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Velocity and diffusion coefficient of a random asymmetricone-dimensional hopping model

C.Aslangul (1, *), N. Pottier (1) and D. Saint-James (2, **)

(1) Groupe de Physique des Solides de lEcole Normale Suprieure (***), Universit Paris VII,2 place Jussieu, 75251 Paris Cedex 05, France

(2) Laboratoire de Physique Statistique, Collge de France, 3 rue dUlm, 75005 Paris, France

(Reu le 25 octobre 1988, accept sous forme dfinitive le Il dcembre 1988)

Rsum. 2014 La vitesse et le coefficient de diffusion dune particule sur un rseau priodiqueunidimensionnel de priode N avec des taux de transfert alatoires et asymtriques sont calculsde manire simple grce une mthode base sur une relation de rcurrence, qui permet dtablirune analogie aux grands temps avec un modle de marche strictement dirige. Les rsultats pourun systme

compltementalatoire sont obtenus en prenant la limite N ~ ~. On montre

quuncalcul, reposant sur une hypothse dchelle dynamique, de la vitesse et du coefficient dediffusion dans un rseau dsordonn infini conduit aux mmes rsultats.

Abstract. 2014 The velocity and the diffusion coefficient of a particle on a periodic one-dimensionallattice of period N with random asymmetric hopping rates are calculated in a simple way througha recursion relation method, which allows for an analogy at large times with a strictly directedwalk. The results for a completely random system are obtained by taking the limitN ~ ~.A dynamical scaling calculation of the velocity and of the diffusion coefficient in aninfinite disordered lattice is shown to yield the same results.

J. Phys. France 50 (1989) 899-921 15AVRIL 1989,

Classification

PhysicsAbstracts05.40 - 71.55J

1. Introduction

The problem of drift and diffusion on one-dimensional lattices with random asymmetrichopping rates has recently received considerable attention [1-11]. Bias appears quite naturallywhen 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 abiased random walk, the diffusing particle drifts in a preferential direction. Depending on theparticular model, either the velocity and the diffusion coefficient are finite, or anomalous driftor diffusion behaviours may appear when the bias varies, thus giving rise, in certain randomstructures, to the so-called

dynamical phasetransitions

[1].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005008089900

http://www.edpsciences.org/http://dx.doi.org/10.1051/jphys:01989005008089900http://dx.doi.org/10.1051/jphys:01989005008089900http://www.edpsciences.org/

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In the present paper we generalize to any random asymmetric hopping-model with non-zerohopping rates a recursion method first proposed by Bernasconi and Schneider [7,8] for thestudy of the drift properties in a particular asymmetric model which they call the diode-model . We show that, in any random asymmetric model with non-zero hopping rates, thedrift velocity (when it is finite) can be calculated by the recursion method, which indeedreveals to be a very natural and instructive technique. Moreover, we show that the recursionmethod 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 biasmodel, in which the asymmetry of the transfer rates has its origin in the existence of anapplied constant bias, and a local random force model, in which the local forces betweenneighbouring 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 steadystate and obtains exact expressions for the velocity and for the diffusion coefficient. Hededuces the corresponding

quantitiesfor 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 therecursion relation method.As in [5], we first study a periodic chain of arbitrary periodN, and we then take the limit of an infinite period. The results we obtain for the velocity andfor the diffusion coefficient indeed coincide with those of [5] : for instance, in the constantbias model, the velocity and the diffusion coefficient are always finite, while in the localrandom 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 theirderivation, which is simple and illustrative, and also gives insight into the behaviour of variousquantities to be used in the dynamical scaling treatment presented in the last section of thepaper.

Let us comment briefly upon Derridas procedure. It can be easily shown by linear algebratechniques [12] that the dynamics at large times, for finite N, is governed by the zeroeigenvalue (linked to the conservation of probability) of the matrix associated with the masterequation of the problem (Eq. (1)). This implies that, after a time larger than the inverse of thesmallest non-zero eigenvalue, Derridas regime (drift and diffusion) is correct up to a finitenumber (= 7V - 1) of exponentially small decaying terms. Otherwise stated, the long-timebehaviour assumed in

[5]is indeed the correct one. The only problem remaining with

Derridas procedure is the question of its validity when one takes the limit N --+ 00 in order topicture an infinite chain. This question is handled in a forthcoming paper [17], in which wedemonstrate that Derridas procedure actually yields the correct V and D, when thesecoefficients are finite.

