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Dynamical properties of a random walk in a one-dimensional random medium
Physica A 171 (1991) 47-68 North-Holland
DYNAMICAL PROPERTIES OF A RANDOM WALK IN A ONE-DIMENSIONAL RANDOM MEDIUM C. A S L A N G U L a, M. BARTHI~LEMY a, N. P O T H E R a and D. SAINT-JAMES "'l "Groupe de Physique des Solides2, Tour 23, Universit~ Paris VII, 2 place Jussieu, 75251 Paris Cedex 05, France bLaboratoire de Physique Statistique, Coll~ge de France, 3 rue d'Ulm, 75231 Paris Cedex 05, France
Received 27 July 1990
The random motion of a particle in a one-dimensional continuous random medium with random forces is investigated by making use of the large-scale equivalence of the properties of the walk with those of a directed random walk on a discrete lattice. When the disorder is weak enough, a normal drift-diffusion regime takes place, whereas for strong disorder anomalous behaviours occur. It is shown that these results may be obtained by introducing a renormalized lattice with a finite spacing. In the normal phase this spacing appears explicitly in the result, while it turns out to be irrelevant in the anomalous ones. The dynamical exponents as well as the prefactors of the power-laws of the particle position and of its average dispersion are explicitly calculated in the anomalous phases.
1. Introduction In the present paper, we consider the problem of the determination of the dynamical exponents for the random walk of a particle in a one-dimensional random medium. The model we study is defined by' the continuous space and time Fokker-Planck equation
Op(x, t) = Do 02p(x, t) 0t
0x 2
1 O 77 0x [F(x)
p(x,
t)l,
(1.1)
in which F(x) denotes a Gaussian space-dependent random force; r/ is the viscosity and Do = k Thl is the diffusion coefficient of the particle in the absence of disorder. Eq. (1.1) thus describes in the viscous limit the Brownian motion of a particle in a disordered medium in which the disorder may be o~"':~ (q,.,~ached) Gaussian random force. It is worthwhile characterized by a ~,,,,,. t Also at Universit6 Paris VII. 2 Laboratoire associ6 au CNRS (U.A. n" 17) ct aux Universit4s Paris VII et Paris VI. 0378-4371/91/$03.50 (~) 1991 - Elsevier Science Publishers B.V. (North-Holland)
48
C. Aslangul et al. t Dynamical properties o f a 1D random walk
to note here that, in order to be properly defined, the continuous model has t o be considered as the continuous limit of a discrete model on a lattice. This model, which has been widely studied for some years [1-14], is known to exhibit interesting non-trivial dynamical properties, generally referred to as dynamical phases. When the mean bias (i.e., the mean value of the force) is varied, a succession of phases is observed, characterized by different drift and diffusion behaviours, normal (i.e., with finite velocity and diffusion coefficients), or not (i.e., with non standard exponents characterizing the time dependence of the average particle position and/or of its average dispersion). The case of a zero mean bias is very peculiar and known as the Sinai model [3]: it leads to an anomalously slow (logarithmic) diffusive behaviour. Using proper reduced variables, one can show that all the physics of the problem is controlled by a single dimensionless parameter ~ (to be defined below), which increases with the bias or the temperature, and decreases when the disorder strength increases. Generally speaking, anomalous phases are found for low values of/~, that is in situations of relatively strong disorder. More precisely, anomalous drift takes place below ~ = 1, while anomalous diffusion is found below ~t = 2. Few exact analytic results are available in the general case. In a previous paper devoted to this model [ 11], we showed that the probability distributions of the energy-dependent transfer rates, as first introduced in [6, 7] by Bernasconi and Schneider, cap. be exactly computed. We also showed that, once these probability distributions are known, it is actually possible to calculate in an exact way the average over disorder of the probability of presence of the particle at its initial position, (p(0, t)). This latter quantity has also been computed by Bouchaud et al. [8,9] by taking advantage of the formal correspondence between the one-dimensional diffusion model and a quantummechanical model of a particle moving in a random one-dimensional potential. When they are finite, the velocity and the diffusion coefficient of the particle can be exactly computed through a variety of techniques [5, 8-10]. It may be shown that, in such cases, the knowledge of the average probability of return to the origin is sufficient to calculate the particle velocity and diffusion • ,-~.J'~,,,III'~i~,.,ilL [ J[UC]I.].
Unfortunately, these methods w~lich allow for the calculation of the velocity and of the diffusion coefficient when they arc finite, cannot bc easily extended to the anomalous phases. New angles of attack have to be found. The questions to be answered are the follow~ng: how is it possible in these regimes to give a determination of the anomalous exponents characterizing the average particle position and its average dispersion? Is it possible to determine the prefactors of these power-laws? Clearly, a central quantity to be analyzed is the average diffusion front
C. Aslangul et al. / Dynamical properties of a I D random walk
49
(p(X, T)), expressed for convenience in appropriate reduced space and time variables X and T, or its Fourier-Laplace transform (P(K, E)), which one can write as 1
(P(K, E ) ) = K2 + 2i/zK + E - ,~(r, E) '
(1.2)
thus defining the self-eaergy ,~(K, E)o First, one might think to try a systematic perturbative treatment of the disorder, foUowing the method given in [12, 15]. However, this perturbative expansion fails down in the anomalous phases; in the normal phase, the velocity is exact at one-loop order, but the same is not true for the diffusion coefficient. Then, one may think to use the available information about the average probability of presence of the particle at its initial position, from which, in the normal phases, the transport coefficients may be extracted. However, it is not a priori clear whether this information would be sufficient in the anomalous regimes. Actually, the answers to all the above questions are known for a simpler version of the model, the so-called directed walk model [8, 9, 14]. In a directed walk, the random variable of interest, here the particle position, can only increase, ~,,,o'-~';'~'-,.s~"""-.--.-. a given initial value. It is believed that the general both-way walk characterized by the parameter g is asymptotically similar to a directed walk on a (possibly renormalized) lattice with hopping rates chosen independently at random in a probability distribution p(W)~. W "-~ for W values smaller than some cut-off [8-10, 13]. The study of the directed walk is thus interesting in that it can generate the basic features of the general problem in a simplified framework. Indeed, the properties of the directed walk are now well understood [8, 9, 14]: in the directed walk, the average particle position and its average dispersion can be explicitly calculated for any value of ~, that is even in the anomalous phases, in terms of the average probability of presence of the particle at its initial point. One can try to exploit this large-scale similarity of the general model with the directed walk model to derive an expression for ~(K, E) for low values of K and E. The aim of the present paper is thus mainly two-fold. In a first step, we review the arguments in favor of the idea of the large-scale similarity of the general walk with a directed walk. This equivalence once made precise, the expression of the average probability of presence of the particle at its initial position (exactly known in the continuous medium) can be used to infer a form of the self-energy, supposedly valid for small K and E. In a second step, using the above-mentioned form of the self-energy, we determine the asymptotic expressions of the average particle position and of its average dispersion for
50
C. Aslangul et al. I Dynamical properties o f a 1 D random walk
any value of it. We are thus able in particular to calculate the dynamical exponents, as weli as the prefactors of the power-laws in the anomalous phases. The paper is organized as follows: in section 2, we recall the expression of the average probability of return to the origin in the general case, as found in [8,9, 11]. In section 3, we sketch out the perturbative expansion of the self-energy and show its failure for the determination of the properties of the anomalous phases. Sections 4 and 5 contain the new results of our paper. In section 4, we review the arguments in favor of the large scale similarity of the general walk with a directed walk; assuming for the self-energy at given energy a series expansion in powers of K, we take advantage of the above-mentioned equivalence with a directed walk to derive an expression for ,~(K, E) for low values of K and E. In section 5, we calculate in all phases the average particle position and its average dispersion. Finally, section 6 contains our conclusions.
2. The average probability of return to the origin 2.1. The m o d e l
Let us first briefly stress some characteristic features of the model. As well known, the continuous space and time Fokker-Planck equation (1.1) is the continuous (weak disorder) limit of the usual master equation describing a random walk on a lattice with random asymmetric hopping rates between neighbouring lattice sites, namely ddt p , = I V . .. + , P.+ , (t) + W . .._ , Pn_l(t) -- (Wn+ ,.. + Wn _ I . . ) p . ( t ) ,
(2.1)
when the hopping rates are chosen independently at random such that
,,,,
W.
+, = --w exp "~" a"
W,,+ ,.,,
Do
2Do { aF,, +,
= ~ exp~ . a 2Do .]
.
(2.2a)
(2.2b)
When the lattice spacing a becomes much smaller than xj = 4D2/or, where or denotes the variance of the Gaussian random force F(x), the master equation (2.1) can be approximated by its continuous limit (x I is the length below which the particle is insensitive to the disordered character of the potential [8, 9]). Actually, in order to be properly defined, the c'~ntinuous model (1.1) has to be thought of as deriving from the discrete model (2.1) when a <~ x t. In the
C. Aslangul et ai. I Dynamical properties of a 1 D random walk
51
following, when dealing with the continuous model, we shall use the reduced position and time variables X and T as defined by X = x/x~ and T = t/2fl, 3 2 where t~ = 8Do/tr is the diffusion time corresponding to x~. One also introduces the dimensionless random force ~,(X), the average value of which is/,. This parameter actually controls the physics of the problem.
2.2. The average probability of return to zhe origin The average over disorder of the probability of presence of the particle on its starting site has been exactly calculated in the continuous limit by several different methods. In [8, 9], a formal correspondence has been established between the present one-dimensional diffusion model and a quantum-mechanical particle moving in a one-dimensional random potential. The average probability of return to the origin (p(0, T)} (expressed for convenience in the reduced variables) has been related to the density of states of a Schr6dinger equation in a random potential. This density of states has been computed through two different approaches: a Dyson-Schmidt technique and the replica method. As a result
(p(0, T)) = f
dE e x p ( - E T ) dN dE'
(2.3)
o
with 2
1
(2.4)
N(E) = ---5 2 1/2) ' ~r J~,2 (EI/2) + N~,(E
where J~, and N~, are Bessel functions of order/z [16]. In [11], use has been made of the so-called equivalent transfer-rates technique, a method first devised for the discrete case in [6, 7] and developed in [10]. One introduces the quantities G+(z)and G~(z) as defined by +
G. (z)P.(z) = W,,+,.,,P.(z)- W..,,+,P.+,(z) , c,,-(z)P_.(z) = w_._, _.P_.(z)-
n >10,
(2.5) n~>0,
where
P,,(z) = f dt exp(- zt)p.(t). 0
(2.6)
52
C. Aslangul et ai. i Dynamical properties o f a I D random walk
The master equation (2.4) is equivalent to the set of equations
zP,,(z) = - G . + ( z ) P . ( z ) + G,_,(z)P,_,(z),+
n = 1, 2 , . . . ,
(2.8)
zPo(z ) - 1 = - G ~ (z)Po(z) - G O(z)Po(z) , z P _ . ( z ) = - G ~ ( z ) P _ . ( z ) + G~_,(z)P_,+,(z),
(2.7)
n = 1,2, . . . .
