Model for Heterogeneous Diffusion

On the modelling of spatially heterogeneous nonlocal diffusion: deciding factors and preferential position of individuals

“…In some sense, the diffusivity matrix does not convey all the information of the motion in the heterogeneous case. This is quite similar to the observation made by two of the present authors in [32] in the kinetic model.…”

“…This provides some new interpretation for equation (1.3). However, as was pointed out by two of the present authors in an earlier work [32], the diffusivity function D(x) may not be sufficient to describe heterogeneous diffusion accurately, because it does not fully cover velocity jump processes given by kinetic equations. In subsection 3.3, we make a related observation that the nonlocal diffusion equation, or position jump process, also leads to a range of local equations wider than (1.3).…”

“…The related question of whether heterogeneous dispersal is advantegeous to a species has been adressed from the evolutionary point of view in several works [6,11,12]. Recently, two of the authors of the present work derived heterogeneous diffusion equations from revertible kinetic systems and compared the properties of diffusion laws using a thought experiment [32]. In particular, they obtained Wereide's diffusion law as a special case.…”

Heterogeneous discrete kinetic model and its diffusion limit

“…It seems that isotropic diffusion equations are always in this form if they are derived from a revertible random walk system (see Section 2). Note that this is the same diffusion equation which has passed the thought experiment test in [12]. The kinetic theory is the foundation of statistical thermodynamics which provides a molecular-level explanation of the classical thermodynamics.…”

“…Kim and Seo [12] introduced a new idea that a particle takes a vector field after a collision and moves along an integral curve of the chosen vector field. The spatial heterogeneity can be included in the vector fields v α as well as in the probability q.…”