior emerges, we have shown that the spin time autocorrelation function decays very little in this time scale, for low enough temperatures. Nevertheless, there are two results concerning this regime that are worth mentioning. First, the relevant KWW function can be obtained in an intermediate time window, matching the result given by the continuous space equation. Second, the parameter P,tr defined in the text decays over auniversal curve in the sense that is independent of the temperature, Rom its short time value P,tr =1to its intermediate time value P,s=1/2. The former corresponds to the initial exponential relaxation and the latter to the KWW function. On the other hand, in this time regime we cannot obtain any information about the final exponential relaxation or, consequently, about the range of validity of the stretched exponential function. It must also be pointed out that almost any model based on amaster equation formulation leads to exponential relaxation in both the short and the long time limits, but with di8'erent characteristic time scales. Then, aKWW function appears in an intermediate time window [3]. We think that difFerent choices of the transition rates simply shift the relevant time regimes (initial exponential, KWW function, and final exponential) to greater or smaller values of the relaxation function under consideration. The same remark remains valid in regard to the relaxation of diferent properties in the same model. For instance, the relaxation of the energy after an homogenous temperature perturbation can also be analyzed in Glauber's model. The continuous space equation governing the dynamics of the system at low temperatures is again Eq. (37), but the initial condition is not the one in Eq. (38), but f(z, 0) =zexp( — z) [17]. The same three time regimes appearing in Eq. (47) show up in this case, but the KWW function extends to smaller values of the relaxation function [10]. ACKNOWLEDGMENT Partial support &om Direccion General de Investigacion Cientifica yTecnica (Spain) through Grant No. PB92-0683 is gratefully acknowledged. [1] G. W. Scherer, Relazation in Glass and Composites (Wiley, New York, 1986); Relazations in Complez Systems, edited by K. L. Ngai and G. B.Wright (U.S. GPO, Washington, DC, 1985). [2] G. H. Fredrickson, Annu. Rev. Phys. Chem. 39, 149 (1988). [3] J. J. Brey, A. Prados, and M. J. Ruiz-Montero, J. NonCryst. Solids 172-1'74, 371 (1994). [4] J.E. Anderson, J. Chem. Phys. 52, 2821 (1970). [5] S. Bozdemir, Phys. Status Solidi B104, 37 (1980); 103, 459 (1981). [6] R. J.Glauber, J. Math. Phys. 4, 294 (1963). [7] C. P. Lindsey and G. D. Patterson, J. Chem. Phys. 73, 3348 (1980). [8] J. L. Skinner, J.Chem. Phys. 79, 1955 (1983). [9] J. Budimir and J. L. Skinner, J. Chem. Phys. 82, 5232 (1985). [10] J.J. Brey and A. Prados, Physica A197, 569 (1993). [11] H. Spohn, Commun. Math. Phys. 125, 3(1989). [12] G. H. Fredrickson and H. C. Andersen, Phys. Rev. Lett. 53, 1244 (1984); J. Jackie and S. Eisinger, Z. Phys. B 84, 115 (1991). [13] N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981). [14] J. E. Shore and R. Zwanzig, J. Chem. Phys. 63, 5445 (1975). [15] The series in Eq. (29) of Ref. [9] can be easily sumined. Note the di8'erence of afactor 2in the definition of the transition rates. [16] P. Harrowell, Phys. Rev. E48, 4359 (1993). [17] J.J. Brey and A. Prados (unpublished).