Velocity and diffusion coefficient of a random asymmetric

Transcription

1 La The J. Phys. France 50 (1989) AVRIL 1989, 899 Classification Physics Abstracts J Velocity and diffusion coefficient of a random asymmetric one-dimensional hopping model C. Aslangul (1, *), N. Pottier (1) and D. Saint-James (2, **) (1) Groupe de Physique des Solides de l Ecole Normale Supérieure (***), Université Paris VII, 2 place Jussieu, Paris Cedex 05, France (2) Laboratoire de Physique Statistique, Collège de France, 3 rue d Ulm, Paris, France (Reçu le 25 octobre 1988, accepté sous forme définitive le Il décembre 1988) 2014 Résumé. vitesse et le coefficient de diffusion d une particule sur un réseau périodique unidimensionnel de période N avec des taux de transfert aléatoires et asymétriques sont calculés de manière simple grâce à une méthode basée sur une relation de récurrence, qui permet d établir une analogie aux grands temps avec un modèle de marche strictement dirigée. Les résultats pour un système complètement aléatoire sont obtenus en prenant la limite N ~ ~. On montre qu un calcul, reposant sur une hypothèse d échelle dynamique, de la vitesse et du coefficient de diffusion dans un réseau désordonné infini conduit aux mêmes résultats Abstract. velocity and the diffusion coefficient of a particle on a periodic one-dimensional lattice of period N with random asymmetric hopping rates are calculated in a simple way through a recursion relation method, which allows for an analogy at large times with a strictly directed walk. The results for a completely random system are obtained by taking the limit N ~ ~. A dynamical scaling calculation of the velocity and of the diffusion coefficient in an infinite disordered lattice is shown to yield the same results. 1. Introduction The problem of drift and diffusion on one-dimensional lattices with random asymmetric hopping rates has recently received considerable attention [1-11]. Bias appears quite naturally when general models with random hopping rates are considered, i. e. when the restriction of symmetry of transitions between sites is relaxed. As underlined for instance in [1], during a biased random walk, the diffusing particle drifts in a preferential direction. Depending on the particular model, either the velocity and the diffusion coefficient are finite, or anomalous drift or diffusion behaviours may appear when the bias varies, thus giving rise, in certain random structures, to the so-called «dynamical phase transitions» [1]. Article published online by EDP Sciences and available at

2 900 In the present paper we generalize to any random asymmetric hopping-model with non-zero hopping rates a recursion method first proposed by Bernasconi and Schneider [7,8] for the study of the drift properties in a particular asymmetric model which they call the «diodemodel». We show that, in any random asymmetric model with non-zero hopping rates, the drift velocity (when it is finite) can be calculated by the recursion method, which indeed reveals to be a very natural and instructive technique. Moreover, we show that the recursion method allows equally for the calculation of the diffusion coefficient when it is finite. As a matter of example, two interesting particular cases may be studied : a constant bias model, in which the asymmetry of the transfer rates has its origin in the existence of an applied constant bias, and a local random force model, in which the local forces between neighbouring sites are random quantities. We consider a periodic one-dimensional hopping model of arbitrary period N, as the one studied by Derrida in [5]. In this paper, Derrida directly calculates the properties of the steady state and obtains exact expressions for the velocity and for the diffusion coefficient. He deduces the corresponding quantities for a random system by taking the limit of an infinite period. In the present paper, we first propose an alternative derivation of the expressions of the velocity and of the diffusion coefficient, when these quantities are finite, based on the recursion relation method. As in [5], we first study a periodic chain of arbitrary period N, and we then take the limit of an infinite period. The results we obtain for the velocity and for the diffusion coefficient indeed coincide with those of [5] : for instance, in the constant bias model, the velocity and the diffusion coefficient are always finite, while in the local random force model, «dynamical phases» (with anomalous drift and diffusion properties) may appear for certain ranges of values of the bias field [1]. However, the main interest of this part of our paper does not lie in these results, which were previously found [5], but lies in their derivation, which is simple and illustrative, and also gives insight into the behaviour of various quantities to be used in the dynamical scaling treatment presented in the last section of the paper. Let us comment briefly upon Derrida s procedure. It can be easily shown by linear algebra techniques [12] that the dynamics at large times, for finite N, is governed by the zero eigenvalue (linked to the conservation of probability) of the matrix associated with the master equation of the problem (Eq. (1)). This implies that, after a time larger than the inverse of the smallest non-zero eigenvalue, Derrida s regime (drift and diffusion) is correct up to a finite number (= 7V - 1) of exponentially small decaying terms. Otherwise stated, the long-time behaviour assumed in [5] is indeed the correct one. The only problem remaining with Derrida s procedure is the question of its validity when one takes the limit N in order to picture an infinite chain. This question is handled in a forthcoming paper [17], in which we demonstrate that Derrida s procedure actually yields the correct V and D, when these coefficients are finite. Secondly, we give a simple recipe, explicitly using the preceding formalism, and based on the dynamical scaling properties of the configuration averaged medium : this leads to a direct computation of the velocity and of the diffusion coefficient, when these quantities are finite, by using only the asmptotic form of the probability of retum to the origin. Interestingly enough, this probability suffices to determine the exact transport coefficients of interest. This derivation, presented here for a discrete lattice, is closely parallel to an independent derivation for a continuous medium presented in reference [11]. To our best knowledge, this simple derivation, introduced in [15], is used for the first time in an asymmetric disordered lattice. The paper is organized as follows : in section 2, we describe the model and we give the

