Mesoscopic description of reactions for anomalous diffusion: a case study

Abstract

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 reactiondiffusion 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 Lévy, 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.

Mathematical Modelling of Subdiffusion-reaction Systems

“…[21], [22], [27]. In a multicomponent system, in the case where the evolution of the spatially homogeneous system is governed by the linear equation…”

“…As mentioned in the introduction, a somewhat controversial point is the age of the molecule transformed into a molecule of another kind due to a chemical reaction without jumping. If the waiting time distribution for jumps is determined by the probability distribution of fluctuations in the environment that make the molecule release possible, it is natural to assume that the age of the molecule, understood as the argument of the waiting time pdf, is not changed due to the participation of the molecule in the reaction [21], [27]. In the latter case, the evolution equation for densities can be written [34] as…”

“…In [23], [24] the particle is "rejuvenated" after it reacts, i.e., its waiting time distribution is set as if the particle has performed a jump. In [21], [27], [28], [25] it is assumed that the reaction does not change the age of the particle. The consequences of both assumptions are discussed in [29].…”

Anomalous subdiffusion with multispecies linear reaction dynamics

“…An equivalent result, using a slightly different formalism, has also been derived by balancing reaction and diffusion fluxes in a CTRW description [32] and this result has been generalized to two species with irreversible linear reaction kinetics [32,33]. An example is the two-species system in which the concentration of one species, A, decays at a constant per capita rate and the concentration of another species, B, increases at the same rate (i.e., A → B) [32,33]. The governing evolution equations for the concentrations can be written as…”

“…(3) but the evolution equation for species B involves a linear combination of both pure fractional diffusive terms as well as reaction modified fractional diffusive terms involving species A. A CTRW formulation of two-species reversible reactions with heavy tailed waiting-time densities was also provided in [33] but no corresponding formulation as a fractional reaction-diffusion equation was obtained for this case. In related work, a set of integrodifferential equations has been introduced to model a multi-species system with anomalous subdiffusion and nonlinear reaction dynamics [34,35].…”