Secondly, we give a simple recipe, explicitly using the preceding formalism, and based onthe dynamical scaling properties of the configuration averaged medium : this leads to a directcomputation 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. Interestinglyenough, this probability suffices to determine the exact transport coefficients of interest. Thisderivation, presented here for a discrete lattice, is closely parallel to an independentderivation for a continuous medium presented in reference [11]. To our best knowledge, thissimple derivation, introduced in [15], is used for the first time in an asymmetric disorderedlattice.

The paper is organized as follows : in section 2, we describe the model and we give the

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general notation for this problem. The master quation which governs the problem is writtendown, and various Laplace transformed quantities, which will be used in the following, areintroduced. In section 3, we show how the recursion relations can be solved in the long-timelimit. In section 4, we calculate the velocity and the diffusion coefficient, when these

quantitiesare 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 ofspacing a, as described by the master equation

in which p,, (t) denotes the probability of finding the particle on the site of index

n at time t 0. We assume that

i.e. at time t = 0 the particle is localized on site n = 0. The transfer rates Wij are randomvariables and are not assumed to be symmetric, in other words Wij is not equal toWji. In addition, the pairs (Wi, i -, 1, Wi , 1, i ) are assumed to be independent from one link tothe other. We exclude here the case where any Wij vanishes. Equation (1) thus describes arandom 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 diffusioncoefficient 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), whichyields

where we have defined

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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 rightare allowed, the quantities G,,- (z) identically vanish whereas the quantities G (z) coincidewith 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 wayeasier to handle, a point upon which we shall come back later) the same information about thedisordered system as the transfer rates do, as soon as the initial condition (2) has beenretained.

We shall now show that, in the long-time limit, a physical solution for the recursionrelations (17) and (18) can easily be obtained, which in tum, using equations (14)-(16), willyield the probabilities pn (t ). This will be done in section 3, in which this solution will be givenin the most general asymmetric system. Of course, one is not sure that no other solution for

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the highly non-linear equations (17) and (18) does exist, but it will be argued that the solutionwe obtain is indeed the physical one.

In order to clarify the discussion, it will be at times worthwhile to refer to the two followinginteresting particular models, in which one considers thermally activated transfer rates.

Model (i ) : constant bias modelIn 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 theabsence 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 forinstance 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

alwaysfinite

while,in

the second one, dynamical phases (with anomalous drift and diffusion properties) may appearfor 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 therecursion 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 asymmetriccase (log (Wn, n + l/Wn + 1, n) > = 0 is excluded from the present study. (This last case has beenstudied by Sinai [13] within the framework of a discrete time model and is known to presentan anomalously slow (logarithmic) diffusive behaviour). Condition (22) will insure inparticular that, if a finite velocity does exist in the system, it will also be directed along then ::> 0 axis. In equation (22), the symbol (... ) denotes the average over the disorderedtransfer rates, i.e. the configuration average.

Just as in [5], it will reveal convenient to consider a periodized chain of periodN, i.e. a chain in which

It is easily seen that, due to the periodicity of the transfer rates, a periodic solution of the

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recursion relations (17) and (18) for the quantities G (z ) and G- (z ) can be found. From nowon, we shall only consider this physically sensible solution, and show that it actually yields thesame results as [5], which are indeed the correct ones, as discussed in the introduction. Notehowever that, since the initial condition (2) is not periodic, the Pn(Z) themselves are not

periodic quantities,even in a

periodizedchain.

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 probabilitiesPn (t ) for finite N are always given by a finite sum of exponentials, their Laplace transforms

Pn (Z) onlyinvolve

integer powersof z. The same conclusion

obviouslyholds for the

quantities G+ (z) and G- (z) (Eqs. (9)-(10)). Moreover, it can be seen on the recursionrelations (17) and (18) that non-integer powers of z would break the translational invarianceof 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 thefollowing terms of the asymptotic expansion, if needed. In addition, it can be seen that thechoice of the periodic solution as well as the expansions (24) and (25) are in accordance withthe particular case of the strictly directed walk model, in which the quantities G+ (z ) coincidewith the periodic Ws whereas the quantities Gn (z) identically vanish.

