Phase transition in a kinetic mean-field game model of inertial self-propelled agents

TL;DR

This work addresses phase transitions in the collective motion of inertial, non-cooperative agents by formulating a kinetic mean-field game with finite-range interactions and inertia. It derives a forward–backward PDE system (nonlinear Fokker–Planck and Hamilton–Jacobi–Bellman) and analyzes the linear stability of the spatially homogeneous ordered state against perturbations, using Fourier–Hermite decomposition and resolvent-based eigenvalue analysis, to identify a critical unit-cost-of-control $r_c$ at which a travelling-wave phase bifurcates. The study shows that for $r>r_c$ the ordered equilibrium is stable, while for $r<r_c$ it becomes unstable and travelling waves emerge, with numerical simulations confirming the transition and the existence of travelling-wave solutions; the analysis leverages Hamiltonian matrices and Riccati equations to certify stability. The work provides a game-theoretic perspective on non-equilibrium collective motion and extends phase-transition phenomena known from phenomenological models to a kinetic MFG framework with inertia and finite-range interactions, offering a principled approach to predictive modeling of bio-inspired collective behavior.

Abstract

The framework of Mean-field Games (MFGs) is used for modelling the collective dynamics of large populations of non-cooperative decision-making agents. We formulate and analyze a kinetic MFG model for an interacting system of non-cooperative motile agents with inertial dynamics and finite-range interactions, where each agent is minimizing a biologically inspired cost function. By analyzing the associated coupled forward-backward in time system of nonlinear Fokker-Planck and Hamilton-Jacobi-Bellman equations, we obtain conditions for closed-loop linear stability of the spatially homogeneous MFG equilibrium that corresponds to an ordered state with non-zero mean speed. Using a combination of analysis and numerical simulations, we show that when energetic cost of control is reduced below a critical value, this equilibrium loses stability, and the system transitions to a travelling wave solution. Our work provides a game-theoretic perspective to the problem of collective motion in non-equilibrium biological and bio-inspired systems.

Figures (6)

• Figure 1: (a) The real (solid) and imaginary (dashed) components of the eigenvalue of $L_{k}$ (for $h=5,k=1$) closest to the imaginary axis, as a function of noise intensity $\sigma$. The system is unstable for $\sigma\leq \sigma_c=1.8$ as the real part is positive in that range. (b) The spectrum of $L_{k}$ at the threshold of stability, $\sigma=\sigma_c$.
• Figure 2: The marginal density $\bigintssss_{ \mathbb{R}} \rho(t,x,v)dv$ of a travelling wave solution of the Czirók model Eq. \ref{['eq:PDE_for']} for $h=5, \sigma=0.8<\sigma_c$. This solution is the steady state reached upon perturbing the unstable spatially homogeneous equilibrium $\rho_{ \xi}(u)$, where $\bigintssss_{ \mathbb{R}} \rho_{\xi}(v)dv= \dfrac{1}{l}=0.1.$
• Figure 3: The spectrum (close to the imaginary axis) of the linearized MFG operator in Eq. \ref{['eq:eigsys_mfg']} as the control cost $r$ is varied, for Fourier mode $k=1$.
• Figure 4: The norm of $Y_1(t)$ (bold) and $Y_2(t)$ (dashed) as a function of time for the unique solution $(Y_1(t),Y_2(t))$ of the BVP of Eq. \ref{['eq:bvpmat1']}. Here, $r=1.4>r_c$, and $k=1$. We choose an arbitrary $Y_1(0)$, and the corresponding value of $Y_2(0)$ is assigned according to Lemma 1.
• Figure 5: The critical unit control cost $r_c$ as a function of $\sigma$, for $h=5$.

Theorems & Definitions (4)

• Definition 1
• Theorem 1
• Lemma 1
• proof