(2.9)
In this formulation, the quantities G+~(z) and G-~(z) play the role of energydependent transfer rates respectively towards the right of the sites of positive index or towards the left of the sites of negative index. Their equivalent in the continuous medium are denoted by G+(X, E) and G - ( - X , E), where E is the reduced variable associated with z. In the disordered medium, these quantities are random variables; their probability distributions can be exactly computed analytically in the continuous limit [11]. From these distributions, using the continuous equivalent of eq. (2.8) P(0, E)[G+(O, E) + G-(O,
E)] =
1
(2.10)
one can deduce the average probability of presence of the particle at its initial site, ( P ( X = 0, E ) ) , which for simplicity we denote as (P(0, E ) ) . As a result
1
i
(P(0, E)) = ~ ~=, ds
K~,
K~,(E2,,z)
,
(2.11)
where K, denotes the modified Bessei function of order /x [16]. By using contour integration, one can show that the above expression yields for (p(0, T)) a result identical to eq. (2.3) with d N / d E as given by eq. (2.4).
2.3. The small-energy expansion of (P(O, E) ) In the following, it will be necessary to use the small-E expansion of (P(0, E ) ) . When n 0), one gets an expansion of the type (P(0, E)) ~ ao(i.t ) + a,(Iz)E + ' . . + a,,_,( l.t)E"-' + C~( tx)E ~-' (2.12) All the coefficients in eq. (2.12) may straightforwardly be computed from eq. (2.11 ) by using standard integration formulas and series expansions of modified Bessel-functions [16]. Let us just quote the results of interest:
C. Aslangul et al. / Dynamical properties of a I D random walk
53
(i) Ix < 1
In this case, one gets at leading order
(P(0, E)) -2'-2 Ix
,
(2.13)
where F(x) denotes the Euler gamma function. (ii) 1 < I x < 2
The first two dominant terms are 1 + 2,_2~/z F ( 1 - Ix) E~_ , (P(0, E ) ) ~ 2(Ix - 1) F(Ix)
(2.14)
(the following term would be of the form a~(Ix)E). The first term of this expansion is just the inverse of the finite velocity [8-10] V= 2(Ix - 1).
(2.15)
v,,J I x > 2 The first three dominant terms are 1
E
(P(0, E)) = 2(Ix - 1)
1
4 (Ix - 1)2(Ix - 2)
+ 21
F(! - Ix) E~,_ 1
-2#Ix F(Ix) (2.16)
(the following term would be of the form a2(Ix)E2). The first two terms correspond to an expansion of the form (P(0, E ) ) =
1
V
2ED
V3 + . . . ,
(2.17)
with the finite diffusion coefficient D as given by refs. [8-10],
Ix-2 When they are finite, the velocity and the diffusion coefficient of the particle which appear in eqs. (2.14) and (2.16) have been exactly computed by various techniques. In [5], the particle position and its dispersion have been calculated on a periodized lattice of large period N; for finite N, whatever the value of Ix, normal drift and diffusion behaviours are found. When the limit N--> ~: is taken, normal drift (for Ix > 1) and diffusion (for Ix > 2) behaviours remain;
54
C. Aslangul et al. / Dynamical properties o f a I D random walk
unfortunately, this method has not been extended, up to now, to a description of the behaviours in the anomalous phases. Another approach has been used in reL [10b], where an infinite lattice is considered from the start; the velocity and the diffusion coefficient of the particle, when they are finite, are calculated through the use of a resummation procedure; this method however cannot be readily extended to the anomalous phases. Since, for/z > 1 (resp. for/x > 2), the velocity (resp. the diffusion coefficient) can be related to the small-E expansion of (P(0, E ) ) [10a], one is logically led to expect to get information from this quantity also in the anomalous phases. In this respect, it is interesting to note that, for any value of/z, the smaU-E expansion of the average probability of return to the origin is analogous to the corresponding expansion for a directed walk on a discrete lattice (see eqs. (6) and (16) in [14a]), which, in that model, contains all the necessary information for the description of the motion, even in the anomalous regimes.
3. Perturbative calculation of the average diffusion front
As indicated in the introduction, the central quantity to be analyzed is the average diffusion front (p(X, T)), or its Fourier-Laplace transform (P(K, E)) (as usual, a Fourier transformation in space and a Laplace transformation in time is carried out). Eq. (1.2) defines the self-energy ,~(K, E). Following refs. [12, 15] (see also ref. [9]), one may attempt to undertake a systematic perturbative treatment of the disorder. The Fokker-Planck equation (1.1), written in its dimensionless form
op(X, r) OT
o p(X, r) -
0X 2
o OX [2~(X) p(X, T)],
(3.1)
is equivalent to the integral equation
P(K,E)= P°ra(K,E) 1 - i K
~2aO(Q)P(K-Q,E)
,
(3.2)
where i "~',-°'a~,,, " e ) = K~ + 2 i ~ K + E
(3.3)
is the Fourier-Laplace of the diffusion front in the absence of disorder and ~¢J(X) = ~ ( X ) - v.
(3.4)
C. Aslangul et al. / Dynamical properties of a 1 D random walk
55
denotes the departure of the (dimensionless) random force from its average value ~. Eq. (3.2) allows for constructing a perturbative expansion in powers of ~$. Since the random force is Gaussian, the average over disorder of the perturbation series deduced from eq. (3.2) is obtained by pairing together the force terms. Let us now examine the first-order contribution to the self-energy.
3.1. First-order contribution to the self-energy As ordinarily, the first-order contribution to the self-energy can be obtained in two ways. One can iterate twice eq. (3.2) and neglect correlations between the terms containing the random force and the diffusion front. One can also iterate eq. (3.2) to infinity and make a partial resummation of the diagrams reducible to one-loop. Using this method, one sees that this expansion provides a convergence criterion which, in the limit K--->0, E---~0, reads /x > 1. At lowest order (one-loop order), the self-energy is given by
~(K, E)= -2 f
dQ ,iT
K ( K - Q) ( K - Q)2 + 2 i ~ ( K - Q) + E
(3.5)
The integration limits are +--~ since we consider the continuous model. An analytic continuation of expression (3.5) can be made in the complex plane for Re E <0. The function ,~(K, E) has a cut on the real negative axis for E < - / z 2. The above integration leads to
2~(K, E)= 2ilzK(E +/.t2)-~/'- ,.
(3~6)•
everywhere outside the cut (the square root is the branch which takes real positive values on the real positive axis). The following expression for ( P(K, E)) is thus obtained" 1
(P(K, E)) = K2 + 2i/xK + E - 2i/xK(E +/x2) -''2 "
(3.7)
The normalization condition 1
(P(K = 0, E ) ) - ~ ,
(3.8)
which implies that Z ( K = O , E) = 0 , is satisfied by the one-loop approximation (3.6) for the self-energy.
(3.9)
56
C. Aslangul et al. I Dynamical properties o f a 1D random walk
3.2. The average particle position From the average diffusion front, one can calculate the average values of the first two moments of the particle position, ( x ( T ) ) and (x 2(T)). The bar denotes the so-called thermal average for a given choice of the random force, as defined by ,-,!,,-"ac
(3.10)
A(T) = f dX A(X) P(X, T) .
As above, the symbol ( - ) stands for the average over disorder. The Laplace transform (x(E)) of ( x - ~ ) is given by 0 ( x - ~ > = i ~ (P(K, E)>Ir= o , or
1(
(x---(~) = ~
2p.
(3.11)
a2~ +i ~-~[K=o).
(3.12)
As a result, one gets at large times for any It, by taking into account the two leading order terms at small E and making the inverse Laplace transformation 1 /.t
(x(T)) ~ 2 ( i t - 1)T + ~ .
(3.13)
Otherwise stated, the one-loop approximation for the self-energy gives a finite velocity equal to 2 ( / z - 1) for any It, a result which is known to be exact for It > 1. Clearly, when It < 1 this result is unacceptable on physical grounds, since the particle cannot move against the field. This indicates a breakdown of the one-loop approximation in this phase [9, 12]. This is linked to the lack of convergence of the geometric series involved in this approximation.
3.3. The average second moment In a similar way, the Laplace transform (x:(E)) of (x2(T)) is given by =
(3.14)
OK" ( P(K, E)>I,,=,,,
or
2
i
(x2(E)) = ~ (2it + i ~-~ K
~
(
2
.
(3.15)
C. Aslangul et al. I Dynamical properties of a 1 D random walk
57
As a result, one gets at large times for any/.~, by taking into account the two leading order terms at small E and making the inverse Laplace transformation /
,,
... 1\
(x2(T)) ~ 4 ( t t - 1)2T 2 + 2~1 + 4 -~---fL/T/~ / .
(3.16)
From this result, one can deduce the so-called average [8, 9] (or annealed [17]) diffusion coefficient D A as defined by ( x E ( r ) ) - ( x - ~ ) 2 ~ 2DA T .
(3.17)
This diffusion coefficient is defined through both thermal and disorder averages over the trajectories. It is generally different from the average over disorder of the diffusion coefficient for a given sample, a quantity denoted by D O (quenched diffusion coefficient) in [17], and defined by (xE(T) - [x(T)l 2) =
2DOT.
(3.18)
One always has DA> Do . Clearly, the knowledge of the average diffusion front yields only eqs. (3.13) and (3.16), one gets D A = D °rd + 2 / x -21 , /x
(3.19) D A.