3.1 LONG-TIME SOLUTION OF THE RECURSION RELATION FOR THE G+ (z). - The recursionrelation (17), when expanded at zero order in z, and iterated for instance between sitesn 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 quantitiesG (0) are well defined, and given by finite sums.

Since we aim at considering the limit of an infinite period (N - oo ), let us investigateequation (27) in this limit. Owing to condition (22), the product appearing in the l.h.s. ofequation (27) can be neglected with probability 1 in the limit N - 00 ([3], [5]). Note that, fora 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 Gs obtaining bythis 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 beconvergent. Since the Wij are uncorrelated, this is the case whenever the condition

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is obeyed. (The notation is obvious : W- replaces W,, -, ,,, and W +- replaces Wn, n + 1). Onethen gets

Note that, as expected, this average value does not depend on the site index n. Conversely, ifcondition (28) is not obeyed, the average value (l/G: is infinite.

Similarly, when expanded at order 1 in z and iterated for instance between sitesn 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 byfinite sums.

The same arguments can be applied to equation (31) as to equation (27), to show that theconfiguration average, for N - oo, is given by

(provided that Eq. (28) is fulfilled). In order to get an explicit expression, it is necessary toanalyse l/ [G+ (11)]2When N --+ ao, equation (27) yields

The configuration average of both sides of equation (33) can easily be taken, provided thatthe 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

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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) isinfinite. Once the average value (1/ [G: (0)]2) is known, equation (32) yields the averagevalue (g: / [G: (0)]2) .3.2 LONG-TIME SOLUTION OF THE RECURSION RELATION FOR THE G; (z).

-

The 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 byequations (24) and (25).As will be demonstrated in the following section, the correspondingvelocity 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 thelocal 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

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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 fromequation (29) .

In the same way, conditions (28) and (34) are automatically obeyed and the configurationaverage (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) quantityWn, n , 1, as defined by equation (19). Similarly, the average value (g; > is always finite ; onegets 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

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The average value (g;) is finite when condition (43) is fulfilled ; in this case one gets fromequation (38)

4. Calculation of the velocity and of the diffusion coefficient.

4.1 VELOCrrY. - Let x(z) denote the Laplace transform of the particle positionXUJ,, 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 periodicsolution 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-timeasymptotic 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,

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Similarly, a careful eximination of equation (51) shows that, when z - 0,

where only leading order terms have to be retained in S+ (z) ; note that the second terminvolving 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

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 discussedin the intrduction, the correct results of reference [5] are recovered, this justifies our choiceof the periodic solution for the quantities G (z ) and G- (z). Expression (60) of the velocity isvalid for a given configuration of the transfer rates ; no average over the transfer rates hasbeen 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 correspondingto 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 conventionalterminology, is a self-averaging quantity.As discussed above, the corresponding behaviour

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would be obtained only at infinite time. However, if a configuration average would have beentaken on an infinite random lattice, it can be seen that this behaviour would be reached atfinite time (provided that the velocity is finite, see below). We shall establish this point in a

forthcoming paper [17].Note the

very simpleform taken

byV, which can be

givena 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 takenfrom 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. Insuch 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

criterion (W_ IW_ > > 1, the statistical weight of the transition rates towards the oppositedirection is important enough, the asymptotic behaviour of x tF) is anomalous. Indeed, it hasbeen 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 modelIn 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 modelIn 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 ofcP, the exponent g is equal to the ratio mlau (a is the lattice spacing) and condition (43) canbe rewritten as

For a given temperature, CP n, n + 1 is proportional to the local random force between sites

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n and n + 1 ; therefore a finite velocity exists provided that the ratio of the average bias - asmeasured bym to the strength of disorder - as measured by o- is high enough. Thevalue of velocity is then given by

4.2 DIFFUSION COEFFICIENT.

4.2.1 Principle of the calculation. - We begin here by briefly sketching the procedure to beworked out for the calculation of the diffusion coefficient, as defined by equation (6). For agiven configuration of the transfer rates, we have to determine the long-time behaviour of thefirst two moments of the particle position, X{t) and X2(t ). This will be done through a Laplacetransform analysis.As discussed in the

introduction, itcan be

readilydemonstrated in various

ways [5], [12],that in a periodized chain of finite period N and in the long-time limit, the quantitiesXl!) and x2(t) behave as

Therefore

This behaviour can also be obtained by making use of the periodic Gs 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 long-time 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.

