Extensions of effective-medium theory of transport in disordered systems

First-passage behavior of the random-barrier model

The previously proposed transport equation for the random-barrier model, which is the diffusion equation with resetting to positions visited in the past, is used here to calculate the first-passage times. The results obtained are compared with those obtained using the normal diffusion equation with an effective diffusion coefficient. It is shown that, under certain conditions, the equation with the effective diffusion coefficient can greatly overestimate the time of the first passage. In particular, the rate constant of a bimolecular diffusion-controlled reaction calculated from this equation can be significantly lower than the actual rate. This result can serve as one of the possible explanations for the high rates of diffusion-controlled reactions observed in an experiment.

The Drude-Smith Equation and Related Equations for the Frequency-Dependent Electrical Conductivity of Materials: Insight from a Memory Function Formalism

The Drude‐Smith equation is widely used for treating the frequency‐dependent electrical conductivity of materials in the terahertz region. An attractive feature is its sparsity of adjustable parameters. A significant improvement over Drude theory for these materials, the theory includes backscattering of the charge carriers. It has nevertheless been criticized, including by Smith himself, because of the arbitrariness of a step in the derivation. We recall a somewhat similar behavior of back scattering in fluids observed in molecular dynamics computations and discussed in terms of memory functions. We show how theories such as Drude‐Smith and Cocker et al. are examples of a broader class of theories by showing how they also arise as particular cases of a memory function formalism that divides the interactions into short and long range.

Subordinated stochastic processes with aged operational time

In this paper, subordinated stochastic processes are considered, where the renewal process acting as the operational time. It is assumed that the observation of the process begins at a certain time after the start of the renewal process. A recurrence formula was derived for calculating the multipoint probability density functions of the aged renewal process. Two-point correlation functions for certain subordinated stochastic processes, particularly for the generalized Ornstein-Uhlenbeck process, were calculated. A model of relaxation in a disordered medium with traps and obstacles is proposed.

Effective-medium approximation for lattice random walks with long-range jumps

We consider the random walk on a lattice with random transition rates and arbitrarily long-range jumps. We employ Bruggeman's effective-medium approximation (EMA) to find the disorder-averaged (coarse-grained) dynamics. The EMA procedure replaces the disordered system with a cleverly guessed reference system in a self-consistent manner. We give necessary conditions on the reference system and discuss possible physical mechanisms of anomalous diffusion. In the case of a power-law scaling between transition rates and distance, lattice variants of Lévy-flights emerge as the effective medium, and the problem is solved analytically, bearing the effective anomalous diffusivity. Finally, we discuss several example distributions and demonstrate very good agreement with numerical simulations.

Encounter times in overlapping domains: application to epidemic spread in a population of territorial animals

We develop an analytical method to calculate encounter times of two random walkers in one dimension when each individual is segregated in its own spatial domain and shares with its neighbor only a fraction of the available space, finding very good agreement with numerically exact calculations. We model a population of susceptible and infected territorial individuals with this spatial arrangement, and which may transmit an epidemic when they meet. We apply the results on encounter times to determine analytically the macroscopic propagation speed of the epidemic as a function of the microscopic characteristics: the confining geometry, the animal diffusion constant, and the infection transmission probability.

Multiscale analysis of collective motion and decision-making in swarms: an advection-diffusion equation with memory approach

We propose a (time) multiscale method for the coarse-grained analysis of collective motion and decision-making in self-propelled particle models of swarms comprising a mixture of 'naïve' and 'informed' individuals. The method is based on projecting the particle configuration onto a single 'meta-particle' that consists of the elongation of the flock together with the mean group velocity and position. We find that the collective states can be associated with the transient and asymptotic transport properties of the random walk followed by the meta-particle, which we assume follows a continuous time random walk (CTRW). These properties can be accurately predicted at the macroscopic level by an advection-diffusion equation with memory (ADEM) whose parameters are obtained from a mean group velocity time series obtained from a single simulation run of the individual-based model.