Year: 2025
Type: article
Abstract: We consider overdamped Langevin dynamics for the attractive log gas on the torus T d {\mathbb {T}}^\mathsf {d} , for d ≥ 1 \mathsf {d}\geq 1 . In dimension d = 2 \mathsf {d}=2 , this model coincides with a periodic version of the parabolic-elliptic Patlak-Keller-Segel model of chemotaxis. The attractive log gas (for our choice of units) is well-known to have a critical inverse temperature β c = 2 d \beta _{\mathrm {c}}={2\mathsf {d}} corresponding to when the free energy is bounded from below. Moreover, it is well-known that the uniform distribution is always a stationary state regardless of the temperature. We identify another temperature threshold β s \beta _{\mathrm {s}} sharply corresponding to the nonlinear stability of the uniform distribution. We show that for β > β s \beta >\beta _{\mathrm {s}} , the uniform distribution does not minimize the free energy and moreover is nonlinearly unstable, while for β > β s \beta >\beta _{\mathrm {s}} , it is stable. We also show that there exists β u \beta _{\mathrm {u}} for which uniqueness of equilibria holds for β >
DOI: 10.1090/cams/55
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Awards: Many-particle Systems with Singular Interactions: Statistical Mechanics and Mean-field Dynamics, G3872290907, G5500258283, G7565540492, FRG: Collaborative Research: New Challenges in the Derivation and Dynamics of Quantum Systems, G8233350650, Mean-Field and Singular Limits of Deterministic and Stochastic Interacting Particle Systems
OpenAlex ID: W4416004123