4.2.2 Periodized chain : long-time behaviour of x (t ). - The Laplace analysis initiated for thecalculation of the velocity has to be pursued to one order further.A careful examination ofequation (51) shows that, when z - 0, it is sufficient to retain terms up to orderllz in the expression

since the second term involving S- (z) in the r.h.s. of equation (51) yields only subdominantquantities. Let us note that

The coefficient is easily obtainable ; however, as will become apparent later, a detailedexpression 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

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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 periodN, 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 ofxi(N) is

4.2.3 Period chain : long-time behaviour of X2(t ). - Let x2(z) denote the Laplace transformof the second moment of the particle position x2(t), as defined by equation (4), for a givenconfiguration of the transfer rates. One has

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) can betransformed into finite sums. One obtains

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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 expansionof 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 periodizedchain of period N, as given by equation (60). Equations (74) and (85) show thatX2(z) behaves, whenz 0, as

which displays the announced behaviour of x 2(t )(Eq. (70)).A (N ) is seen to be equal toV 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, adiffusive regime exists. The expression of the diffusion coefficient follows from equation (72)

D (N ) can be given the remarkably simple form

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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], buthere they take a much more condensed form, as for the velocity, related to the fact that, forthe considered initial condition, the quantities GI (z ) and G- (z ) are easier to handle than thetransfer rates themselves. We shall come bacl to this point later. Let us emphasize that, just asV (N ), D (N ) is obtained for a given configuration of the transfer rates.A discussion of therange of validity in time of the diffusive regime could be modelled on the correspondingdiscussion for the drift regime.

4.2.4 Chain of infinite periode - In 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 thereforeoccur, depending on whether the various average values are finite or not.

In this case , the average values 1 ] 2 and /[?,) are finite . D is thus alsofinite, and equal to

In this case, the average values (1/ [Gt (0)]2) and (gt / [Gt (0)]2) are infinite. D istherefore infinite, which indicates an anomalous diffusion behaviour. Physically speaking, this

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means that the particle may remain trapped on some favourable sites of the lattices, with ananomalously large dispersion of its position. Indeed, it has been found in a similar continuousmedium model [11] that, in this range of values of the bias field

with g as defined by equation (65).4.2.4.2 Zero velocity regime W_ IW_ > 1). - The average values (1/ [Gt (0)]2) and(gt / [Gt (0)]2) are infinite. Since the denominator in equation (91) also tends towardsinfinity, the value of D cannot be directly extracted from this equation. One therefore has toresort to other arguments. For a similar continuous medium model [11], it has been found thatD is equal to zero in some domain containing the marginally asymmetric case

(log (W+-/W-. = 0. Remind that this case (excluded from the present study) is thecontinuous time analog of Sinais model [13], which is known to present an ultraslow increaseof

X2(t ), correspondingto 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 modelIn 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

(We have restored the explicit dependence on the lattice parameter.) The absolute valuereflects the fact that D must be an even function of a.

In an ordered biased lattice, the second term in equation (95) would disappear andV 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

supplementarycontribution

doesappear in the diffusion coefficient, due to the fluctuation of the transfer rates, as revealed bythe bias field. However, in the small bias regime, this term is negligible, so that the standardEinstein relation remains valid.

Model (i i ) : local random force modelJust 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 exponentIL is equal to the ratio m / a (1 .

In the finite velocity regime (IL > 1 ), the diffusion coefficient is finite when IL >- 2 andinfinite when 1 g 2. For IL > 2 one gets

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In the zero velocity regime (1 : 1), the value of D cannot be directly extracted fromequation (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 infinitewhen 1/2 : IL : 1. It is not known whether this result can be exactly extended to the presentdiscrete lattice model.

Obviously enough, the question of the existence of an Einstein relation betweenV and D only makes sense when the drift and diffusion properties are normal, that is, whenIL > 2. If the lattice were ordered, that is, if 0 have no dispersion around its mean valuem, 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 velocityvanishes 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 thepresence of disorder in this model.

