Mesoscopic description of reactions under anomalous diffusion: A case study

Abstract

Reaction-diffusion equations deliver a versatile tool for the description of reactions in inhomogeneous systems under the assumption that the characteristic reaction scales and the scales of the inhomogeneities in the reactant concentrations separate. In the present work we discuss the possibilities of a generalization of reaction-diffusion equations to the case of anomalous diffusion described by continuous-time random walks with decoupled step length and waiting time probability densities, the first being Gaussian or Levy, the second one being an exponential or a power-law lacking the first moment. We consider a special case of an irreversible or reversible A ->B conversion and show that only in the Markovian case of an exponential waiting time distribution the diffusion- and the reaction-term can be decoupled. In all other cases, the properties of the reaction affect the transport operator, so that the form of the corresponding reaction-anomalous diffusion equations does not closely follow the form of the usual reaction-diffusion equations.

References in corpus (10)

• On an explicit finite difference method for fractional diffusion equations
• Optimal target search on a fast folding polymer chain with volume exchange
• Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach
• Fractional Reaction-Diffusion Equation
• On reaction-subdiffusion equations
• Subdiffusion-limited reactions
• Levy Flights in Inhomogeneous Media
• Superfast front propagation in reactive systems with anomalous diffusion
• Trapping reactions with subdiffusive traps and particles characterized by different anomalous diffusion exponents
• Particle Dispersion on Rapidly Folding Random Hetero-Polymers

Cited by in corpus (9)

• Reaction-subdiffusion and reaction-superdiffusion equations for evanescent particles performing continuous time random walks
• On subdiffusive continuous time random walks with stochastic resetting
• On tempered fractional calculus with respect to functions and the associated fractional differential equations
• Continuous-time random-walk approach to normal and anomalous reaction-diffusion processes
• Subdiffusion-limited fractional reaction-subdiffusion equations with affine reactions: Solution, stochastic paths, and applications
• Anomalous reaction-diffusion equations for linear reactions
• Reaction-subdiffusion on moving fluids
• Demographic fluctuations in a population of anomalously diffusing individuals
• Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations