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

Abstract

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 latticebased 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.

Lane Formation by Side-Stepping

“…[7]. We would like to remark that the lengthy Taylor expansion and formal limiting procedure can be accomplished automatically using computer algebra techniques, even for more general classes of models, see [23].…”

“…The limit τ → 0. Let (r k , b k ) be a sequence of solutions to (23). We define r τ (x, y, t) = r k (x, y) and b τ (x, y, t) = b k (x, y) for (x, y) ∈ Ω and t ∈ ((k − 1)τ, kτ ].…”

Micro- and macroscopic modeling of crowding and pushing in corridors

“…One advantage of CA approaches is that the formal passage from the microscopic to the macroscopic level is rather straight-forward based on a Taylor expansion of the respective transition rates. This can for example be done systematically using tools from symbolic computation, see [22]. CA approaches have been used successfully to describe lane formation, as for example in [30], or evacuation situations, such as in [21].…”

“…Here, we use a Taylor expansion to develop the transition rates and functions in x ± ∆x and y ± ∆x. This rather tedious calculation can be done in a systematic manner using a similar approach as discussed in [22]. 4.1.…”