4.2.5A few additional comments. - Before concluding this section, let us first brieflycomment 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, forany 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 thedetails of the particular configuration ; in other words, as stated above, in this limitV 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, andB 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 self-averaging contributions which, even in the limit N - ao, continue to fluctuate from one

configuration to another one. Interestingly enough, these non self-averaging contributionscancel 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 thelimit N --+ oo, is another indication of the fact that the time required to reach the drift regimefor a given configuration is itself infinite in this limit, as already noted.A similar conclusioncould probably be drawn for the time necessary to reach the diffusive regime, if one could goone step further in the expansions. However, as indicated above, if a configuration averagewould have been taken on an infinite random lattice, it is physically plausible that thisbehaviour 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 coefficientif 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

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N ---* 00 in expressions (60) and (89) (or (90)). Clearly, in order to assert that V andD, as obtained in this framework, represent the proper velocity and diffusion coefficient ofthe infinite random system, it is necessary to demonstrate that the two limits t - 00 andN --> oo actually commute. In fact, a direct calculation of the velocity and of the diffusioncoefficient in the infinite random

system isnot so

easyto achieve. For instance the

long-timebehaviour of the first two moments of the particle position x (t ) and X2(t) might well not be inan infinite system of the form (69) and (70). From a technical point of view, the periodizationtrick allows for transforming the infinite series S (z) and S:!:. ( g, z) into finite sums ; thebehaviours (69) and (70) of x(t) and x2(t ) are thus safely obtained through expansions of thefinite sums S (z) and aS+ (e, z)lae 1 e = 1. It is not a priori obvious whether such behavioursstill hold when one works on infinite series. In a forthcoming paper [17], we shall considerfrom the start an infinite random system ; we calculate V and D, which tum out to be actuallygiven 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, thatis, require the knowledge of all the probabilities Pn (t ). However, it is well-known that, in thecase of a biased ordered lattice (O.L.) (in which the velocity and the diffusion coefficient arealways 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 ofpO.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 easilyextended to the biased case in the following way : one assumes that, for a random system witha finite velocity and a finite diffusion coefficient, the particle at large time obeys in theaverage an ordinary diffusion equation as in a biased ordered lattice. In this approach, thebasic quantity tums out to be the configuration average (Po(z)) of the Laplace transform ofthe 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 thedetermination of V and D, and indeed yields the correct values. However, it is interesting tosee whether in the present model such an assumption can be a priori justified to some extentfor the present case of an asymmetric random hopping model ; this can be done by comparing

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the configuration averaged quantities (P n (z) and (P - n (z (n = 1, 2, ... ) in the long-timelimit with the corresponding quantities in an ordered lattice (we shall restrict this examinationto the leading contributions (P n (z = 0 )) and (P_ n (z = 0 )) . The average values of interestin the infinite random medium will be borrowed from section 3, where they have been

computedby taking the limit N - ao in th

correspondingexpressions for a

periodizedchain

of period N.Actually, we have demonstrated [17] that this procedure gives the correct resultsfor an infinite lattice.

5.1 CALCULATION OF THE LONG-TIME BEHAVIOUR OF (Po(z. - This quantity is particu-larly easy to compute. Indeed, by looking at the expression (14) of Po(z), one gets, up toorder z

As can be seen on the recursion relations (17) and (18) for the quantities GI (z) andGn (z), there are no correlations between 96 and Gt (); 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 discussionof the conditions under which this is the case has been given in section 3. In particular, whenthe two conditions (28) and (34) are simultaneously obeyed, all the average values appearingin 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. Inother words, this demonstrates that (Po(z)), expanded up to order z, suffices to determineV 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 justifiedto some

extent.

5.2 CALCULATION OF THE LONG-TIME LIMITS OF (Pn(zAND (P -n(Z (n = 1,2, ...).-Since an average bias field, directed along the n :::. 0 axis, has been assumed to exist, we have

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to discuss separately the long-time behaviour of the probabilities of finding the particle onsites of positive or negative index n.

5.2.1 Calculation of (P nez = 0 (n :::. 0). - We begin by calculating (Pl (z = 0)) ; thequantities (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 similarto the corresponding one in an ordered lattice, which can be deduced from equation (102).

5.2.2 Calculation of (P -n(z = 0 (n:> 0).- We begin by calculating (P -l(Z = 0 ) > ; thequantities (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, -1i 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-timelimit is actually similar to the corresponding one in an ordered lattice, which can be deducedfrom equation (103).