3 901 general notation for this problem. The master équation which governs the problem is written down, and various Laplace transformed quantities, which will be used in the following, are introduced. In section 3, we show how the recursion relations can be solved in the long-time limit. In section 4, we calculate the velocity and the diffusion coefficient, when these quantities are finite. Finally, in section 5, the dynamical scaling approach is developed. 2. Model and general notations. We consider here the random motion of a particle on a regular one-dimensional lattice of spacing a, as described by the master equation in which p,, (t) n at time t ± 0. We assume that denotes the probability of finding the particle on the site of index i.e. at time t 0 the particle is localized = on site n 0. The transfer rates Wij = variables and are not assumed to be symmetric, in other words Wij is are random not equal to Wji. In addition, the pairs (Wi, i -, 1, Wi, 1, i ) are assumed to be independent from one link to the other. We exclude here the case where any Wij vanishes. Equation (1) thus describes a random walk in a random environment, the so-called «random random walk». We are interested in the long-time asymptotic behaviour of the first two moments of the particle position, x (t ) and X2(t ), defined, for a given configuration of the transfer rates, as (x (t ) and x2(t ) are measured in lattice units). To calculate the velocity V and the diffusion coefficient D, we use the standard definitions At this stage V and D are expressed for a given configuration of the transfer rates and no average over the transfer rates is carried out. It is convenient to perform a Laplace transformation of the master equation (1), which yields where we have defined

4 902 Following Bernasconi and Schneider [7, 8], we associate the quantities G (z ) as defined by with the sites of indexes n (n ± 0) Similarly, we associate the quantities Gn (z) as defined by with the sites of indexes - n (n -- 0). The master equation (7) is easily seen to be equivalent to the set of equations In the particular case of a strictly directed walk model, in which only steps towards the right are allowed, the quantities G,,- (z) identically vanish whereas the quantities G (z) coincide with the transfer rates, and are thus simple constants. Let us now return to the general case. The quantities Pn(Z), which are solutions of the master equation (7), can in turn be expressed as [8] : The G (z) are infinite continued fractions, recursively defined by Similarly, the Gn (z) also are infinite continued fractions, recursively defined by As shown by equations (17) and (18), the quantities G+ (z) and G- (z) contain (in a way easier to handle, a point upon which we shall come back later) the same information about the disordered system as the transfer rates do, as soon as the initial condition (2) has been retained. We shall now show that, in the long-time limit, a physical solution for the recursion relations (17) and (18) can easily be obtained, which in tum, using equations (14)-(16), will yield the probabilities pn (t ). This will be done in section 3, in which this solution will be given in the most general asymmetric system. Of course, one is not sure that no other solution for

5 903 the highly non-linear equations (17) and (18) does exist, but it will be argued that the solution we obtain is indeed the physical one. In order to clarify the discussion, it will be at times worthwhile to refer to the two following interesting particular models, in which one considers thermally activated transfer rates. Model (i ) : constant bias model In this model, the asymmetry of the transfer rates has its origin in the existence of an applied constant bias, i.e. one has Wn, n + 1 denotes the random (symmetric) hopping rate between sites n and n + 1 in the absence of the external bias field E. The bias energy is assumed to be negative, so that in the in the long-time limit a positive velocity is expected. Model (i i ) : local random forces model Another interesting model is that in which the local forces are random variables, i.e. 2 kb T cf> n, n + 1 denotes the local random force between sites n and n + 1, which one may for instance assume to be Gaussianly distributed, with a positive mean value. As we shall see in section 4, in the first of these models, V and D are always finite while, in the second one, dynamical phases (with anomalous drift and diffusion properties) may appear for certain ranges of values of the bias field. 3. Solution of the recursion relations. Let us for the present time consider the most general asymmetric model and solve the recursion relations (17) and (18). We shall assume from now on that a condition which means that the average bias field is directed along then n > 0 axis. Clearly, all the results are easy to transpose to the opposite case ; however, the marginally asymmetric case (log (Wn, n + l/wn + 1, n) > = 0 is excluded from the present study. (This last case has been studied by Sinai [13] within the framework of a discrete time model and is known to present an anomalously slow (logarithmic) diffusive behaviour). Condition (22) will insure in particular that, if a finite velocity does exist in the system, it will also be directed along the n ::> 0 axis. In equation (22), the symbol (... ) denotes the average over the disordered transfer rates, i.e. the configuration average. Just as in [5], it will reveal convenient to consider a periodized chain of period N, i.e. a chain in which It is easily seen that, due to the periodicity of the transfer rates, a periodic solution of the