From
(3.20)
where D °rd= 1 is the diffusion coefficient in the absence of disorder. Otherwise stated, the one-loop approximation for the self-energy gives a finite diffusion coefficient for any/.t. However, even in the normal phase/x > 2, this result is not satisfying, since in this phase the exact result is [8, 9] DA =
/X -~ .
(3.21)
Therefore, the situation for the diffusion coefficient is not the same as for the velocity, all the more since eq. (3.20) does not display any clear-cut indication of the expected onset of anomalous diffusion regimes below/~ = 2. Therefore, eq. (3.20) is unacceptable f o r / z < 2 . In addition, one can remark eq. (3.20) cannot be acceptable below /x = 1, since it would lead to a value of the diffusion coefficient lower than the one in the corresponding orderd lattice and even possibly negative.
C. Aslangul et al. ! Dynamical properties of a I D random walk
58
The one-loop results are thus very far from being satisfactory. Actually, looking again at expressions (3.12) and (3.15), one sees that the first two moments of the particle position require the knowledge of the first two derivatives of ,~(K, E) with respect to K (for K = 0), which are not correctly calculated in the one-loop approximation. Terms of any order in the loop expansion of the self-energy do contribute to O~(K,E)/OKIr,= o and a2,~(K, E ) ! OK21r=o . In other words, the dynamical properties of interest in the anomalous phases cannot be properly described within an approximation which is basically a weak-disorder one. Some other method must be proposed. Coming back to the integral equation (3.2~ ,~,ae sees that a series expansion of ,~(K, E) "n powers of K is not incompatible with the structure of the equation. This is the reason why, from now on, we shall assume the existence of such an expansion. To find its coefficients, we shall exploit the large-scale similarity of the general walk with a directed walk. As will be shown below, in the directed walk, a series expansion of the self-energy in powers of K can be found. By comparison, this will lead us to propose an expression for Z(K, E) for small values of K and E in the general case.
4. Large-scale equivalence with a directed walk model
Various arguments have been put forward in favor of the idea of the large-scale similarity of the general walk with a directed walk [8-10, 13]. Let us briefly recall the main features of the directed walk model and sum up these arguments. 4.1. The model
It is defined by the following master equation: dD~ dt
(4.1)
W.p. + W._,p._, .
which describes a random waik in which the position of the particle can only increase, starting from a given initial value, since the possibility of jumps towards the left is suppressed. The properties of the directed walk are well understood [8, 9, 14]. For example, one can choose the hopping rates independently at random in a given probability distribution
(w)
p(W) = C,W~-'f~ ~
tel
,
a >0,
(4.2)
C. As!angul et al. / Dynamical properties of a I D random walk
59
where C, is a normalization constant and fc a cut-off function basically specified by the fixed frequency W,.. The smaller a, the higher the probability to find a qtzasi-broken link. If T denotes a dimensionless time variable as defined by T= Wmt, the average particle position behaves as follows: (x-('~) ~ T ° ,
0 < a < 1,
(4.3)
(x--~) ~ T,
1 < a.
(4.4)
A phase of anomalous drift is seen to appear for 0 < a < 1. One can define also the average dispersion
(AxZ(T))A = (x2(T)) - ( x ( T ) ) 2 ,
(4.5)
an extension to the anomalous phases of the definition of the average (or annealed) diffusion coefficient D A [8, 9, 17]. This quantity behaves as (Ax2(T)) a ~ T2" ,
0< a < 1 ,
(4.6)
(Ax2(T))A ~ T 3-" ,
l
(4.7)
(aZ(T))
2<
T,
,
(4.8)
Anomalous diffusion thus takes place below a = 2. The following important point desexes to be underlined: in the directed walk model as described by eq. (4.1), the probability distribution of the hopping rates is chosen, for sufficiently low values of a (actually a < 1), to diverge in the limit of small hopping rates (eq. (4.2)). The physical origin of the anomalous phases has to be traced back to the existence of a relatively great weight of quasi-broken links [14].
4.2. Large-scale equivalence of the general walk with a directed walk In ref. [13], Bernasconi and Schneider applied a real-space renormalization procedure to a discrete-time random walk analogous to the continuous-time random walk described by eq. (2.1). Through a Monte Carlo procedure, they showed that the renormalized random walk approaches a directed random walk with uncorrelated successive transition times (or inverse hopping rates) with a broad distribution. This idea has been further developed in [8, 9l by Bouchaud et al. Let us recall here the main line of their argument. Clearly, in the problem at hand.
60
C. Aslangul et al. I Dynamical properties o f a 1 D random walk
which is a Brownian-motion problem in the presence of a Gaussian random force, the asymptotic regime is characterized by the competition between the bias term - F o x and the slowing-down effect of the very large fluctuations of the potential. These high barriers act as trapping regions between which the motion is convective. High barriers are exponentially rare, but the time needed to overcome one of them depends exponentially on its height. The net effect is a broad distribution of the local trapping times. We provided in [10a] complementary analytical arguments in favour of this equivalence in the discrete case. We showed that the quantities G+~(z) and G~(z) display the following behaviours at small z:
(z)--- an+ G-~ (z) ~ g~ z .
+ g2 z,
(4.9) (4.113)
Eqs. (4.9) and (4.10)-together with eqs. ( 2 . 7 ) - ( 2 . 9 ) - support the following interpretation: the long-time asymptotic properties of the walk are similar to the properties of a strictly directed walk, in which only steps towards the right are allowed. The role of the equivalent transfer rates towards the right of this strictly directed walk is played by the quantities G~+w). In this interpretation, 1/G+~ ~°~ appears to represent the time needed by the particle to leave the site of index n. These equivalent transfer rates however are not uncorrelated quantities. To get rid of these correlation effects, one could try to proceed to a real-space renormalization, with the hope to use new effective uncorrelated hopping rates, the correlation effects being included in lattice spacing and possibly time renormalization. After an infinite number of renormalization steps, and at infinite time, one hopes thus that the walk renormalizes towards a directed walk with independent random hopping rates. The role of the hopping rates should be played by the quantities G +t0), considered now as uncorrelated, but on a lattice with a renormalized spacing which takes into account the former correlations. For the moment, let us note that the probability distribution of these quantities has been calculated in a discrete lattice [I1], and shown to behave like [G~t°~] "-1 for small values of the variable. In other words, the sojourn times (or local trapping times) distribution, which in the present framework is tl~e distribution oi the quantities 1/G~+t°~, is a broad law, as expected on physical grounds [8, 9]. One can expect that the general random walk as described by eq. (2.1) would be equivalent, as far as its large-scale properties are concerned, to a directed walk on a renormalized lattice with independent random hopping rates, distributed according to a law of the form (4.2), in
C. Aslangul etal. I Dynamical properties of a 1D random walk
61
which one takes a = It. Similarly, the continuous limit of the general random walk is expected to renormalize towards a directed walk on a lattice with a finite spacing d. Note that the consideration of a finite lattice spacing is necessary, since in its continuous limit, eq. (4.1) does not contain diffusion effects. One is thus led to expect, for the general case, behaviours with exponents identical to those of eqs. (4.3)-(4.4) for the average particle position and to those of eqs. (4.6)-.(4.8) for its average dispersion. A precise determination of the coefficients would in particular imply the knowledge of the lattice spacing. Supplementary information can actually be extracted from the average probability of return to the origin, as we now show.
4.3. Approximate form of the average diffusion front for low values of K and E Let us consider on a lattice of spacing d a directed walk with independent hopping rates. A convenient way to deal with the properties at large space scale is to look for small values of q at the Fourier transform (Pq(z)> of (P,(z) >, as defined by ~c
= ~ e-aq"a
(4.11)
n =1)
One easily calculates
1W>
(4.12)
= ( z + ,,
W
1 e iq It may be seen on the above formula that (Pq(Z) > depends only of the quantity (1/(z + W)> = (P,=0(z)). In terms of this last quantity, one has
(Pq(Z)>= ( P,,__o(z)>1 -- e-iqd(1
-- Z(P,,-_o(z) >)
(4.13)
The question now arises: how must we choose the lattice spacing d on the one hand and (P,,=0(z)> on the other hand in order to describe at best the directed walk for which the large-scale properties of the average diffusion front are similar to those of the continuous problem (eq. (1.i))? A reasonable choice for (P,,=0(z)> valid for large times (i.e., for small z and E) seems to be a
x, o,,
o, E)>,
(4.14)
62
C. Aslangul et al. ! Dynamical properties o f a 1 D random walk
where ( P ( X = 0 , E)) is the Laplace transform of the average probability of return to the origin in the continuous model, expressed in reduced variables. Eq. (4.14) means that one identifies the average probabilities of return to the origin in the directed lattice of spacing d on the one hand, and in the continuous model on the other hand. Note that such an identification has equally been proposed in [8, 9]. In the continuous model, the average probability of return to t:~, )rigin is exactly known; its expression for any value of E is given by eq. (2.11); its small-energy expansion depends on the value of and can be found in eqs. (2.13)-(2.16). Implicitly, the reduced time variables have been identified in the two models, which in particular leads to the identification of Wm with (2tl) -1. The lattice spacing d is as yet undetermined. We shall come back to this question later. An expression for the average diffusion front ( P ( K , E)) in the continuous model is easily obtained from eqs. (4.13) and (4.14). Indeed, assuming that, according to the arguments developed in the two preceding paragraphs, this average diffusion frant may, for large scales of space and time, be identified to the one in the discrete model, one is led to d -- ( P ( X = O, E)) x,
(P(K,E))= :-
Xi/
XI
,
(4.15)
= o,
in which the small-E expansion of ( P ( X =0, E ) ) has to be used. Since we compute the first two moments of the particle position, which require the knowledge of the first two derivatives of the self-energy with respect to K (for K = 0), it is sufficient to have an expression of this quantity up to order K 2. In other words, comparing eq. (4.15) with eq. (1.2), we propose as an expression of the self-energy ,~(K, E) for low values of K and E ~V(K, E)~-.2iIxK-iK
_E )_
1 (P(0, E))
X 1
+
.r
K"[I'
A\a 2 x~ "\ ( t ( 0 , E)) - E "--~'t], XI/ ! d /
!