5.3 DYNAMICAL SCALINGASSUMPTION : DISCUSSION OF THE RESULTS FOR THE VELOCITY

AND FOR THE DIFFUSION COEFFICIENT. - The results (113) and (117) indicate that, in the

long-time limit, the average values of the probabilities of finding the particle on a given sitebehave in a way similar to that in an ordered lattice. One must emphasize that these resultsclearly 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 thediffusion coefficient, when these quantities are finite.

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All the results which can be deduced from the present approach are strictly identical tothose derived in section 4. Note that the derivation based on the scaling hypothesis presentedhere for a discrete lattice is closely parallel to a derivation for a continuous medium presentedin reference [11].

6. Conclusion.

In this paper we showed how the velocity and the diffusion coefficient in any randomasymmetric model with non-zero hopping rates can be calculated, when they are finite, withthe help of a recursion method first proposed by Bemasconi and Schneider [7, 8] for the studyof 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 thesolution 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) andGn (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 periodizedchains of finite period N, we argued that - assuming an average bias field directed along then > 0 axis -, the physical quantities of interest Gn (z ) and Gn (z ) may be taken periodic andin the long-time limit (t --+ 00, Le. z -+ 0) respectively tend towards finite constantsG;- (0) or behave proportionally to z (Eqs. (24) and (25)). This assumption actually yields thecorrect values for V and D. Therefore one may in some sense consider that the long-timeasymptotic properties of the walk are similar to the properties of a strictly directed walk, inwhich only steps towards the right are allowed [11]. The role of the equivalent transfer ratestowards the right of this strictly directed walk is played by the quantities Gn (0); theseequivalent 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 renormalizationcalculation by Bernasconi and Schneider [16]. In this interpretation, 11G+ () appears torepresent - in periodized chains of finite period N - the typical time (for a givenconfiguration of the transfer rates) needed by the particle to leave the site of indexn ; this interpretation can equally be deduced from the following equality

The value (63) of the velocity- when it is finite- is in accordance with this interpretation.Note that the introduction of the Gs allows for a

straightforwardobtention of V. The

analogyat large times with a strictly directed walk may probably be extended. Indeed the diffusioncoefficient, 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

expressionwould be absent.

Obviouslythis detailed balance factor takes into account the

weight of the transitions towards the left and the retarded character of the motion. This maybe interpreted by considering a strictly directed walk on a space-renormalized lattice, in whichthe distance between the new sites takes into account the possibility of retums towards the leftin the initial lattice.

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In addition, we have shown that for this problem the knowledge of (Po(z suffices fordetermining the physical transport coefficients of interest when they are finite, which extendsthe validity of the scaling hypothesis [15] to the asymmetric case.

Acknowledgments.We are indebted to Drs. J. P. Bouchaud andA. Georges for helpful comments anddiscussions.

Rfrences

[1] See for instance the reviews by HAUS J. W. and KEHR K. W., Phys. Rep. 150 (1987) 263, and byHAVLIN S. and BEN-AVRAHAM D.,Adv. Phys. 36 (1987) 695 and references therein.

[2] STEPHEN M. J., J. Phys. C 14 (1981) L 1077.

[3]DERRIDA B. and POMEAU

Y., Phys.Rev. Lett. 48

(1982)627.

[4] DERRIDA B. and ORBACH R., Phys. Rev. B 27 (1983) 4694.[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., POLITIA. and ZANNETTI, M., Phys. Rev. Lett. 56 (1986) 2341.

[10] BOUCHAUD J. P., COMTETA., GEORGESA. and LE DOUSSAL P., Europhys. Lett. 3 (1987) 653,Phys. Rev.A, in preparation.

[11] GEORGESA., Thesis, University Paris-Sud, unpublished (1988) ;BOUCHAUD J. P. and GEORGESA., Phys. Rep., in preparation.

[12]ASLANGUL C., POTTIER N. and SAINT-JAMES D., Submitted to

PhysicaA.

[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.) 1986.

[17]ASLANGUL C., BOUCHAUD J. P., GEORGESA., POTTIER N. and SAINT-JAMES D., to be publishedin J. Stat. Phys.