6 904 recursion relations (17) and (18) for the quantities G (z ) and G- (z ) can be found. From now on, we shall only consider this physically sensible solution, and show that it actually yields the same results as [5], which are indeed the correct ones, as discussed in the introduction. Note however that, since the initial condition (2) is not periodic, the Pn(Z) themselves are not periodic quantities, even in a periodized chain. We shall now demonstrate that, in such a periodized chain, and in the long-time limit (t - oo, i. e. z - 0), the physical quantities GI (z ) and G- (z ) display the following behaviours (once the choice (22) has been made) : where the coefficients G+ (0), 9+ and g- are independent of z. Indeed, since the probabilities Pn (t ) for finite N are always given by a finite sum of exponentials, their Laplace transforms Pn (Z) only involve integer powers of z. The same conclusion obviously holds for the quantities G+ (z) and G- (z) (Eqs. (9)-(10)). Moreover, it can be seen on the recursion relations (17) and (18) that non-integer powers of z would break the translational invariance of the configuration averages. As will be shown below, the dominant balance method indeed yields unique values for the coefficients G ( ), g and g-, and could allow for generating the following terms of the asymptotic expansion, if needed. In addition, it can be seen that the choice of the periodic solution as well as the expansions (24) and (25) are in accordance with the particular case of the strictly directed walk model, in which the quantities G+ (z ) coincide with the periodic Ws whereas the quantities Gn (z) identically vanish. 3.1 LONG-TIME SOLUTION OF THE RECURSION RELATION FOR THE G+ (z). - The recursion relation (17), when expanded at zero order in z, and iterated for instance between sites n and n + N (n -- 0), yields For the periodic solution, equation (26) amounts to Thus, whatever the period N of the chain, provided that it is finite, the quantities G (0) are well defined, and given by finite sums. Since we aim at considering the limit of an infinite period (N - oo ), let us investigate equation (27) in this limit. Owing to condition (22), the product appearing in the l.h.s. of equation (27) can be neglected with probability 1 in the limit N - 00 ([3], [5]). Note that, for a given configuration of the transfer rates, the infinite series in the r.h.s. converges almost everywhere, provided that condition (22) is satisfied. This allows to use the G s obtaining by this limiting procedure in an infinite medium (see sect. 5). For an infinite period, it is easy to take the configuration average of both sides of equation (27). The resulting averaged infinite series which then appears in the r.h.s. may be convergent. Since the Wij are uncorrelated, this is the case whenever the condition

7 905 is obeyed. (The notation is obvious : W- replaces W,, -,,,, and W +- replaces Wn, n + 1). One then gets Note that, as expected, this average value does not depend on the site index n. Conversely, if condition (28) is not obeyed, the average value (l/g: is infinite. Similarly, when expanded at order 1 in z and iterated for instance between sites n and n + N (n -- 0), the recursion relation (17) yields Because of the assumed periodicity, equation (30) reads : Again, provided that N is finite, the quantities g,, + / [Gn (0)]2 are well defined and given by finite sums. The same arguments can be applied to equation (31) as to equation (27), configuration average, for N - oo, is given by to show that the (provided that Eq. (28) is fulfilled). In order to get an explicit expression, it is necessary to analyse l/ [G+ (11)]2 When N --+ ao, equation (27) yields The configuration average of both sides of equation (33) can easily be taken, provided that the correlations between transfer rates on the same link are properly taken into account. As a result, the series which appear in the r.h.s. are convergent only when condition (28) and

8 The 906 are simultaneously obeyed. When this is the case, one gets Il one of the two requirements (28) or (34) is violated, the average value (l/[g,i (0)]2) is infinite. Once the average value (1/ [G: (0)]2) is known, equation (32) yields the average value (g: / [G: (0)]2) LONG-TIME SOLUTION OF THE RECURSION RELATION FOR THE G; (z). recursion relation (18), when expanded at lowest order and iterated for instance between sites - n and - n - N (n. 0), yields Because of the assumed periodicity, equation (36) amounts to Here again, for N finite, the quantities gn are well defined, and given by finite sums. Following the same line of arguments as above, one gets, in the limit of an infinite period provided that condition (28) is obeyed. Otherwise, the average value (g;;) is infinite. In summary, we have shown that, in periodized chains of finite period N, the long-time behaviour of the periodic solution G (z) and G- (z) is indeed of the form displayed by equations (24) and (25). As will be demonstrated in the following section, the corresponding velocity V (N ) and diffusion coefficient D (N ) are always finite. On the contrary, when the limit N - ao is taken, various behaviours may exist, depending on the model. For instance, in the constant bias model V and D are always finite, while in the local random force model, dynamical phases (with anomalous drift and diffusion properties) may appear for certain ranges of values of the bias field [1]. This will be investigated in the next section. Let us now specialize the preceding results for the two particular models quoted above. 3.3 CHAIN OF INFINITE PERIOD : CONSTANT BIAS MODEL. - In this model, since the bias

9 energy E is assumed to be negative, the convergence condition (28) is automatically obeyed, so that the configuration average (11G+ (0» is always finite. As a result, one gets from equation (29). 907 In the same way, conditions (28) and (34) are automatically obeyed and the configuration average (1/ [G,i (0)]2) is always finite. Equations (32) and (35) yield and g - 1 and 9-2 denote the first two inverse moments of the random (symmetric) quantity Wn, n, 1, as defined by equation (19). Similarly, the average value (g; > is always finite ; one gets from equation (38) 3.4 CHAIN OF INFINITE PERIOD : LOCAL RANDOM FORCES MODEL. - In this model, condition (28) reads where 0 stands for CPn, n + 1. When this condition is obeyed, one gets from equation (29) As for condition (34), it reads When conditions (43) and (45) are both fulfilled, the configuration averages (1/[G; (0)]2) and (g / [G; (0)]2) are finite and given by and

10 908 The average value (g;) is finite when condition (43) is fulfilled ; in this case one gets from equation (38) 4. Calculation of the velocity and of the diffusion coefficient. 4.1 VELOCrrY. - Let x(z) denote the Laplace transform of the particle position XUJ, as defined by equation (3), for a given configuration of the transfer rates. One has By taking equations (11)-(13) into account, one can rewrite equation (49) as or, equivalently The quantities S± are infinite series defined by with For the present time, let us consider a periodized chain of period N. Using the periodic solution G (z) and G- (z) greatly facilitates the study of the a priori infinite series S (z) and S- (z), since one can transform them into finite sums. Indeed, one obtains By applying general theorems about inverse Laplace transformation [14], the long-time asymptotic behaviour of X(t) can be deduced from that of its Laplace transform flfi when z - 0. Combining equations (14), (24) and (25), it is seen that, when z --+ 0,

11 909 Similarly, a careful eximination of equation (51) shows that, when z - 0, where only leading order terms have to be retained in S+ (z) ; involving S- (z) in the r.h.s. of equation (51) yields only subdominant quantities. For a given period N, the finite sum S+ (z) behaves like note that the second term Hence, for a given configuration of the periodized chain This in turn implies that, in the long-time limit t --+ oo, so that, as long as N is finite, a finite velocity V (N ) exists, equal to a result which is indeed in accordance with equation (49) of reference [5]. Since, as discussed in the intrôduction, the correct results of reference [5] are recovered, this justifies our choice of the periodic solution for the quantities G (z ) and G- (z). Expression (60) of the velocity is valid for a given configuration of the transfer rates ; no average over the transfer rates has been carried out. However, since all the G" s of a cell appear in equation (60) ; the drift regime for a given configuration makes sense once the particle has covered a distance corresponding to the cell length. Let us now consider the limit N --+ oo. One finds Note that the denominator in equation (61) introduces the average of l/gt (0) Therefore, in the limit N -+> oo, V no longer fluctuates and, according to the conventional terminology, is a self-averaging quantity. As discussed above, the corresponding behaviour

12 910 would be obtained only at infinite time. However, if a configuration average would have been taken on an infinite random lattice, it can be seen that this behaviour would be reached at finite time (provided that the velocity is finite, see below). We shall establish this point in a forthcoming paper [17]. Note the very simple form taken by V, which can be given a direct physical interpretation. Indeed, using expansions (24)-(25) in equations (11)-(13), one obtains a directed walk model, for which V takes the form (60) for a periodic system or (61) for a non-periodic system taken from the start [17]. The average value (I/Gt (0» has been calculated in section 3. As a result, for a given configuration of a disordered chain, a finite velocity does exist if condition (28) is obeyed. In such a case, using equation (29), V is seen to be equal to Otherwise, velocity vanishes, which indicates an anomalous drift behaviour [11]. Physically, this means that, although an average bias does exist in the system, if, as measured by the 1, the statistical weight of the transition rates towards the opposite criterion (W_ IW_ > >» direction is important enough, the asymptotic behaviour of x tf) is anomalous. Indeed, it has been found in similar models ([3, 11]) that, in this range of values of the bias field with the exponent g as given by Let us now examine in more detail models (i) and (ii) as described in section 2. Model (i) : constant bias model In this model, the configuration average (11G:l (0» is always finite and given by equation (39). One obtains a normal drift regime with a finite velocity (where the explicit dependence on the lattice parameter has been restored). Model (ii) : local random forces model In this model, the configuration average (11G: - (0» is finite (and given by equation (44)) when condition (43) is obeyed ; the velocity is then finite. Otherwise, it vanishes. It is interesting to make criterion (43) precise for a finite velocity when 0 is a Gaussian random variable. If m and u respectively denote the mean value (positive) and the variance of cp, the exponent g is equal to the ratio mlau (a is the lattice spacing) and condition (43) can be rewritten as For a given temperature, CP n, n + 1 is proportional to the local random force between sites