(4.16)
in which for simplicity ( P(0, E) > stands for ( P(X = 0, E)). Eq. (4.16) has to be thought of as the beginning of a series expansion of 2(K, E) in powers of K. The proposed identification of the general walk with a directed one allowed us to make precise the first two terms. However, let us emphasize the fact that the proposed identification, if plausible and leading to sensible physical results, has, to the best of our knowledge, not yet been proved.
C. A~ ~angul et ai. / Dynamical properties o f a I D random walk
63
5. The average particle position and its average dispersion The small-energy behaviour of the average probability of return to the origin strongly depends on the value of/~. The same will be true of the average diffusion-front. Note that this is in marked opposition with the perturbative calculation of section 3, in which, at one-loop order, the value of the self-energy at small E does not depend on/~. Thus, it will be convenient to examine separately the three cases p, < 1, 1 < / z < 2 and/.~ > 2.
5.1. t~
1
(
1 d
1
(P(O,E)) +K'~ 1 2 x~ (P(O,E)))"
(5.1)
From eq. (3.11), one immediately sees that in this phase the knowledge of the lattice spacing d is not required for the determination of the average particle position, a point which has also been noted in [8,9]. Remind, however, that the lattice spacing is certainly not equal to zero. As a result, one gets 21-2•aF(1- a ) E~'+~ "
(5.2)
By inverse Laplace transformation, one obtains at large times ( x - ~ ) ~ 2 2"-'
1
/r(l-
T".
(5.3)
This result appears also in refs. [8, 9], with the same prefactor, varying like ! / / 2 for small values of ~t., a dependence expected on physical grounds [8, 9]. By analogy with an ordinary directed-walk, one may expect that such a dependence upon time would also be valid for a given sample, with a coefficient taken at random in a known probability law [14b]. In a similar way, the Laplace transform of the average second-moment of the "r,~ -:, partl,,~ position can be calculated using eq. (3.14). One readily sees that the leading-order term is governed soieiy by the first derivative of the self-energy at K = 0. In other words, the knowledge of the lattice spacing d is not required for the determination of the average second-moment of the particle position.
64
C. Aslangul et al. / Dynamical properties o f a I D random walk
One gets 2
2
2,-2.t
r(1_ Ix)
(5.4)
"
By inverse Laplace transformation, one obtains at large times
(x2(T)) ~ 2(22~,_1 /-,(p,) )2 IXF(1- it)
T2~
(5.5)
F(EIX+ 1) "
From this result, one can deduce the average dispersion as defined by eq. (4.5). Since this quantity grows with time a s T 2~, one can analyze the time-independent ratio (x:frD
22~ - I
- 1,
(5.6)
which decreases from 1 to 0 when Ix increases from 0 to 1. Note that this ratio takes the same value as the (different) quantity 8 defined in ref. [14a] for a directed walk on a lattice witt~ hopping rates chosen at random in a probability distribution o(W) ~ W ~ - t by (5.7) This is related to the fact that the leading order terms of (x2(T)) (involved in the average dispersion) and of ([x-~)] 2) are the same. The quantity 8 is linked to the sample-to-sample fluctuations of ( x ( T ) ) . A non-zero 8 means a lack of self-averaging of the particle position. This property, established in ref. [14a] for the directed walk model, thus still holds in the general case, in the framework of the present treatment, that is as far as the large-scale equivalence with a directed walk is considered as valid.
5.2. l < I x < 2 We proceed along the same lines. Using formula (2.14) for (P(0, E ) ) , one first sees that, in the combination (1/(P(O, E))) - E(d/x~), the dominant term at small E is the first one, as in the case Ix < 1. The self-energy may thus be approximated by eq. (5.1). The Laplace transform ( x ( E ) ) of (x(T)) is deduced from the first derivative of the self-energy at K = 0, and therefore, as in the case Ix < 1, the knowledge of the lattice spacing d is not required for the determination of the average
C. Aslangul et al. / Dynamical properties o f a I D random walk
65
particle position. As a result, taking into account the two leading-order terms at small E, one gets 1
(x(E)) .~ - ~ 2(0` - 1)-40`(0` - 1)22'-2~' / ' ( 1 - / z ) E~,_ 1
(5.8)
r(t,)
By inverse Laplace transformation, one obtains at large times
(x(T))
2(/z - I ) T + 4 0 ` ( 0 ` - 1)21-2"
1
T2_ .
(5.9)
"
The calculation of the average second-moment of the particle position proceeds along the same lines as in the case 0` < 1. Here again note that the knowledge of the lattice spacing d is not required for the determination of the average second m o m e n t (x2(T)). As a result, taking into account the two leading-order terms at small E (x2(E)) -~ ~2 4 ( 0 ` - 1) 2
32 (0` - 1)30,21-2~' F(1 - 0`) E" i E3
(5.10)
By inverse Laplace transformation, one obtains at large times ~21-2u
(xE(T)) ~4(0` - 1)ET2 + 32(gt - 1) 2
( 3 - / ~ ) ( 2 - 0,)F(0,)
T 3-u .
(5.11)
The average dispersion in this phase is given by (AxE(T)) g = 16 (4-- 0,)(0, -- 1)30,21-2" (3-- 0 , ) ( 2 - 0,)F(~) T3-"
(5.12)
5.3. 2
66
C. Aslangul et al. I Dynamical properties o f a 1 D random walk
.~(K, E ) = 2 i / ~ K - i K [ 2 ( / ~ - 1)
+ K2
+E( 1
d)]
l,-2
x1
{ 1 21 X~ d [2(/z_l)+E ( p, 1-- 2
d)]}
X~
(5.13)
"
The two leading order terms of the Laplace transform (x--~) of (x-~) are 1
1(
(x-~)~--~2(/x-1)+~
1
d)
/x-2
(5.14)
xI "
By inverse Laplace transformation, one obtains at large times
(x(T)) -~ 2 ( / x -
1)T +
(]
d)
tt - 2
(5.15)
x1
The Laplace transform of the average second moment of the particle position can be calculated using eq. (3.14). One gets
(xZ(E))=~4(/x-1
+~
4
3(/x-l)
/x-2
(5.16)
.
By inverse Laplace transformation, one obtains at large times
(x"(T))~4(/.t-I)ZT2+2( 4/2-1/.t-2
3 ( / x - 1 ) d ) T.
(5.17)
The average diffusion coefficient is given by D A = 2 tt - 1 /~-2
d (/z - 1) x---~"
(5.18)
Formula (5.18) appears as a clue for obtaining the equivalent lattice spacing d. Indeed, the average diffusion-coefficient is exactly known in the continuous medium and its value is given by eq. (3.22). A convenient choice of the lattice spacing is therefore d-
xI
W-I
'
(5.19)
in accordance with refs. [8, 9]. This spacing is equal to twice the continuous limit of the correlation length of the local trapping times 1/G,+, [8-10], in accordance with the ideas developed above about the renormalized equivalent lattice. Note that d is finite.
~°)
C. Aslangul et al. / Dynamical properties o] a 1 D random walk
67
6. Conclusion The problem studied in the present paper is that of the Brownian motion of a particle in a one-dimensional disordered medium in which the disorder may be characterized by a static Gaussian random force. This Brownian motion is studied in the viscous limit, that is when inertial effects may be neglected. All the physics of the problem is controlled by ~t single dimensionless parameter g, which increases with the bias or the temperature, and decreases when the disorder strength increases. The question of the determination of the dynamical exponents and of the prefactors for the random walk has been addressed through the iarge-scale similarity of the general model with a model of a directed walk on a lattice with random uncorrelated hopping rates. This last model is simpler, and all its properties are fully understood. The calculation makes large use of the average probability of return to the origin, which is exactly known in the continuous medium. A central quantity of inter2st is the average diffusion front (p(_¥, T ) ) , or its Fourier-Laplace transform (P(K, E)). Assuming for the self-energy ,~(K, E) a series expansion in powers of K, we identify the first two terms of this expansion with the corresponding ones for a directed walk on a lattice. The average particle position and its average dispersion for a directed walk only depend on the average probability of return to the origin, and the same is true of the first two terms of the expansion of the self-energy. We thus identify the average probabilities of return to the origin in the directed lattice and in the continuous model. This allows us to derive an expression of 2~(K, E) for low values of K and E. However, the lattice spacing is left undetermined. The exponents and the prefactors of the average particle-position and of its average (or annealed) dispersion are then calculated in all phases, that is for any value of/L. As for the spacing of the equivalent lattice, it does not enter the expressions of these quantities and therefore has not to be determined in the anomalous phases, that is for/L < 2. In the normal phase however, where the average particle-position and its average dispersion can be directly calculated, this spacing is found to be equal to twice the correlation length of the local, tra.p...Din__, times, which provides a physically clear representation of the equivalent directed lattice. The use of the large-scale similarity with a directed walk model with random uncorrelated hopping rates avoids to resort to the calculation of the correlations of the energy-dependent transfer rates, which would be required by a direct calculation in the continuous medium, and which, in the anomalous phases, are most likely hard to obtain.
