Mean-field and Monte Carlo Analysis of Multi-Species Dynamics of agents

TL;DR

This paper develops a mean-field approximation and compares it with Monte Carlo simulations to analyze multi-species particle dynamics on a lattice, revealing a transient Gaussian-to-Gaussian behavior in spatial distributions.

Contribution

The study introduces a combined mean-field and Monte Carlo approach for multi-species dynamics, including a renormalization technique and analysis of distribution transformations.

Findings

Mean-field and Monte Carlo results show good agreement with optimized parameters.

Identified a transient Gaussian-to-Gaussian distribution behavior.

Analyzed effects of density and bias on spatial distribution fluctuations.

Abstract

We propose a mean-field (MF) approximation for the recurrence relation governing the dynamics of $m$ species of particles on a square lattice, and we simultaneously perform Monte Carlo (MC) simulations under identical initial conditions to emulate the intricate motion observed in environments such as subway corridors and scramble crossings in large cities. Each species moves according to transition probabilities influenced by its respective static floor field and the state of neighboring cells. To illustrate the methodology, we analyze statistical fluctuations in the spatial distribution for $m = 1$, $m = 2$, and $m = 4$ and for different regimes of average density and biased movement. A numerical comparison is conducted to determine the best agreement between the MC simulations and the MF approximation considering a renormalization exponent $\beta$ that optimizes the fit between methods. Finally, we report a phenomenon we term "Gaussian-to-Gaussian" behavior, in which an initially normal distribution of particles becomes distorted due to interactions among same and opposing species, passes through a transient regime, and eventually returns to a Gaussian-like profile in the steady state, after multiple rounds of motion under periodic boundary conditions.