13 We 911 n and n + 1 ; therefore a finite velocity exists provided that the ratio of the average bias - as measured by m to the strength of disorder - as measured by o- is high enough. The value of velocity is then given by 4.2 DIFFUSION COEFFICIENT Principle of the calculation. - begin here by briefly sketching the procedure to be worked out for the calculation of the diffusion coefficient, as defined by equation (6). For a given configuration of the transfer rates, we have to determine the long-time behaviour of the first two moments of the particle position, X{t) and X2(t ). This will be done through a Laplace transform analysis. As discussed in the introduction, it can be readily demonstrated in various ways [5], [12], that in a periodized chain of finite period N and in the long-time limit, the quantities Xl!) and x2(t) behave as Therefore This behaviour can also be obtained by making use of the periodic G s and of equations (24)- (25), which is indeed a proof of the correctness of these expansions. Using the explicit expressions of V (N ), xl (N ), A (N ) and B (N ), we shall verify that, as expected, a compensation of the kinematical terms does occur, thus giving rise, in the longtime limit, to a normal diffusive regime with a finite diffusion coefficient At the end of the calculation, the limit N --> oo will be taken. In this limit, various behaviours, normal or not, may exist, depending on the model Periodized chain : long-time behaviour of x (t ). - The Laplace analysis initiated for the calculation of the velocity has to be pursued to one order further. A careful examination of equation (51) shows that, when z - 0, it is sufficient to retain terms up to order llz in the expression since the second term involving S- (z) in the r.h.s. of equation (51) yields only subdominant quantities. Let us note that The coefficient À is easily obtainable ; however, as will become apparent later, a detailed expression of this quantity is not necessary for the calculation of the diffusion coefficient. It is straightforward to obtain from equation (54) the two dominant terms in the expansion

14 Let 912 of the finite sum S+ (z). As a result, one gets where we have introduced the finite velocity V (N ) of the periodized chain of period N, as given by equation (60). Equations (74)-(75) show that, when z - 0 which displays the announced behaviour of x(!) (Eq. (69)). The explicit expression of xi(n) is x2(z) denote the Laplace transform of the second moment of the particle position x2(t), as defined by equation (4), for a given configuration of the transfer rates. One has Period chain : long-time behaviour of X2(t ). - By taking equations (11)-(13) into account, one can rewrite equation (78) as It is useful to introduce the infinite series with the ut (z) as defined by equation (53). Note the obvious identity As above, for a periodized chain, the infinite series S (e, z) and S- (e, z) transformed into finite sums. One obtains can be

15 913 Equation (79) can be rewritten equivalently as A careful examination of equation (83) shows that, up to order 1 Iz2 If is straightforward to obtain from equation (82) the two dominant terms in the expansion of the finite sum as (e, z)lae 1 e = 1. As a result one gets where, as in equation (75), we have introduced the finite velocity V (N ) of the periodized chain of period N, as given by equation (60). Equations (74) and (85) show that X2(z) behaves, when z 0, as which displays the announced behaviour of x 2(t )(Eq. (70)). A (N ) is seen to be equal to V 2(N) ; the explicit expression of B (N ) is As expected on physical grounds, a compensation of the kinematical terms does occur in the quantity x 2(t ) _ (X (t))2 in the long-time limit ; this means that, as long as N is finite, a diffusive regime exists. The expression of the diffusion coefficient follows from equation (72) D (N ) can be given the remarkably simple form

16 In 914 or, equivalently, by making use of the expression (60) of V (N ) The above formulas (89) or (90) for D (N ) correspond to equation (47) of reference [5], but here they take a much more condensed form, as for the velocity, related to the fact that, for the considered initial condition, the quantities GI (z ) and G- (z ) are easier to handle than the transfer rates themselves. We shall come bacl to this point later. Let us emphasize that, just as V (N ), D (N ) is obtained for a given configuration of the transfer rates. A discussion of the range of validity in time of the diffusive regime could be modelled on the corresponding discussion for the drift regime Chain of infinite periode - the limit N -+ oo, one gets Thus, in this limit, D no longer fluctuates, and, like V, is a self-averaging quantity. The numerator in equation (91) is equal to the average of (1+2 gt )/2[Gt (1)]2 while the limit of the denominator is given by equation (62). Several situations may therefore occur, depending on whether the various average values are finite or not. In this case, the average values 1 ] 2 and /[?, ) finite, and equal to are finite. D is thus also In this case, the average values (1/ [Gt (0)]2) and (gt / [Gt (0)]2) are infinite. D is therefore infinite, which indicates an anomalous diffusion behaviour. Physically speaking, this