68
C. Aslangul et al. / Dynamical properties o f a 1 D random walk
References [1] F. Solomon, Ann. Prob. 3 (1975) 1. [2] H. Kesten, M. Koslov and F. Spitzer, Compos. Math. 30 (1975) 145. [3] Ya.G. Sinai, in: Lecture Notes in Physics 153, R. Schrader, R. Seiler and D.A. Uhlenbrock, eds. (Springer, Berlin, 1981); Theor. Prob. Appl. 27 (1982) 256. [4] B. Derrida and Y. Pomeau, Phys. Rev. Lett. 48 (1982) 627. [5] B. Denida, J. Stat. Phys. 31 (1983) 433. [6] J. Bernasconi and W.R. Schneider, J. Phys. A 15 (1983) L 729. [7] J. Bemasconi and W.R. Schneider, Helv. Phys. Acta 58 (1985) 597. [8a] J.P. Bouchaud, A. Comtet, A. Georges and P. Le Doussal, Europhys. Lett. 3 (1987) 653. [8b] J.P. Bouchaud, A. Comtet, A. Georges and E Le Doussa!, Ann. Phys. submitted. [9a] A. Georges, Thesis, University Paris-Sud, unpublished (1988). J.P. Bouchaud and A. Georges, Phys. Rep., to be published. [10a I C. Aslangul, N. Pottier and D. Saint-James, J. Phys. (Paris) 50 (1989) 899; J. Phys. Paris 50 (1989) 1581 (E). [10b] C. Aslangui, J.P. Bouchaud, A. Georges, N. Pottier and D. Saint-James, J. Stat. Phys. 55 (1989) 461. [11] C. Aslangul, N. Pottier and D. Saint-James, Physica A 164 (1990) 52. [12] J.M. Luck, Nucl. Phys. B 225 (1983) 169. [13] J. Bernasconi and W.R. Schneider, in: Fractals in Physics, L. Pietronero and E. Tosatti, eds. (Elsevier, Amsterdam, 1986). [14a] C. Aslangul, M. Barth~16my, N. Pottier and D. Saint-James, J. Stat. Phys. 59 (1990) 11. [14b] C. Aslangul, M. Barth61~my, N. Pottier and D. Saint-James, J. Stat. Phys. 61 (1990) 399. [15] J.A. Aronovitz and D.R. Nelson, Phys. Rev. A 30 (1984) 1948. [16] G.N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, 1966). [17] P. Le Doussal and J. Machta, preprint (1989).
DYNAMICAL PROPERTIES OF A RANDOM WALK IN A ONE-DIMENSIONAL RANDOM MEDIUM C. A S L A N G U L a, M. BARTHI~LEMY a, N. P O T H E R a and D. SAINT-JAMES "'l "Groupe de Physique des Solides2, Tour 23, Universit~ Paris VII, 2 place Jussieu, 75251 Paris Cedex 05, France bLaboratoire de Physique Statistique, Coll~ge de France, 3 rue d'Ulm, 75231 Paris Cedex 05, France
Received 27 July 1990
The random motion of a particle in a one-dimensional continuous random medium with random forces is investigated by making use of the large-scale equivalence of the properties of the walk with those of a directed random walk on a discrete lattice. When the disorder is weak enough, a normal drift-diffusion regime takes place, whereas for strong disorder anomalous behaviours occur. It is shown that these results may be obtained by introducing a renormalized lattice with a finite spacing. In the normal phase this spacing appears explicitly in the result, while it turns out to be irrelevant in the anomalous ones. The dynamical exponents as well as the prefactors of the power-laws of the particle position and of its average dispersion are explicitly calculated in the anomalous phases.
1. Introduction In the present paper, we consider the problem of the determination of the dynamical exponents for the random walk of a particle in a one-dimensional random medium. The model we study is defined by' the continuous space and time Fokker-Planck equation
Op(x, t) = Do 02p(x, t) 0t
0x 2
1 O 77 0x [F(x)
p(x,
t)l,
(1.1)
in which F(x) denotes a Gaussian space-dependent random force; r/ is the viscosity and Do = k Thl is the diffusion coefficient of the particle in the absence of disorder. Eq. (1.1) thus describes in the viscous limit the Brownian motion of a particle in a disordered medium in which the disorder may be o~"':~ (q,.,~ached) Gaussian random force. It is worthwhile characterized by a ~,,,,,. t Also at Universit6 Paris VII. 2 Laboratoire associ6 au CNRS (U.A. n" 17) ct aux Universit4s Paris VII et Paris VI. 0378-4371/91/$03.50 (~) 1991 - Elsevier Science Publishers B.V. (North-Holland)
48
C. Aslangul et al. t Dynamical properties o f a 1D random walk
to note here that, in order to be properly defined, the continuous model has t o be considered as the continuous limit of a discrete model on a lattice. This model, which has been widely studied for some years [1-14], is known to exhibit interesting non-trivial dynamical properties, generally referred to as dynamical phases. When the mean bias (i.e., the mean value of the force) is varied, a succession of phases is observed, characterized by different drift and diffusion behaviours, normal (i.e., with finite velocity and diffusion coefficients), or not (i.e., with non standard exponents characterizing the time dependence of the average particle position and/or of its average dispersion). The case of a zero mean bias is very peculiar and known as the Sinai model [3]: it leads to an anomalously slow (logarithmic) diffusive behaviour. Using proper reduced variables, one can show that all the physics of the problem is controlled by a single dimensionless parameter ~ (to be defined below), which increases with the bias or the temperature, and decreases when the disorder strength increases. Generally speaking, anomalous phases are found for low values of/~, that is in situations of relatively strong disorder. More precisely, anomalous drift takes place below ~ = 1, while anomalous diffusion is found below ~t = 2. Few exact analytic results are available in the general case. In a previous paper devoted to this model [ 11], we showed that the probability distributions of the energy-dependent transfer rates, as first introduced in [6, 7] by Bernasconi and Schneider, cap. be exactly computed. We also showed that, once these probability distributions are known, it is actually possible to calculate in an exact way the average over disorder of the probability of presence of the particle at its initial position, (p(0, t)). This latter quantity has also been computed by Bouchaud et al. [8,9] by taking advantage of the formal correspondence between the one-dimensional diffusion model and a quantummechanical model of a particle moving in a random one-dimensional potential. When they are finite, the velocity and the diffusion coefficient of the particle can be exactly computed through a variety of techniques [5, 8-10]. It may be shown that, in such cases, the knowledge of the average probability of return to the origin is sufficient to calculate the particle velocity and diffusion • ,-~.J'~,,,III'~i~,.,ilL [ J[UC]I.].
Unfortunately, these methods w~lich allow for the calculation of the velocity and of the diffusion coefficient when they arc finite, cannot bc easily extended to the anomalous phases. New angles of attack have to be found. The questions to be answered are the follow~ng: how is it possible in these regimes to give a determination of the anomalous exponents characterizing the average particle position and its average dispersion? Is it possible to determine the prefactors of these power-laws? Clearly, a central quantity to be analyzed is the average diffusion front
C. Aslangul et al. / Dynamical properties of a I D random walk
49
(p(X, T)), expressed for convenience in appropriate reduced space and time variables X and T, or its Fourier-Laplace transform (P(K, E)), which one can write as 1
(P(K, E ) ) = K2 + 2i/zK + E - ,~(r, E) '
(1.2)
thus defining the self-eaergy ,~(K, E)o First, one might think to try a systematic perturbative treatment of the disorder, foUowing the method given in [12, 15]. However, this perturbative expansion fails down in the anomalous phases; in the normal phase, the velocity is exact at one-loop order, but the same is not true for the diffusion coefficient. Then, one may think to use the available information about the average probability of presence of the particle at its initial position, from which, in the normal phases, the transport coefficients may be extracted. However, it is not a priori clear whether this information would be sufficient in the anomalous regimes. Actually, the answers to all the above questions are known for a simpler version of the model, the so-called directed walk model [8, 9, 14]. In a directed walk, the random variable of interest, here the particle position, can only increase, ~,,,o'-~';'~'-,.s~"""-.--.-. a given initial value. It is believed that the general both-way walk characterized by the parameter g is asymptotically similar to a directed walk on a (possibly renormalized) lattice with hopping rates chosen independently at random in a probability distribution p(W)~. W "-~ for W values smaller than some cut-off [8-10, 13]. The study of the directed walk is thus interesting in that it can generate the basic features of the general problem in a simplified framework. Indeed, the properties of the directed walk are now well understood [8, 9, 14]: in the directed walk, the average particle position and its average dispersion can be explicitly calculated for any value of ~, that is even in the anomalous phases, in terms of the average probability of presence of the particle at its initial point. One can try to exploit this large-scale similarity of the general model with the directed walk model to derive an expression for ~(K, E) for low values of K and E. The aim of the present paper is thus mainly two-fold. In a first step, we review the arguments in favor of the idea of the large-scale similarity of the general walk with a directed walk. This equivalence once made precise, the expression of the average probability of presence of the particle at its initial position (exactly known in the continuous medium) can be used to infer a form of the self-energy, supposedly valid for small K and E. In a second step, using the above-mentioned form of the self-energy, we determine the asymptotic expressions of the average particle position and of its average dispersion for
50
C. Aslangul et al. I Dynamical properties o f a 1 D random walk
any value of it. We are thus able in particular to calculate the dynamical exponents, as weli as the prefactors of the power-laws in the anomalous phases. The paper is organized as follows: in section 2, we recall the expression of the average probability of return to the origin in the general case, as found in [8,9, 11]. In section 3, we sketch out the perturbative expansion of the self-energy and show its failure for the determination of the properties of the anomalous phases. Sections 4 and 5 contain the new results of our paper. In section 4, we review the arguments in favor of the large scale similarity of the general walk with a directed walk; assuming for the self-energy at given energy a series expansion in powers of K, we take advantage of the above-mentioned equivalence with a directed walk to derive an expression for ,~(K, E) for low values of K and E. In section 5, we calculate in all phases the average particle position and its average dispersion. Finally, section 6 contains our conclusions.
2. The average probability of return to the origin 2.1. The m o d e l
Let us first briefly stress some characteristic features of the model. As well known, the continuous space and time Fokker-Planck equation (1.1) is the continuous (weak disorder) limit of the usual master equation describing a random walk on a lattice with random asymmetric hopping rates between neighbouring lattice sites, namely ddt p , = I V . .. + , P.+ , (t) + W . .._ , Pn_l(t) -- (Wn+ ,.. + Wn _ I . . ) p . ( t ) ,
(2.1)
when the hopping rates are chosen independently at random such that
,,,,
W.
+, = --w exp "~" a"
W,,+ ,.,,
Do
2Do { aF,, +,
= ~ exp~ . a 2Do .]
.
(2.2a)
(2.2b)
When the lattice spacing a becomes much smaller than xj = 4D2/or, where or denotes the variance of the Gaussian random force F(x), the master equation (2.1) can be approximated by its continuous limit (x I is the length below which the particle is insensitive to the disordered character of the potential [8, 9]). Actually, in order to be properly defined, the c'~ntinuous model (1.1) has to be thought of as deriving from the discrete model (2.1) when a <~ x t. In the
C. Aslangul et ai. I Dynamical properties of a 1 D random walk
51
following, when dealing with the continuous model, we shall use the reduced position and time variables X and T as defined by X = x/x~ and T = t/2fl, 3 2 where t~ = 8Do/tr is the diffusion time corresponding to x~. One also introduces the dimensionless random force ~,(X), the average value of which is/,. This parameter actually controls the physics of the problem.
