Diffusion in a random medium: A Monte Carlo study

Biased and greedy random walks on two-dimensional lattices with quenched randomness: the greedy ant within a disordered environment

The main characteristics of biased greedy random walks (BGRWs) on two-dimensional lattices with real-valued quenched disorder on the lattice edges are studied. Here the disorder allows for negative edge weights. In previous studies, considering the negative-weight percolation (NWP) problem, this was shown to change the universality class of the existing, static percolation transition. In the presented study, four different types of BGRWs and an algorithm based on the ant colony optimization heuristic were considered. Regarding the BGRWs, the precise configurations of the lattice walks constructed during the numerical simulations were influenced by two parameters: a disorder parameter ρ that controls the amount of negative edge weights on the lattice and a bias strength B that governs the drift of the walkers along a certain lattice direction. The random walks are "greedy" in the sense that the local optimal choice of the walker is to preferentially traverse edges with a negative weight (associated with a net gain of "energy" for the walker). Here, the pivotal observable is the probability that, after termination, a lattice walk exhibits a total negative weight, which is here considered as percolating. The behavior of this observable as function of ρ for different bias strengths B is put under scrutiny. Upon tuning ρ, the probability to find such a feasible lattice walk increases from zero to 1. This is the key feature of the percolation transition in the NWP model. Here, we address the question how well the transition point ρ(c), resulting from numerically exact and "static" simulations in terms of the NWP model, can be resolved using simple dynamic algorithms that have only local information available, one of the basic questions in the physics of glassy systems.

Effect of the transition of networks from floppy to rigid on the diffusion coefficient

We investigate a tracer particle moving on a two-dimensional square lattice created by network formers (NF). The positions of all network formers are randomly displaced by a small amount from the nodes of the network. Each NF can be in a "floppy" or "rigid" state, depending on the number of bonds connecting it to neighboring network formers. The NF that have more than a specified number of m bonds are in rigid state, the remaining ones are in a floppy state. The energy of the tracer particle depends on its distance from those of the four nearest NF that are in "rigid" state. The NF in floppy state do not contribute to the energy. We here demonstrate that the a priori increase in the diffusion coefficient with the concentration of the floppy states goes through a crossover point, after which the increase is much sharper.

Kubo-Anderson oscillator and NMR of solid state

The analytical solution for the Kubo-Anderson oscillator with a fluctuating frequency omega for arbitrary distribution function p(omega) has been obtained. The obtained theoretical expression has been applied to consideration of some dynamical problems of solid state NMR, namely (1) dynamical transformation of NMR line shape and spin-echo signal and (2) the temperature transformation of the second moment of NMR line for the case, when the potential barrier for the mobility of magnetic nuclei is a stochastic function of time.

A biological interpretation of transient anomalous subdiffusion. I. Qualitative model

Anomalous subdiffusion has been reported for two-dimensional diffusion in the plasma membrane and three-dimensional diffusion in the nucleus and cytoplasm. If a particle diffuses in a suitable infinite hierarchy of binding sites, diffusion is well known to be anomalous at all times. But if the hierarchy is finite, diffusion is anomalous at short times and normal at long times. For a prescribed set of binding sites, Monte Carlo calculations yield the anomalous diffusion exponent and the average time over which diffusion is anomalous. If even a single binding site is present, there is a very short, almost artifactual, period of anomalous subdiffusion, but a hierarchy of binding sites extends the anomalous regime considerably. As is well known, an essential requirement for anomalous subdiffusion due to binding is that the diffusing particle cannot be in thermal equilibrium with the binding sites; an equilibrated particle diffuses normally at all times. Anomalous subdiffusion due to barriers, however, still occurs at thermal equilibrium, and anomalous subdiffusion due to a combination of binding sites and barriers is reduced but not eliminated on equilibration. This physical model is translated directly into a plausible biological model testable by single-particle tracking.

Diffusion and percolation in anisotropic random barrier models

An anisotropic random barrier model is presented, in which the transition probabilities in different directions have different probability density functions. At low temperatures, the anisotropic long-time diffusion coefficients, obtained using an effective medium approximation, follow an Arrhenius temperature dependence, with the same activation energy for each direction. Such activation energy is related to the anisotropic percolation properties of the lattice, and can be analyzed in terms of the critical percolation path approximation. The anisotropic effective medium approximation is shown to predict the correct percolation threshold for an anisotropic two-dimensional square lattice. In addition, results are compared with numerical simulations using a fast kinetic Monte Carlo algorithm.