Diffusion in systems with static disorder (original) (raw)

Long time tails in stationary random media. I. Theory

J Statist Phys, 1984

Diffusion of moving particles in stationary disordered media is studied using a phenomenological mode-coupling theory. The presence of disorder leads to a generalized diffusion equation, with memory kernels having power law long time tails. The velocity autocorrelation function is found to decay like t-(d/2+1), while the time correlation function associated with the super-Burnett coefficient decays like t -d/2 for long times. The theory is applicable to a wide variety of dynamical and stochastic systems including the Lorentz gas and hopping models. We find new, general expressions for the coefficients of the long time tails which agree with previous results for exactly solvable hopping models and with the low-density results obtained for the Lorentz gas. Finally we mention that if the moving particles are charged, then the long time tails imply that there is an ω d/2 contribution to the low-frequency part of the frequency-dependent electrical conductivity.

Random Time-Scale Invariant Diffusion and Transport Coefficients

Physical Review Letters, 2008

Single particle tracking of mRNA molecules and lipid granules in living cells shows that the time averaged mean squared displacement δ 2 of individual particles remains a random variable while indicating that the particle motion is subdiffusive. We investigate this type of ergodicity breaking within the continuous time random walk model and show that δ 2 differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable δ 2 . Similarly we quantify the response to a constant external field, revealing a generalization of the Einstein relation. Consequences for the interpretation of single molecule tracking data are discussed. 05.40.Fb,87.10.Mn An ensemble of non interacting Brownian particles spreads according to Fick's law as a Gaussian packet. The ensemble averaged mean square displacement (MSD) is x 2 (t) = 2D 1 t where D 1 is the diffusion constant. By an Einstein relation D 1 is expressed in terms of statistical properties of the microscopic jumps according to D 1 = δx 2 /2 τ where τ is the average time between jumps and δx 2 is the variance of the jump lengths. Instead one can analyze the time series x(t) of the particle trajectory and determine the time averaged (TA) MSD

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

Journal de Physique, 1989

2014 La vitesse et le coefficient de diffusion d'une particule sur un réseau périodique unidimensionnel de période N avec des taux de transfert aléatoires et asymétriques sont calculés de manière simple grâce à une méthode basée sur une relation de récurrence, qui permet d'établir une analogie aux grands temps avec un modèle de marche strictement dirigée. Les résultats pour un système complètement aléatoire sont obtenus en prenant la limite N ~ ~. On montre qu'un calcul, reposant sur une hypothèse d'échelle dynamique, de la vitesse et du coefficient de diffusion dans un réseau désordonné infini conduit aux mêmes résultats.

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.

Normal-to-anomalous diffusion transition in disordered correlated potentials: From the central limit theorem to stable laws

Physical Review E, 2013

We study the diffusion of an ensemble of overdamped particles sliding over a tilted random potential (produced by the interaction of a particle with a random polymer) with long-range correlations. We found that the diffusion properties of such a system are closely related to the correlation function of the corresponding potential. We model the substrate as a symbolic trajectory of a shift space which enables us to obtain a general formula for the diffusion coefficient when normal diffusion occurs. The total time that the particle takes to travel through n monomers can be seen as an ergodic sum to which we can apply the central limit theorem. The latter can be implemented if the correlations decay fast enough in order for the central limit theorem to be valid. On the other hand, we presume that when the central limit theorem breaks down the system give rise to anomalous diffusion. We give two examples exhibiting a transition from normal to anomalous diffusion due to this mechanism. We also give analytical expressions for the diffusion exponents in both cases by assuming convergence to a stable law. Finally we test our predictions by means of numerical simulations.

Classical diffusion and quantum level velocities: systematic deviations from random matrix theory

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1999

We study the response of the quasienergy levels in the context of quantized chaotic systems through the level velocity variance and relate them to classical diffusion coefficients using detailed semiclassical analysis. The systematic deviations from random matrix theory, assuming independence of eigenvectors from eigenvalues, are shown to be connected to classical higher-order time correlations of the chaotic system. We study the standard map as a specific example, and thus the well-known oscillatory behavior of the diffusion coefficient with respect to the parameter is reflected exactly in the oscillations of the variance of the level velocities. We study the case of mixed phase-space dynamics as well and note a transition in the scaling properties of the variance that occurs along with the classical transition to chaos.

Diffusion in periodic, correlated random forcing landscapes

Journal of Physics A: Mathematical and Theoretical, 2014

We study the dynamics of a Brownian particle in a strongly correlated quenched random potential defined as a periodically-extended (with period L) finite trajectory of a fractional Brownian motion with arbitrary Hurst exponent H ∈ (0, 1). While the periodicity ensures that the ultimate long-time behavior is diffusive, the generalised Sinai potential considered here leads to a strong logarithmic confinement of particle trajectories at intermediate times. These two competing trends lead to dynamical frustration and result in a rich statistical behavior of the diffusion coefficient DL: Although one has the typical value D typ L ∼ exp(−βL H ), we show via an exact analytical approach that the positive moments