2.2. The average probability of return to zhe origin The average over disorder of the probability of presence of the particle on its starting site has been exactly calculated in the continuous limit by several different methods. In [8, 9], a formal correspondence has been established between the present one-dimensional diffusion model and a quantum-mechanical particle moving in a one-dimensional random potential. The average probability of return to the origin (p(0, T)} (expressed for convenience in the reduced variables) has been related to the density of states of a Schr6dinger equation in a random potential. This density of states has been computed through two different approaches: a Dyson-Schmidt technique and the replica method. As a result
(p(0, T)) = f
dE e x p ( - E T ) dN dE'
(2.3)
o
with 2
1
(2.4)
N(E) = ---5 2 1/2) ' ~r J~,2 (EI/2) + N~,(E
where J~, and N~, are Bessel functions of order/z [16]. In [11], use has been made of the so-called equivalent transfer-rates technique, a method first devised for the discrete case in [6, 7] and developed in [10]. One introduces the quantities G+(z)and G~(z) as defined by +
G. (z)P.(z) = W,,+,.,,P.(z)- W..,,+,P.+,(z) , c,,-(z)P_.(z) = w_._, _.P_.(z)-
n >10,
(2.5) n~>0,
where
P,,(z) = f dt exp(- zt)p.(t). 0
(2.6)
52
C. Aslangul et ai. i Dynamical properties o f a I D random walk
The master equation (2.4) is equivalent to the set of equations
zP,,(z) = - G . + ( z ) P . ( z ) + G,_,(z)P,_,(z),+
n = 1, 2 , . . . ,
(2.8)
zPo(z ) - 1 = - G ~ (z)Po(z) - G O(z)Po(z) , z P _ . ( z ) = - G ~ ( z ) P _ . ( z ) + G~_,(z)P_,+,(z),
(2.7)
n = 1,2, . . . .
(2.9)
In this formulation, the quantities G+~(z) and G-~(z) play the role of energydependent transfer rates respectively towards the right of the sites of positive index or towards the left of the sites of negative index. Their equivalent in the continuous medium are denoted by G+(X, E) and G - ( - X , E), where E is the reduced variable associated with z. In the disordered medium, these quantities are random variables; their probability distributions can be exactly computed analytically in the continuous limit [11]. From these distributions, using the continuous equivalent of eq. (2.8) P(0, E)[G+(O, E) + G-(O,
E)] =
1
(2.10)
one can deduce the average probability of presence of the particle at its initial site, ( P ( X = 0, E ) ) , which for simplicity we denote as (P(0, E ) ) . As a result
1
i
(P(0, E)) = ~ ~=, ds
K~,
K~,(E2,,z)
,
(2.11)
where K, denotes the modified Bessei function of order /x [16]. By using contour integration, one can show that the above expression yields for (p(0, T)) a result identical to eq. (2.3) with d N / d E as given by eq. (2.4).
2.3. The small-energy expansion of (P(O, E) ) In the following, it will be necessary to use the small-E expansion of (P(0, E ) ) . When n 0), one gets an expansion of the type (P(0, E)) ~ ao(i.t ) + a,(Iz)E + ' . . + a,,_,( l.t)E"-' + C~( tx)E ~-' (2.12) All the coefficients in eq. (2.12) may straightforwardly be computed from eq. (2.11 ) by using standard integration formulas and series expansions of modified Bessel-functions [16]. Let us just quote the results of interest:
C. Aslangul et al. / Dynamical properties of a I D random walk
53
(i) Ix < 1
In this case, one gets at leading order
(P(0, E)) -2'-2 Ix
,
(2.13)
where F(x) denotes the Euler gamma function. (ii) 1 < I x < 2
The first two dominant terms are 1 + 2,_2~/z F ( 1 - Ix) E~_ , (P(0, E ) ) ~ 2(Ix - 1) F(Ix)
(2.14)
(the following term would be of the form a~(Ix)E). The first term of this expansion is just the inverse of the finite velocity [8-10] V= 2(Ix - 1).
(2.15)
v,,J I x > 2 The first three dominant terms are 1
E
(P(0, E)) = 2(Ix - 1)
1
4 (Ix - 1)2(Ix - 2)
+ 21
F(! - Ix) E~,_ 1
-2#Ix F(Ix) (2.16)
(the following term would be of the form a2(Ix)E2). The first two terms correspond to an expansion of the form (P(0, E ) ) =
1
V
2ED
V3 + . . . ,
(2.17)
with the finite diffusion coefficient D as given by refs. [8-10],
Ix-2 When they are finite, the velocity and the diffusion coefficient of the particle which appear in eqs. (2.14) and (2.16) have been exactly computed by various techniques. In [5], the particle position and its dispersion have been calculated on a periodized lattice of large period N; for finite N, whatever the value of Ix, normal drift and diffusion behaviours are found. When the limit N--> ~: is taken, normal drift (for Ix > 1) and diffusion (for Ix > 2) behaviours remain;
54
C. Aslangul et al. / Dynamical properties o f a I D random walk
unfortunately, this method has not been extended, up to now, to a description of the behaviours in the anomalous phases. Another approach has been used in reL [10b], where an infinite lattice is considered from the start; the velocity and the diffusion coefficient of the particle, when they are finite, are calculated through the use of a resummation procedure; this method however cannot be readily extended to the anomalous phases. Since, for/z > 1 (resp. for/x > 2), the velocity (resp. the diffusion coefficient) can be related to the small-E expansion of (P(0, E ) ) [10a], one is logically led to expect to get information from this quantity also in the anomalous phases. In this respect, it is interesting to note that, for any value of/z, the smaU-E expansion of the average probability of return to the origin is analogous to the corresponding expansion for a directed walk on a discrete lattice (see eqs. (6) and (16) in [14a]), which, in that model, contains all the necessary information for the description of the motion, even in the anomalous regimes.
3. Perturbative calculation of the average diffusion front
As indicated in the introduction, the central quantity to be analyzed is the average diffusion front (p(X, T)), or its Fourier-Laplace transform (P(K, E)) (as usual, a Fourier transformation in space and a Laplace transformation in time is carried out). Eq. (1.2) defines the self-energy ,~(K, E). Following refs. [12, 15] (see also ref. [9]), one may attempt to undertake a systematic perturbative treatment of the disorder. The Fokker-Planck equation (1.1), written in its dimensionless form
op(X, r) OT
o p(X, r) -
0X 2
o OX [2~(X) p(X, T)],
(3.1)
is equivalent to the integral equation
P(K,E)= P°ra(K,E) 1 - i K
~2aO(Q)P(K-Q,E)
,
(3.2)
where i "~',-°'a~,,, " e ) = K~ + 2 i ~ K + E
(3.3)
is the Fourier-Laplace of the diffusion front in the absence of disorder and ~¢J(X) = ~ ( X ) - v.
(3.4)
C. Aslangul et al. / Dynamical properties of a 1 D random walk
55
denotes the departure of the (dimensionless) random force from its average value ~. Eq. (3.2) allows for constructing a perturbative expansion in powers of ~$. Since the random force is Gaussian, the average over disorder of the perturbation series deduced from eq. (3.2) is obtained by pairing together the force terms. Let us now examine the first-order contribution to the self-energy.
3.1. First-order contribution to the self-energy As ordinarily, the first-order contribution to the self-energy can be obtained in two ways. One can iterate twice eq. (3.2) and neglect correlations between the terms containing the random force and the diffusion front. One can also iterate eq. (3.2) to infinity and make a partial resummation of the diagrams reducible to one-loop. Using this method, one sees that this expansion provides a convergence criterion which, in the limit K--->0, E---~0, reads /x > 1. At lowest order (one-loop order), the self-energy is given by
~(K, E)= -2 f
dQ ,iT
K ( K - Q) ( K - Q)2 + 2 i ~ ( K - Q) + E
(3.5)
The integration limits are +--~ since we consider the continuous model. An analytic continuation of expression (3.5) can be made in the complex plane for Re E <0. The function ,~(K, E) has a cut on the real negative axis for E < - / z 2. The above integration leads to
2~(K, E)= 2ilzK(E +/.t2)-~/'- ,.
(3~6)•
everywhere outside the cut (the square root is the branch which takes real positive values on the real positive axis). The following expression for ( P(K, E)) is thus obtained" 1
(P(K, E)) = K2 + 2i/xK + E - 2i/xK(E +/x2) -''2 "
(3.7)
The normalization condition 1
(P(K = 0, E ) ) - ~ ,
(3.8)
which implies that Z ( K = O , E) = 0 , is satisfied by the one-loop approximation (3.6) for the self-energy.
(3.9)
56
C. Aslangul et al. I Dynamical properties o f a 1D random walk
3.2. The average particle position From the average diffusion front, one can calculate the average values of the first two moments of the particle position, ( x ( T ) ) and (x 2(T)). The bar denotes the so-called thermal average for a given choice of the random force, as defined by ,-,!,,-"ac
(3.10)
A(T) = f dX A(X) P(X, T) .
As above, the symbol ( - ) stands for the average over disorder. The Laplace transform (x(E)) of ( x - ~ ) is given by 0 ( x - ~ > = i ~ (P(K, E)>Ir= o , or
1(
(x---(~) = ~
2p.
(3.11)
a2~ +i ~-~[K=o).
(3.12)
As a result, one gets at large times for any It, by taking into account the two leading order terms at small E and making the inverse Laplace transformation 1 /.t
(x(T)) ~ 2 ( i t - 1)T + ~ .
(3.13)
Otherwise stated, the one-loop approximation for the self-energy gives a finite velocity equal to 2 ( / z - 1) for any It, a result which is known to be exact for It > 1. Clearly, when It < 1 this result is unacceptable on physical grounds, since the particle cannot move against the field. This indicates a breakdown of the one-loop approximation in this phase [9, 12]. This is linked to the lack of convergence of the geometric series involved in this approximation.