17 915 means that the particle may remain trapped on some favourable sites of the lattices, with an anomalously large dispersion of its position. Indeed, it has been found in a similar continuous medium model [11] that, in this range of values of the bias field with g as defined by equation (65) Zero velocity regime W_ IW_ > 1). - The average values (1/ [Gt (0)]2) (gt / [Gt (0)]2) and are infinite. Since the denominator in equation (91) also tends towards infinity, the value of D cannot be directly extracted from this equation. One therefore has to resort to other arguments. For a similar continuous medium model [11], it has been found that D is equal to zero in some domain containing the marginally asymmetric case (log (W+-/W-.» = Remind that this case (excluded from the present study) is the continuous time analog of Sinai s model [13], which is known to present an ultraslow increase of X2(t ), corresponding to a zero value of the diffusion coefficient. Let us now turn to models (i) and (ii) as described in section 2. Model (i ) : constant bias model In this model, the configuration averages are always finite and respectively given by equations (39)-(41). One obtains a normal diffusive regime, with the diffusion coefficient 0. (We have restored the explicit dependence on the lattice parameter.) The absolute value reflects the fact that D must be an even function of a. In an ordered biased lattice, the second term in equation (95) would disappear and V and D would obey a generalized fluctuation-dissipation relation which, in the small bias regime, reduces to the standard Einstein form The result (95) indicates that, in a disordered lattice, a supplementary contribution does appear in the diffusion coefficient, due to the fluctuation of the transfer rates, as revealed by the bias field. However, in the small bias regime, this term is negligible, so that the standard Einstein relation remains valid. Model (i i ) : local random force model Just as for the velocity, we discuss the diffusion in this model when 0 is a Gaussian random variable of (positive) mean value m and of variance cr. We recall that the exponent IL is equal to the ratio m / a (1. In the finite velocity regime (IL > 1 ), the diffusion coefficient is finite when IL >- 2 and infinite when 1 g 2. For IL > 2 one gets

18 Before 916 In the zero velocity regime (1 : 1), the value of D cannot be directly extracted from equation (91). In a similar continuous medium model, it has been shown by other arguments [11] that D is equal to zero in a range of values of g extending up to g = 1/2, and infinite when 1/2 : IL : 1. It is not known whether this result can be exactly extended to the present discrete lattice model. Obviously enough, the question of the existence of an Einstein relation between V and D only makes sense when the drift and diffusion properties are normal, that is, when IL > 2. If the lattice were ordered, that is, if 0 have no dispersion around its mean value m, V and D would obey a generalized fluctuation-dissipation relation.vhich, in the small bias regime, reduces to the standard Einstein form since 2 kbtm denotes the average bias force. It is easy to verify that, in a disordered lattice, where the dispersion of 0 around its mean value has to be taken into account, a supplementary contribution does appear in the diffusion coefficient. Since the velocity vanishes up to g = 1, the response to the average bias is not linear, even in the small bias regime. Thus, even the standard Einstein relation in the small bias regime is violated in the presence of disorder in this model A few additional comments. concluding this section, let us first briefly comment on the procedure used to calculate the velocity and the diffusion coefficient. Following reference [5], we first consider periodized chains of period N. In such chains, for any given configuration, V (N ) and D (N ) are always finite and respectively given by formulas (60) and (89) (or (90)). When the limit N --+ oo is taken, V and D become independent of the details of the particular configuration ; in other words, as stated above, in this limit V and D are self-averaging quantities. However, it is worthwhile to point out that this property of the transport coefficients V and D as a whole is not separately shared by the auxiliary quantities x, and B introduced in equations (69) and (70) : indeed, it can be seen on the expressions (77) and (87) of x, (N ) and B (N ) that these quantities, considered separately, contain non selfaveraging contributions which, even in the limit N - ao, continue to fluctuate from one configuration to another one. Interestingly enough, these non self-averaging contributions cancel each other even at finite N in the combination of interest D (N) == B (N ) - V (N ) xl (N ). The fluctuation of the subdominant term xl (N ), which diverges in the limit N --+ oo, is another indication of the fact that the time required to reach the drift regime for a given configuration is itself infinite in this limit, as already noted. A similar conclusion could probably be drawn for the time necessary to reach the diffusive regime, if one could go one step further in the expansions. However, as indicated above, if a configuration average would have been taken on an infinite random lattice, it is physically plausible that this behaviour would be reached at finite time (provided that the diffusion coefficient is finite). Our second comment (which is not independent of the foregoing one) concerns the following question : what would have one found for the velocity and the diffusion coefficient if one would have considered prima facie an infinite random system? Indeed, as underlined in reference [5], it is not obvious that the velocity and the diffusion coefficient of the infinite random system are the same as found by taking the limit

