On the Field Dependence of Random Walks in the Presence of Random Fields (original) (raw)

Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications

Physics reports, 1990

The subject of this paper is the evolution of Browman particles m disordered environments The "Ariadne's clew" we follow is understanding of the general statistical mechanisms which may generate "anomalous" (non-Brownlan) diffusion laws, this allows us to develop simple arguments to obtain a qualitative (but often qmte accurate) picture of most situations Several analytical techniques -such as the Green function formalism and renormahzatlon group methods-are also exposed Care is devoted to the problem of sample to sample fluctuations, particularly acute here We consider the specific effects of a bias on anomalous diffusion, and discuss the generahzat~ons of Einstein's relation m the presence of disorder An effort is made to illustrate the theoreucal models by describing many physical situations where anomalous d~ffuslon laws have been-or could be -observed (1 16) ~>2 ~GAU$SIAN J -P Bouchaud and A Georges, Anomalous d(fuston m dzsordered media 1 2 2 2 Polymer adsorption and self-avoiding Levy flights The structure of an adsorbed polymer is shown m fig 1 4 it Is made of pomts m &rect contact with an attractive wall, separated by large loops in the bulk The main point is that the *~ This is of course only a rough approximation, since complicated correlations between successive lumps exist in this deterministic motion They are, however, very likely to be short ranged and thus will not change the time dependence of X, (see section 1 3) A Comgho, M Daoud and H M DonsKer ana a varadhan, Lommun IJure Appl Math 32 (1979) 721 P Le Doussal, Thesis, Umv Pans 6 (1987), unpubhshed P Le Doussal, Phys Rev B 39 (1988) 881 P Le Doussal and J Machta, Phys Rev B 40 (1989) 9427 P Le Doussal, Phys Rev Lett 62 (1989) 3097 P LeDoussal, private communication, and m preparation P G Doyle and

Generalization of the Sinai Anomalous Diffusion Law

Sinai has considered a novel one-dimensional walk with a random bias field E on each site. He has shown that when the field is taken with equal probability to be +E,, or -E , the R M S displacement R [(x2)]"' increases with time t by the anomalously slow law R -(log t)'. Here we introduce long-range correlation between the random fields on each site by choosing a 'string' of k sites to have the same value of E, where k is chosen from the power law distribution P ( k ) = k-O. We find that the Sinai law is generalised to the form R -(log I ) ' , where J sticks at the Sinai value y = 2 for p 2. However, for 1 < p < 2, y varies continuously with p as y = p / ( p -1). We interpret this result physically

Anomalous Diffusion in Random-Media of Any Dimensionality

Journal De Physique, 1987

2014 Nous montrons que la diffusion est anormale en toute dimension dans les milieux aléatoires où les forces présentent des corrélations à longue portée. Ce résultat est obtenu à la fois par des arguments physiques et par une analyse de groupe de renormalisation. Les comportements obtenus sont en général surdiffusifs sauf lorsque la force aléatoire dérive d'un potentiel. Dans cette situation on obtient un comportement sousdiffusif. Dans le cas critique supérieur (D = 2 pour des corrélations à courte portée), celui-ci est caractérisé par un exposant dépendant continûment du désordre. La raison en est l'annulation de la fonction 03B2 qui est démontrée à tous les ordres de la théorie des perturbations. Dans le cas général, des arguments simples suggèrent qu'une force potentielle conduit à des diffusions logarithmiques (c'est-à-dire à du bruit en 1/f).