3.3. The average second moment In a similar way, the Laplace transform (x:(E)) of (x2(T)) is given by =
(3.14)
OK" ( P(K, E)>I,,=,,,
or
2
i
(x2(E)) = ~ (2it + i ~-~ K
~
(
2
.
(3.15)
C. Aslangul et al. I Dynamical properties of a 1 D random walk
57
As a result, one gets at large times for any/.~, by taking into account the two leading order terms at small E and making the inverse Laplace transformation /
,,
... 1\
(x2(T)) ~ 4 ( t t - 1)2T 2 + 2~1 + 4 -~---fL/T/~ / .
(3.16)
From this result, one can deduce the so-called average [8, 9] (or annealed [17]) diffusion coefficient D A as defined by ( x E ( r ) ) - ( x - ~ ) 2 ~ 2DA T .
(3.17)
This diffusion coefficient is defined through both thermal and disorder averages over the trajectories. It is generally different from the average over disorder of the diffusion coefficient for a given sample, a quantity denoted by D O (quenched diffusion coefficient) in [17], and defined by (xE(T) - [x(T)l 2) =
2DOT.
(3.18)
One always has DA> Do . Clearly, the knowledge of the average diffusion front yields only eqs. (3.13) and (3.16), one gets D A = D °rd + 2 / x -21 , /x
(3.19) D A.
From
(3.20)
where D °rd= 1 is the diffusion coefficient in the absence of disorder. Otherwise stated, the one-loop approximation for the self-energy gives a finite diffusion coefficient for any/.t. However, even in the normal phase/x > 2, this result is not satisfying, since in this phase the exact result is [8, 9] DA =
/X -~ .
(3.21)
Therefore, the situation for the diffusion coefficient is not the same as for the velocity, all the more since eq. (3.20) does not display any clear-cut indication of the expected onset of anomalous diffusion regimes below/~ = 2. Therefore, eq. (3.20) is unacceptable f o r / z < 2 . In addition, one can remark eq. (3.20) cannot be acceptable below /x = 1, since it would lead to a value of the diffusion coefficient lower than the one in the corresponding orderd lattice and even possibly negative.
C. Aslangul et al. ! Dynamical properties of a I D random walk
58
The one-loop results are thus very far from being satisfactory. Actually, looking again at expressions (3.12) and (3.15), one sees that the first two moments of the particle position require the knowledge of the first two derivatives of ,~(K, E) with respect to K (for K = 0), which are not correctly calculated in the one-loop approximation. Terms of any order in the loop expansion of the self-energy do contribute to O~(K,E)/OKIr,= o and a2,~(K, E ) ! OK21r=o . In other words, the dynamical properties of interest in the anomalous phases cannot be properly described within an approximation which is basically a weak-disorder one. Some other method must be proposed. Coming back to the integral equation (3.2~ ,~,ae sees that a series expansion of ,~(K, E) "n powers of K is not incompatible with the structure of the equation. This is the reason why, from now on, we shall assume the existence of such an expansion. To find its coefficients, we shall exploit the large-scale similarity of the general walk with a directed walk. As will be shown below, in the directed walk, a series expansion of the self-energy in powers of K can be found. By comparison, this will lead us to propose an expression for Z(K, E) for small values of K and E in the general case.
4. Large-scale equivalence with a directed walk model
Various arguments have been put forward in favor of the idea of the large-scale similarity of the general walk with a directed walk [8-10, 13]. Let us briefly recall the main features of the directed walk model and sum up these arguments. 4.1. The model
It is defined by the following master equation: dD~ dt
(4.1)
W.p. + W._,p._, .
which describes a random waik in which the position of the particle can only increase, starting from a given initial value, since the possibility of jumps towards the left is suppressed. The properties of the directed walk are well understood [8, 9, 14]. For example, one can choose the hopping rates independently at random in a given probability distribution
(w)
p(W) = C,W~-'f~ ~
tel
,
a >0,
(4.2)
C. As!angul et al. / Dynamical properties of a I D random walk
59
where C, is a normalization constant and fc a cut-off function basically specified by the fixed frequency W,.. The smaller a, the higher the probability to find a qtzasi-broken link. If T denotes a dimensionless time variable as defined by T= Wmt, the average particle position behaves as follows: (x-('~) ~ T ° ,
0 < a < 1,
(4.3)
(x--~) ~ T,
1 < a.
(4.4)
A phase of anomalous drift is seen to appear for 0 < a < 1. One can define also the average dispersion
(AxZ(T))A = (x2(T)) - ( x ( T ) ) 2 ,
(4.5)
an extension to the anomalous phases of the definition of the average (or annealed) diffusion coefficient D A [8, 9, 17]. This quantity behaves as (Ax2(T)) a ~ T2" ,
0< a < 1 ,
(4.6)
(Ax2(T))A ~ T 3-" ,
l
(4.7)
(aZ(T))
2<
T,
,
(4.8)
Anomalous diffusion thus takes place below a = 2. The following important point desexes to be underlined: in the directed walk model as described by eq. (4.1), the probability distribution of the hopping rates is chosen, for sufficiently low values of a (actually a < 1), to diverge in the limit of small hopping rates (eq. (4.2)). The physical origin of the anomalous phases has to be traced back to the existence of a relatively great weight of quasi-broken links [14].
4.2. Large-scale equivalence of the general walk with a directed walk In ref. [13], Bernasconi and Schneider applied a real-space renormalization procedure to a discrete-time random walk analogous to the continuous-time random walk described by eq. (2.1). Through a Monte Carlo procedure, they showed that the renormalized random walk approaches a directed random walk with uncorrelated successive transition times (or inverse hopping rates) with a broad distribution. This idea has been further developed in [8, 9l by Bouchaud et al. Let us recall here the main line of their argument. Clearly, in the problem at hand.
60
C. Aslangul et al. I Dynamical properties o f a 1 D random walk
which is a Brownian-motion problem in the presence of a Gaussian random force, the asymptotic regime is characterized by the competition between the bias term - F o x and the slowing-down effect of the very large fluctuations of the potential. These high barriers act as trapping regions between which the motion is convective. High barriers are exponentially rare, but the time needed to overcome one of them depends exponentially on its height. The net effect is a broad distribution of the local trapping times. We provided in [10a] complementary analytical arguments in favour of this equivalence in the discrete case. We showed that the quantities G+~(z) and G~(z) display the following behaviours at small z:
(z)--- an+ G-~ (z) ~ g~ z .
+ g2 z,
(4.9) (4.113)
Eqs. (4.9) and (4.10)-together with eqs. ( 2 . 7 ) - ( 2 . 9 ) - support the following interpretation: the long-time asymptotic properties of the walk are similar to the properties of a strictly directed walk, in which only steps towards the right are allowed. The role of the equivalent transfer rates towards the right of this strictly directed walk is played by the quantities G~+w). In this interpretation, 1/G+~ ~°~ appears to represent the time needed by the particle to leave the site of index n. These equivalent transfer rates however are not uncorrelated quantities. To get rid of these correlation effects, one could try to proceed to a real-space renormalization, with the hope to use new effective uncorrelated hopping rates, the correlation effects being included in lattice spacing and possibly time renormalization. After an infinite number of renormalization steps, and at infinite time, one hopes thus that the walk renormalizes towards a directed walk with independent random hopping rates. The role of the hopping rates should be played by the quantities G +t0), considered now as uncorrelated, but on a lattice with a renormalized spacing which takes into account the former correlations. For the moment, let us note that the probability distribution of these quantities has been calculated in a discrete lattice [I1], and shown to behave like [G~t°~] "-1 for small values of the variable. In other words, the sojourn times (or local trapping times) distribution, which in the present framework is tl~e distribution oi the quantities 1/G~+t°~, is a broad law, as expected on physical grounds [8, 9]. One can expect that the general random walk as described by eq. (2.1) would be equivalent, as far as its large-scale properties are concerned, to a directed walk on a renormalized lattice with independent random hopping rates, distributed according to a law of the form (4.2), in
C. Aslangul etal. I Dynamical properties of a 1D random walk
61
which one takes a = It. Similarly, the continuous limit of the general random walk is expected to renormalize towards a directed walk on a lattice with a finite spacing d. Note that the consideration of a finite lattice spacing is necessary, since in its continuous limit, eq. (4.1) does not contain diffusion effects. One is thus led to expect, for the general case, behaviours with exponents identical to those of eqs. (4.3)-(4.4) for the average particle position and to those of eqs. (4.6)-.(4.8) for its average dispersion. A precise determination of the coefficients would in particular imply the knowledge of the lattice spacing. Supplementary information can actually be extracted from the average probability of return to the origin, as we now show.
4.3. Approximate form of the average diffusion front for low values of K and E Let us consider on a lattice of spacing d a directed walk with independent hopping rates. A convenient way to deal with the properties at large space scale is to look for small values of q at the Fourier transform (Pq(z)> of (P,(z) >, as defined by ~c
(4.11)
n =1)
One easily calculates
1W>
(4.12)
W
1 e iq
(Pq(Z)>= ( P,,__o(z)>1 -- e-iqd(1
-- Z(P,,-_o(z) >)
(4.13)
The question now arises: how must we choose the lattice spacing d on the one hand and (P,,=0(z)> on the other hand in order to describe at best the directed walk for which the large-scale properties of the average diffusion front are similar to those of the continuous problem (eq. (1.i))? A reasonable choice for (P,,=0(z)> valid for large times (i.e., for small z and E) seems to be a
x, o,,
o, E)>,
(4.14)
62
C. Aslangul et al. ! Dynamical properties o f a 1 D random walk
where ( P ( X = 0 , E)) is the Laplace transform of the average probability of return to the origin in the continuous model, expressed in reduced variables. Eq. (4.14) means that one identifies the average probabilities of return to the origin in the directed lattice of spacing d on the one hand, and in the continuous model on the other hand. Note that such an identification has equally been proposed in [8, 9]. In the continuous model, the average probability of return to t:~, )rigin is exactly known; its expression for any value of E is given by eq. (2.11); its small-energy expansion depends on the value of and can be found in eqs. (2.13)-(2.16). Implicitly, the reduced time variables have been identified in the two models, which in particular leads to the identification of Wm with (2tl) -1. The lattice spacing d is as yet undetermined. We shall come back to this question later. An expression for the average diffusion front ( P ( K , E)) in the continuous model is easily obtained from eqs. (4.13) and (4.14). Indeed, assuming that, according to the arguments developed in the two preceding paragraphs, this average diffusion frant may, for large scales of space and time, be identified to the one in the discrete model, one is led to d -- ( P ( X = O, E)) x,
(P(K,E))= :-
Xi/
XI
,
(4.15)
= o,
in which the small-E expansion of ( P ( X =0, E ) ) has to be used. Since we compute the first two moments of the particle position, which require the knowledge of the first two derivatives of the self-energy with respect to K (for K = 0), it is sufficient to have an expression of this quantity up to order K 2. In other words, comparing eq. (4.15) with eq. (1.2), we propose as an expression of the self-energy ,~(K, E) for low values of K and E ~V(K, E)~-.2iIxK-iK
_E )_
1 (P(0, E))
X 1
+
.r
K"[I'
A\a 2 x~ "\ ( t ( 0 , E)) - E "--~'t], XI/ ! d /
!
