Rigorous mean-field limit and cross-diffusion

Abstract

The mean-field limit in a weakly interacting stochastic many-particle system for multiple population species in the whole space is proved. The limiting system consists of cross-diffusion equations, modeling the segregation of populations. The mean-field limit is performed in two steps: First, the many-particle system leads in the large population limit to an intermediate nonlocal diffusion system. The local cross-diffusion system is then obtained from the nonlocal system when the interaction potentials approach the Dirac delta distribution. The global existence of the limiting and the intermediate diffusion systems is shown for small initial data, and an error estimate is given.

Rigorous Derivation of Population Cross-Diffusion Systems from Moderately Interacting Particle Systems

“…Our results are valid for functions f i depending on the species type, but we choose the same function for all species to simplify the presentation. This paper extends the many-particle limit of Chen et al (2019) leading to the cross-diffusion system…”

“…Finally, we remark that the cross-diffusion models (1) and ( 2) have quite different structural properties; also see (Burger et al 2020a, b). First, system (2) has a formal gradient-flow structure for each species separately, while system (1) can be written, under the detailed-balance condition (Chen et al 2019), only in a vector-valued gradient-flow form. Second, the segregation behavior of both models is different, i.e., segregation is stronger for the solutions to (2) than for model (1); see the numerical experiments in Sect.…”

Random-batch method for multi-species stochastic interacting particle systems

“…Equations (1) can be used to model the information flow through social networks [2], the dynamics of opinions [8], the herding of sheep by dogs [26], or the segregation behavior of populations [6]. Stochastic gradient descent can be interpreted as the evolution of interacting particle systems governed by a potential related to the objective function used to train neural networks [25].…”

Structure and Stability of Steady State Bifurcation in a Cannibalism Model with Cross-Diffusion

“…e possibility of crossdiffusion terms in multicomponent systems was proposed by Onsager and Fuoss [2], while Baldwin et al [3] undertook the experimental verification of the existence of crossdiffusion and also observed that the crossdiffusion coefficients can be quite significant. Since then, various crossdiffusion mathematical models have been suggested to interpret and predict many interesting features of natural multicomponent dynamics [4][5][6][7][8][9][10].…”