Effect of temperature on biased random walks in disordered media

Physical Review E, 1997

We study diffusion on an energetically disordered lattice, where each bond between sites is characterized as a random energy barrier. In such a model it had previously been observed that the mean square displacement is sublinear with time at early times, but eventually reaches the classical linear behavior at long times, as a strong function of the temperature. In the current work we add the effect of directional bias in the random walk motion, in which along one axis only, motion in one direction is assigned a higher probability while along the opposite direction a reduced probability. We observe that for low temperatures a ballistic character dominates, as shown by a slope of 2 in the R 2 vs time plot, while at high temperatures the slope reverts to 1, manifesting that the effect of the bias parameter is obliterated. Thus, we show that for a biased random walk diffusion may proceed faster at lower temperatures. The details of how this crossover takes place, and the scaling law of the crossover temperature as a function of the bias are also given. ͓S1063-651X͑97͒51207-4͔

Biased diffusion in anisotropic disordered systems

Physical Review E, 2000

We investigate a diffusion process into an anisotropic disordered medium in the presence of a bias. The medium is modeled by a two-dimensional square lattice in which the anisotropic disorder is represented by a bond percolation model with different occupation probabilities on each direction. The biased diffusion process is mapped by a random walk with unequal transition probabilities along and against the field ͑in the ͓1,1͔ direction͒ by performing Monte Carlo simulations. We observe a transition from pure to drift diffusion when the bias reaches a threshold B c. In order to estimate this B c , an effective exponent is used to characterize the diffusion process. This B c is also compared with another estimation for the critical field.

Anomalous biased diffusion in a randomly layered medium

Physical Review E, 2010

We present analytical results for the biased diffusion of particles moving under a constant force in a randomly layered medium. The influence of this medium on the particle dynamics is modeled by a piecewise constant random force. The long-time behavior of the particle position is studied in the frame of a continuous-time random walk on a semi-infinite one-dimensional lattice. We formulate the conditions for anomalous diffusion, derive the diffusion laws and analyze their dependence on the particle mass and the distribution of the random force.

Intrinsic randomness of transport coefficient in subdiffusion with static disorder

Physical Review E, 2011

Fluctuations in the time-averaged mean-square displacement for random walks on hypercubic lattices with static disorder are investigated. It is analytically shown that the diffusion coefficient becomes a random variable as a manifestation of weak ergodicity breaking. For two- and higher- dimensional systems, the distribution function of the diffusion coefficient is found to be the Mittag-Leffler distribution, which is the same as for the continuous-time random walk, whereas for one-dimensional systems a different distribution (a modified Mittag-Leffler distribution) arises. We also present a comparison of these two distributions in terms of an ergodicity-breaking parameter and show that the modified Mittag-Leffler distribution has a larger deviation from ergodicity. Some remarks on similarities between these results and observations in biological experiments are presented.

Corrections to scaling for diffusion in disordered media

Journal of Physics A: Mathematical and General, 1989

We study the diffusion of a particle in a &dimensional lattice where disorder arises from a random distribution of waiting times associated with each site of the lattice. Using scaling arguments we derive, in addition to the leading asymptotic behaviour, the correction-to-scaling terms for the mean square displacement. We also perform detailed Monte Carlo simulations for one, two and three dimensions which give results in substantial agreement with the scaling argument predictions.

Markov chain analysis of random walks in disordered media

Physical Review E, 1994

We study the dynamical exponents dwd_{w}dw and dsd_{s}ds for a particle diffusing in a disordered medium (modeled by a percolation cluster), from the regime of extreme disorder (i.e., when the percolation cluster is a fractal at p=pcp=p_{c}p=pc) to the Lorentz gas regime when the cluster has weak disorder at p>pcp>p_{c}p>pc and the leading behavior is standard diffusion. A new technique of relating the velocity autocorrelation function and the return to the starting point probability to the asymptotic spectral properties of the hopping transition probability matrix of the diffusing particle is used, and the latter is numerically analyzed using the Arnoldi-Saad algorithm. We also present evidence for a new scaling relation for the second largest eigenvalue in terms of the size of the cluster, ∣lnlambdamax∣simS−dw/df|\ln{\lambda}_{max}|\sim S^{-d_w/d_f}∣lnlambdamax∣simS−dw/df, which provides a very efficient and accurate method of extracting the spectral dimension dsd_sds where ds=2df/dwd_s=2d_f/d_wds=2df/dw.