19 917 N ---* 00 in expressions (60) and (89) (or (90)). Clearly, in order to assert that V and D, as obtained in this framework, represent the proper velocity and diffusion coefficient of the infinite random system, it is necessary to demonstrate that the two limits t - 00 and N --> oo actually commute. In fact, a direct calculation of the velocity and of the diffusion coefficient in the infinite random system is not so easy to achieve. For instance the long-time behaviour of the first two moments of the particle position x (t ) and X2(t) might well not be in an infinite system of the form (69) and (70). From a technical point of view, the periodization trick allows for transforming the infinite series S (z) and S:!:. ( g, z) into finite sums ; the behaviours (69) and (70) of x(t) and x2(t ) are thus safely obtained through expansions of the finite sums S (z) and as+ (e, z)lae 1 e = 1. It is not a priori obvious whether such behaviours still hold when one works on infinite series. In a forthcoming paper [17], we shall consider from the start an infinite random system ; we calculate V and D, which tum out to be actually given by the same expressions as above, which justifies the periodization trick [5]. In addition, we shall equally demonstrate that both V and D are self-averaging quantities. 5. A direct dynamical scaling approach. The calculations of the preceding sections make use of the dynamics on the whole lattice, that is, require the knowledge of all the probabilities Pn (t ). However, it is well-known that, in the case of a biased ordered lattice (O.L.) (in which the velocity and the diffusion coefficient are always finite), one has [1, 11] In the long-time limit, equation (101), when expanded up to order z, reduces to Thus in a biased ordered lattice, the knowledge of po.l.(z) up to order z is sufficient for the determination of V and D. The dynamical scaling approach, first introduced for the unbiased case in [15], can be easily extended to the biased case in the following way : one assumes that, for a random system with a finite velocity and a finite diffusion coefficient, the particle at large time obeys in the average an ordinary diffusion equation as in a biased ordered lattice. In this approach, the basic quantity tums out to be the configuration average (Po(z)) of the Laplace transform of the probability of return to the origin, calculated in the long-time limit. Indeed one has Thus in this framework the knowledge of (Po(z) up to order z is sufficient for the determination of V and D, and indeed yields the correct values. However, it is interesting to see whether in the present model such an assumption can be a priori justified to some extent for the present case of an asymmetric random hopping model ; this can be done by comparing

20 918 the configuration averaged quantities (P n (z) and (P - n (z» (n 1, 2,... ) in the long-time = limit with the corresponding quantities in an ordered lattice (we shall restrict this examination to the = leading contributions (P n (z = 0 )) and (P_ n (z 0 )). The average values of interest in the infinite random medium will be borrowed from section 3, where they have been computed by taking the limit N - ao in thé corresponding expressions for a periodized chain of period N. Actually, we have demonstrated [17] that this procedure gives the correct results for an infinite lattice. 5.1 CALCULATION OF THE LONG-TIME BEHAVIOUR OF (Po(z». - This quantity is particularly easy to compute. Indeed, by looking at the expression (14) of Po(z), one gets, up to order z As can be seen on the recursion relations (17) and (18) for the quantities GI (z) and Gn (z), there are no correlations between 96 and Gt (û); therefore The expansion up to order z of (Po(z)) is thus given by Clearly, this only makes sense when the different coefficients are finite. A detailed discussion of the conditions under which this is the case has been given in section 3. In particular, when the two conditions (28) and (34) are simultaneously obeyed, all the average values appearing in equation (108) are finite. It is interesting to note the equality so that one finally obtains Clearly, equation (110) yields for V and D the same formal expressions as obtained above. In other words, this demonstrates that (Po(z)), expanded up to order z, suffices to determine V and D, as in an ordered biased lattice. Let us now see, by examining the long-time limits of the configuration averaged quantities (P"(z)) and =... (P - n (z» (n 1, 2, ), how the scaling hypothesis can be justified to some extent. 5.2 CALCULATION OF THE LONG-TIME LIMITS OF (Pn(z» AND (P -n(z» (n = 1,2,...).- Since an average bias field, directed along the n :::. 0 axis, has been assumed to exist, we have

21 The We We 919 to discuss separately the long-time behaviour of the probabilities of finding the particle on sites of positive or negative index n Calculation = = of (P nez 0» (n :::. 0). begin by calculating (Pl (z 0)) ; the = quantities (Pn(z 0)) with rc =A 1 will then follow from it step by step. By looking at equation (15) for Pl(z) one gets Therefore so that, step by step, it follows that Thus (P" (z =0)) does not depend on the site index n ; this long-time limit is actually similar to the corresponding one in an ordered lattice, which can be deduced from equation (102) Calculation = = of (P -n(z 0» (n:> 0). begin by calculating (P -l(z 0 ) > ; the quantities (P - n (z = 0)) with n :0 1 will the follow from it step by step. By looking at equation (16) for P- 1 (z) one gets There are no correlations between the ratio W -1, o/wo, -1 i and G 0 + (0), so that Therefore, at leading order so that, step by step, it follows that Thus (P -,, (z = 0)) decreases exponentially with the distance n to the origin ; this long-time limit is actually similar to the corresponding one in an ordered lattice, which can be deduced from equation (103). 5.3 DYNAMICAL SCALING ASSUMPTION : DISCUSSION OF THE RESULTS FOR THE VELOCITY - AND FOR THE DIFFUSION COEFFICIENT. results (113) and (117) indicate that, in the long-time limit, the average values of the probabilities of finding the particle on a given site behave in a way similar to that in an ordered lattice. One must emphasize that these results clearly cannot per se be considered as a sufficient proof of the dynamical scaling hypothesis. However, this hypothesis allows for a simple recipe for computing the velocity and the diffusion coefficient, when these quantities are finite.

