Symbolic Derivation of Mean-Field PDEs from Lattice-Based Models

TL;DR

This paper introduces a symbolic computation method to derive mean-field PDEs from lattice-based models, enabling analysis of transportation processes like crowd motion and cell motility.

Contribution

It presents a novel symbolic approach to derive PDEs from microscopic lattice models, including implementation details and applications in crowd dynamics.

Findings

Successfully derives mean-field PDEs from lattice models.

Demonstrates application in crowd motion analysis.

Provides a general framework applicable to various transportation processes.

Abstract

Transportation processes, which play a prominent role in the life and social sciences, are typically described by discrete models on lattices. For studying their dynamics a continuous formulation of the problem via partial differential equations (PDE) is employed. In this paper we propose a symbolic computation approach to derive mean-field PDEs from a lattice-based model. We start with the microscopic equations, which state the probability to find a particle at a given lattice site. Then the PDEs are formally derived by Taylor expansions of the probability densities and by passing to an appropriate limit as the time steps and the distances between lattice sites tend to zero. We present an implementation in a computer algebra system that performs this transition for a general class of models. In order to rewrite the mean-field PDEs in a conservative formulation, we adapt and implement symbolic integration methods that can handle unspecified functions in several variables. To illustrate our approach, we consider an application in crowd motion analysis where the dynamics of bidirectional flows are studied. However, the presented approach can be applied to various transportation processes of multiple species with variable size in any dimension, for example, to confirm several proposed mean-field models for cell motility.