Velocity and diffusion coefficient of a random asymmetric one-dimensional hopping model

HAL Id: jpa-00210967

https://hal.archives-ouvertes.fr/jpa-00210967

Submitted on 1 Jan 1989

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Velocity and diffusion coefficient of a random asymmetric one-dimensional hopping model

C. Aslangul, N. Pottier, D. Saint-James

To cite this version:

C. Aslangul, N. Pottier, D. Saint-James. Velocity and diffusion coefficient of a random asymmetric one-dimensional hopping model. Journal de Physique, 1989, 50 (8), pp.899-921.

�10.1051/jphys:01989005008089900�. �jpa-00210967�

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, 75251 Paris Cedex 05, ^France

(2) Laboratoire de Physique Statistique, Collège ^{de France, 3 rue} d’Ulm, ⁷⁵⁰⁰⁵ Paris, ^France

(Reçu le 25 octobre 1988, accepté ^sous forme définitive le Il décembre 1988)

Résumé.

La 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.

The 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.}

Classification

Physics ^Abstracts

05.40

71.55J

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

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 « diode- model

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 --+ 00 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

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) 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 random}

variables and are not assumed to be symmetric, in other words Wij ^{is 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

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

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

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

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), ^to show that the

configuration average, for N - ^oo, is given by

(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

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) .

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 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

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)

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

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,}

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 term

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

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

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

criterion (W_ IW_ >

1, the statistical weight of the transition rates towards the opposite

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

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.

4.2.1 Principle of ^the calculation.

We 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} 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 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

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

4.2.3 Period chain : long-time ^behaviour of X2(t ).

^Let 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

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 be}

transformed into finite sums. One obtains

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

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.

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 therefore occur, 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 also}

finite, ^and equal ^to

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

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).

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 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-.»

0. 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

(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

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.

4.2.5 A few additional comments.

Before 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 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 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}

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

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 particu-}

larly 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}

to discuss separately ^the long-time behaviour of the probabilities ^of finding ^the particle ^on

sites of positive ^or negative ^{index n.}

5.2.1 Calculation of (P nez

0» (n :::. 0).

^We 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).

5.2.2 Calculation of (P -n(z

0» (n:> 0).

^We 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.

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 ^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.}