Effect of temperature on biased random walks in disordered media

Abstract

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. ͓S1063-651X͑97͒51207-4͔PACS number͑s͒: 05.40.ϩj, 05.60.ϩw Biased random walk is a prototype model for studies of particle diffusion in disorded solids ͓1,2͔ where kinetic problems are concerned, such as conduction, viscous flow, polymer dynamics, etc. The characteristic of the bias implies a preferential direction for the motion, as opposed to purely stochastic motion. The problem becomes much more complicated when the underlying space is not a simple lattice, but contains a certain degree of randomness itself, such as, for example, a rugged energy landscape model that we recently introduced ͓3-5͔. Every lattice site has its own energy ͑usually chosen randomly, from a random number distribution͒. Thus, the problem is not directly amenable to exact analytical theories, except mean field arguments. In the present study we continue the investigation of such systems, with the inclusion now of the bias characteristics.The model used here is the one used in our previous studies ͓3-5͔, in which we now incorporate an external field ͑bias͒. Briefly, a square lattice is generated. Each bond between any two sites is a barrier with a height that is randomly chosen from a given distribution. The height of the barriers depends on the mean value ͗E͘ and on the dispersion parameter of the E distribution in the following way:

Kubo–Anderson oscillator and NMR of solid state

“…In the most works in which the concept of the correlation times distribution has been discussed, it was assumed that the randomness of correlation times appears due to a static set of random barriers and an atom (or molecule) moves without changing them. However, randomness of potential barriers must be dynamic rather than static, since, due to the collective character of the diffusion process, the fluctuations of the potential barriers would occur simultaneously with each jump of a moving particle [39][40][41]. Now we consider the transformations of the second moment of NMR line shape for the case, when the potential barrier for the mobility of magnetic nuclei is the stochastic function of the time.…”

Particle Separation Using High Frequency Alternating External Fields

“…Earlier, 10,11 we found that if a bias is applied, then there is a crossover time after which the mode of motion switches from a predominantly random walk to a ballistic one. The transition time depends on the jump probability and the height of the bias.…”

“…In many models, the nonsymmetric behavior is due to the retention of memory of the previous step, resulting in enhanced diffusion. In our previous work, 10,11 this was due to an external field leading to a ballistic motion in the long time limit. On the other hand, when the bias alternates at a given frequency ν ) 1/2τ (τ is the time it takes to change the direction of the bias) the mean square displacement was found to be proportional to time t, a behavior typical of a stochastic random walk.…”

Random walks interacting with evolving energy landscapes

“…In the latter case, the lattice sites (or bonds) are assigned different energy states and, consequently, the walker is biased towards sites corresponding to potential wells (or small energy barriers). Moreover, disorder can be deterministic or random and it can be dynamic (the environment is renewed at each jump of the walker) [1,10], or static (the environment is frozen in a particular configuration) [11,12]. Particles diffusing on such structures can also be endowed with memory effects [13,14], or be influenced by the distribution of other diffusing particles on the same structure [15].…”