Subdiffusion of mixed origin with chemical reactions

We derive a reaction-subdiffusion equation that takes into account two microscopic mechanisms responsible for subdiffusion in real media. We show that the concentration profiles in media with identical subdiffusion exponents but with different microscopic structures can differ significantly at the same chemical kinetics.     

Model of superdiffusion

A locally nonequilibrium model of superdiffusion is proposed that is based on the partition of the set of diffusing particles into groups according to the flight length of these particles. The process of diffusion is described in terms of partial concentrations of particles belonging to different groups. As special limit cases, the model yields equations with fractional time derivative and the so-called porous medium equation. The basic equations of the model are Markov equations; therefore, they easily include reaction terms. The model can be applied to describing the types of diffusion in which the diffusing particles are in free flight most of the time.     

Mass transfer in nonuniform media

The lattice model was used to derive equations describing diffusion in a nonuniform medium in the absence of local equilibrium at nonzero temperature gradients. Equations are obtained that extend a diffusion equation with fractional derivative in time and a nonlinear diffusion equation, which was previously obtained within the framework of a generalized thermodynamic approach, to cover diffusion in a medium with a nonuniform temperature field.     

An extension of the multiple-trapping model

The hopping charge transport in disordered semiconductors is considered. Using the concept of the transport energy level, macroscopic equations are derived that extend a multiple-trapping model to the case of semiconductors with both energy and spatial disorders. It is shown that, although both types of disorder can cause dispersive transport, the frequency dependence of conductivity is determined exclusively by the spatial disorder.     

Subdiffusion with the disappearance of particles at the time of a jump

The reaction-subdiffusion equations corresponding to a monomolecular chemical reaction at the time of a diffusion jump are derived. It is shown that the approach to deriving such equations suggested previously in [8] gives the correct result in the case of asymptotic subdiffusion but is inapplicable in the practically important case of transient subdiffusion.     

A necessary condition for the emergence of diffusion instability in media with nonclassical diffusion

A necessary condition is derived for the emergence of diffusion instability in media in which diffusion does not obey classical Fick laws. The equations derived by Yadav and Horsthemke [Phys. Rev. E 74, 066118 (2006)] using the continuous-time random walk model are employed as equations simulating reaction—diffusion processes. The waiting-time distribution function is represented by the sum of a finite number of exponents. It is shown that passage to the diffusion limit in the time variable is an incorrect operation if it is used to analyze diffusion instability in media with a distribution function that differs from the Poisson distribution function.     

Continuous-Time Random Walks under Finite Concentrations

Journal of Experimental and Theoretical Physics - Nonlinear equations generalizing the continuous-time random walk model to the case of finite concentrations have been derived. The equations take...     

Equations for the distributions of functionals of a random-walk trajectory in an inhomogeneous medium

Based on the random-trap model and using the mean-field approximation, we derive an equation that allows the distribution of a functional of the trajectory of a particle making random walks over inhomogeneous-lattice site to be calculated. The derived equation is a generalization of the Feynman-Kac equation to an inhomogeneous medium. We also derive a backward equation in which not the final position of the particle but its position at the initial time is used as an independent variable. As an example of applying the derived equations, we consider the one-dimensional problem of calculating the first-passage time distribution. We show that the average first-passage times for homogeneous and inhomogeneous media with identical diffusion coefficients coincide, but the variance of the distribution for an inhomogeneous medium can be many times larger than that for a homogeneous one.     

A model of anomalous transport

A lattice model is used to derive a system of equations describing anomalous transport in the case of low tracer concentration. In the adopted model, anomalous transport is due to nonequilibrium distribution of tracer particles over sites in an inhomogeneous lattice. It is shown that a well-known time-fractional differential equation can be derived from the lattice equations under certain additional assumptions.     

Transport Equation for Subdiffusion of Mixed Origin

Journal of Experimental and Theoretical Physics - The transport equation is proposed for the model, which is a combination of the random barrier model and the multiple trapping model. Negative...