22 in when is 920 All the results which can be deduced from the present approach are strictly identical to those derived in section 4. Note that the derivation based on the scaling hypothesis presented here for a discrete lattice is closely parallel to a derivation for a continuous medium presented in reference [11]. 6. Conclusion. In this paper we showed how the velocity and the diffusion coefficient in any random asymmetric model with non-zero hopping rates can be calculated, when they are finite, with the help of a recursion method first proposed by Bemasconi and Schneider [7, 8] for the study of the drift properties in the so-called «diode-model». The basic quantities G (z) and Gn (z ) of the recursion relation method proved indeed to be extremely efficient tools for the solution of this problem, so that a few comments on this point are of valuable interest. Clearly, as soon as the initial condition (2) has been retained, the quantities G (z) and Gn (z) contain the same information about the disordered system as do the transfer rates. Their physical meaning is most easily seen by considering equations (11)-(13). In periodized chains of finite period N, we argued that - assuming an average bias field directed along the n > 0 axis -, the physical quantities of interest Gn (z ) and Gn (z ) may be taken periodic and in the long-time limit (t , Le. z -+ 0) respectively tend towards finite constants G;- (0) or behave proportionally to z (Eqs. (24) and (25)). This assumption actually yields the correct values for V and D. Therefore one may in some sense consider that the long-time asymptotic properties of the walk are similar to the properties of a strictly directed walk, in which only steps towards the right are allowed [11]. The role of the equivalent transfer rates towards the right of this strictly directed walk is played by the quantities Gn (0); these equivalent transfer rates however are not uncorrelated quantities. This remark about the long-time asymptotic properties of the walk is in accordance with a real-space renormalization calculation by Bernasconi and Schneider [16]. In this interpretation, 11G+ ( ) appears to periodized chains of finite period N - the typical time (for a given - represent configuration of the transfer rates) needed by the particle to leave the site of index n ; this interpretation can equally be deduced from the following equality - - The value (63) of the velocity it is finite in accordance with this interpretation. Note that the introduction of the G s allows for a straightforward obtention of V. The analogy at large times with a strictly directed walk may probably be extended. Indeed the diffusion coefficient, as given by equation (93) when it is finite, is equal tao In the strictly directed walk with equivalent transfer rates G (0), the last factor in the above expression would be absent. Obviously this detailed balance factor takes into account the weight of the transitions towards the left and the retarded character of the motion. This may be interpreted by considering a strictly directed walk on a space-renormalized lattice, in which the distance between the new sites takes into account the possibility of retums towards the left in the initial lattice.

23 In addition, we have shown that for this problem the knowledge of (Po(z» suffices for determining the physical transport coefficients of interest when they are finite, which extends the validity of the scaling hypothesis [15] to the asymmetric case. 921 Acknowledgments. We are indebted to Drs. J. P. Bouchaud and A. Georges for helpful discussions. comments and Références [1] See for instance the reviews by HAUS J. W. and KEHR K. W., Phys. Rep. 150 (1987) 263, and by HAVLIN S. and BEN-AVRAHAM D., Adv. Phys. 36 (1987) 695 and references therein. [2] STEPHEN M. J., J. Phys. C 14 (1981) L [3] DERRIDA B. and POMEAU Y., Phys. Rev. Lett. 48 (1982) 627. [4] DERRIDA B. and ORBACH R., Phys. Rev. B 27 (1983) [5] DERRIDA B., J. Stat. Phys. 31 (1983) 433. [6] BILLER R., Z. Phys. B 55 (1984) 7. [7] BERNASCONI J. and SCHNEIDER W. R., J. Phys. A 15 (1983) L 729. [8] BERNASCONI J. and SCHNEIDER W. R., Helv. Phys. Acta 58 (1985) 597. [9] SCHNEIDER T., SOERENSEN M. P., POLITI A. and ZANNETTI, M., Phys. Rev. Lett. 56 (1986) [10] BOUCHAUD J. P., COMTET A., GEORGES A. and LE DOUSSAL P., Europhys. Lett. 3 (1987) 653, Phys. Rev. A, in preparation. [11] GEORGES A., Thesis, University Paris-Sud, unpublished (1988) ; BOUCHAUD J. P. and GEORGES A., Phys. Rep., in preparation. [12] ASLANGUL C., POTTIER N. and SAINT-JAMES D., Submitted to Physica A. [13] SINAI Ya. G., Lect. Notes Phys., Eds. R. Schrader, R. Seiler and D. A. Uhlenbrock (Springer, Berlin) 153 (1981). [14] DOETSCH G., Guide to the applications of Laplace transforms, D. Van Nostrand (1961). [15] ALEXANDER S., BERNASCONI J., SCHNEIDER W. R. and ORBACH R., Rev. Mod. Phys. 53 (1981) 175. [16] BERNASCONI J. and SCHNEIDER W. R., Fractals in Physics, Eds. L. Pietronero and E. Tosatti (Elsevier Science Publishers B.V.) [17] ASLANGUL C., BOUCHAUD J. P., GEORGES A., POTTIER N. and SAINT-JAMES D., to be published in J. Stat. Phys.