(4.16)
in which for simplicity ( P(0, E) > stands for ( P(X = 0, E)). Eq. (4.16) has to be thought of as the beginning of a series expansion of 2(K, E) in powers of K. The proposed identification of the general walk with a directed one allowed us to make precise the first two terms. However, let us emphasize the fact that the proposed identification, if plausible and leading to sensible physical results, has, to the best of our knowledge, not yet been proved.
C. A~ ~angul et ai. / Dynamical properties o f a I D random walk
63
5. The average particle position and its average dispersion The small-energy behaviour of the average probability of return to the origin strongly depends on the value of/~. The same will be true of the average diffusion-front. Note that this is in marked opposition with the perturbative calculation of section 3, in which, at one-loop order, the value of the self-energy at small E does not depend on/~. Thus, it will be convenient to examine separately the three cases p, < 1, 1 < / z < 2 and/.~ > 2.
5.1. t~
1
(
1 d
1
(P(O,E)) +K'~ 1 2 x~ (P(O,E)))"
(5.1)
From eq. (3.11), one immediately sees that in this phase the knowledge of the lattice spacing d is not required for the determination of the average particle position, a point which has also been noted in [8,9]. Remind, however, that the lattice spacing is certainly not equal to zero. As a result, one gets 21-2•aF(1- a ) E~'+~ "
(5.2)
By inverse Laplace transformation, one obtains at large times ( x - ~ ) ~ 2 2"-'
1
/r(l-
T".
(5.3)
This result appears also in refs. [8, 9], with the same prefactor, varying like ! / / 2 for small values of ~t., a dependence expected on physical grounds [8, 9]. By analogy with an ordinary directed-walk, one may expect that such a dependence upon time would also be valid for a given sample, with a coefficient taken at random in a known probability law [14b]. In a similar way, the Laplace transform of the average second-moment of the "r,~ -:, partl,,~ position can be calculated using eq. (3.14). One readily sees that the leading-order term is governed soieiy by the first derivative of the self-energy at K = 0. In other words, the knowledge of the lattice spacing d is not required for the determination of the average second-moment of the particle position.
64
C. Aslangul et al. / Dynamical properties o f a I D random walk
One gets 2
2
2,-2.t
r(1_ Ix)
(5.4)
"
By inverse Laplace transformation, one obtains at large times
(x2(T)) ~ 2(22~,_1 /-,(p,) )2 IXF(1- it)
T2~
(5.5)
F(EIX+ 1) "
From this result, one can deduce the average dispersion as defined by eq. (4.5). Since this quantity grows with time a s T 2~, one can analyze the time-independent ratio (x:frD
22~ - I
- 1,
(5.6)
which decreases from 1 to 0 when Ix increases from 0 to 1. Note that this ratio takes the same value as the (different) quantity 8 defined in ref. [14a] for a directed walk on a lattice witt~ hopping rates chosen at random in a probability distribution o(W) ~ W ~ - t by (5.7) This is related to the fact that the leading order terms of (x2(T)) (involved in the average dispersion) and of ([x-~)] 2) are the same. The quantity 8 is linked to the sample-to-sample fluctuations of ( x ( T ) ) . A non-zero 8 means a lack of self-averaging of the particle position. This property, established in ref. [14a] for the directed walk model, thus still holds in the general case, in the framework of the present treatment, that is as far as the large-scale equivalence with a directed walk is considered as valid.
5.2. l < I x < 2 We proceed along the same lines. Using formula (2.14) for (P(0, E ) ) , one first sees that, in the combination (1/(P(O, E))) - E(d/x~), the dominant term at small E is the first one, as in the case Ix < 1. The self-energy may thus be approximated by eq. (5.1). The Laplace transform ( x ( E ) ) of (x(T)) is deduced from the first derivative of the self-energy at K = 0, and therefore, as in the case Ix < 1, the knowledge of the lattice spacing d is not required for the determination of the average
C. Aslangul et al. / Dynamical properties o f a I D random walk
65
particle position. As a result, taking into account the two leading-order terms at small E, one gets 1
(x(E)) .~ - ~ 2(0` - 1)-40`(0` - 1)22'-2~' / ' ( 1 - / z ) E~,_ 1
(5.8)
r(t,)
By inverse Laplace transformation, one obtains at large times
(x(T))
2(/z - I ) T + 4 0 ` ( 0 ` - 1)21-2"
1
T2_ .
(5.9)
"
The calculation of the average second-moment of the particle position proceeds along the same lines as in the case 0` < 1. Here again note that the knowledge of the lattice spacing d is not required for the determination of the average second m o m e n t (x2(T)). As a result, taking into account the two leading-order terms at small E (x2(E)) -~ ~2 4 ( 0 ` - 1) 2
32 (0` - 1)30,21-2~' F(1 - 0`) E" i E3
(5.10)
By inverse Laplace transformation, one obtains at large times ~21-2u
(xE(T)) ~4(0` - 1)ET2 + 32(gt - 1) 2
( 3 - / ~ ) ( 2 - 0,)F(0,)
T 3-u .
(5.11)
The average dispersion in this phase is given by (AxE(T)) g = 16 (4-- 0,)(0, -- 1)30,21-2" (3-- 0 , ) ( 2 - 0,)F(~) T3-"
(5.12)
5.3. 2
66
C. Aslangul et al. I Dynamical properties o f a 1 D random walk
.~(K, E ) = 2 i / ~ K - i K [ 2 ( / ~ - 1)
+ K2
+E( 1
d)]
l,-2
x1
{ 1 21 X~ d [2(/z_l)+E ( p, 1-- 2
d)]}
X~
(5.13)
"
The two leading order terms of the Laplace transform (x--~) of (x-~) are 1
1(
(x-~)~--~2(/x-1)+~
1
d)
/x-2
(5.14)
xI "
By inverse Laplace transformation, one obtains at large times
(x(T)) -~ 2 ( / x -
1)T +
(]
d)
tt - 2
(5.15)
x1
The Laplace transform of the average second moment of the particle position can be calculated using eq. (3.14). One gets
(xZ(E))=~4(/x-1
+~
4
3(/x-l)
/x-2
(5.16)
.
By inverse Laplace transformation, one obtains at large times
(x"(T))~4(/.t-I)ZT2+2( 4/2-1/.t-2
3 ( / x - 1 ) d ) T.
(5.17)
The average diffusion coefficient is given by D A = 2 tt - 1 /~-2
d (/z - 1) x---~"
(5.18)
Formula (5.18) appears as a clue for obtaining the equivalent lattice spacing d. Indeed, the average diffusion-coefficient is exactly known in the continuous medium and its value is given by eq. (3.22). A convenient choice of the lattice spacing is therefore d-
xI
W-I
'
(5.19)
in accordance with refs. [8, 9]. This spacing is equal to twice the continuous limit of the correlation length of the local trapping times 1/G,+, [8-10], in accordance with the ideas developed above about the renormalized equivalent lattice. Note that d is finite.
~°)
C. Aslangul et al. / Dynamical properties o] a 1 D random walk
67
6. Conclusion The problem studied in the present paper is that of the Brownian motion of a particle in a one-dimensional disordered medium in which the disorder may be characterized by a static Gaussian random force. This Brownian motion is studied in the viscous limit, that is when inertial effects may be neglected. All the physics of the problem is controlled by ~t single dimensionless parameter g, which increases with the bias or the temperature, and decreases when the disorder strength increases. The question of the determination of the dynamical exponents and of the prefactors for the random walk has been addressed through the iarge-scale similarity of the general model with a model of a directed walk on a lattice with random uncorrelated hopping rates. This last model is simpler, and all its properties are fully understood. The calculation makes large use of the average probability of return to the origin, which is exactly known in the continuous medium. A central quantity of inter2st is the average diffusion front (p(_¥, T ) ) , or its Fourier-Laplace transform (P(K, E)). Assuming for the self-energy ,~(K, E) a series expansion in powers of K, we identify the first two terms of this expansion with the corresponding ones for a directed walk on a lattice. The average particle position and its average dispersion for a directed walk only depend on the average probability of return to the origin, and the same is true of the first two terms of the expansion of the self-energy. We thus identify the average probabilities of return to the origin in the directed lattice and in the continuous model. This allows us to derive an expression of 2~(K, E) for low values of K and E. However, the lattice spacing is left undetermined. The exponents and the prefactors of the average particle-position and of its average (or annealed) dispersion are then calculated in all phases, that is for any value of/L. As for the spacing of the equivalent lattice, it does not enter the expressions of these quantities and therefore has not to be determined in the anomalous phases, that is for/L < 2. In the normal phase however, where the average particle-position and its average dispersion can be directly calculated, this spacing is found to be equal to twice the correlation length of the local, tra.p...Din__, times, which provides a physically clear representation of the equivalent directed lattice. The use of the large-scale similarity with a directed walk model with random uncorrelated hopping rates avoids to resort to the calculation of the correlations of the energy-dependent transfer rates, which would be required by a direct calculation in the continuous medium, and which, in the anomalous phases, are most likely hard to obtain.
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C. Aslangul et al. / Dynamical properties o f a 1 D random walk
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