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Ergodicity and asymptotic limits for Langevin interacting systems with singular forces and multiplicative noises

Manh Hong Duong, Hung Dang Nguyen, Wenxuan Tao

TL;DR

This work analyzes N-particle Langevin systems with singular interactions and multiplicative noise in both classical and relativistic regimes. It develops a unified Lyapunov-Hörmander framework to prove ergodicity and quantifies convergence rates, obtaining exponential mixing for the classical model and algebraic mixing for the relativistic model. It also derives robust limit results: a small-mass overdamped limit toward Gibbs–Boltzmann equilibrium and a Newtonian limit for the relativistic dynamics, including convergence statements for multi- and single-particle cases. The findings advance the mathematical understanding of dissipative particle systems with singular forces by handling state-dependent friction and multiplicative diffusion, with potential implications for coarse-grained molecular dynamics and relativistic stochastic modeling.

Abstract

In this paper, we study systems of $N$ interacting particles described by the classical and relativistic Langevin dynamics with singular forces and multiplicative noises. For the classical model, we prove the ergodicity, obtaining an exponential rate of convergence to the invariant Boltzmann-Gibbs distribution, and the small-mass limit, recovering the $N$-particle interacting overdamped Langevin dynamics. For the relativistic model, we establish the ergodicity, obtaining an algebraic mixing rate of any order to the Maxwell-Jüttner distribution, and the Newtonian limit (that is when the speed of light tends to infinity), approximating a system of underdamped Langevin dynamics. The proofs rely on the construction of Lyapunov functions that account for irregular potentials and multiplicative noises.

Ergodicity and asymptotic limits for Langevin interacting systems with singular forces and multiplicative noises

TL;DR

This work analyzes N-particle Langevin systems with singular interactions and multiplicative noise in both classical and relativistic regimes. It develops a unified Lyapunov-Hörmander framework to prove ergodicity and quantifies convergence rates, obtaining exponential mixing for the classical model and algebraic mixing for the relativistic model. It also derives robust limit results: a small-mass overdamped limit toward Gibbs–Boltzmann equilibrium and a Newtonian limit for the relativistic dynamics, including convergence statements for multi- and single-particle cases. The findings advance the mathematical understanding of dissipative particle systems with singular forces by handling state-dependent friction and multiplicative diffusion, with potential implications for coarse-grained molecular dynamics and relativistic stochastic modeling.

Abstract

In this paper, we study systems of interacting particles described by the classical and relativistic Langevin dynamics with singular forces and multiplicative noises. For the classical model, we prove the ergodicity, obtaining an exponential rate of convergence to the invariant Boltzmann-Gibbs distribution, and the small-mass limit, recovering the -particle interacting overdamped Langevin dynamics. For the relativistic model, we establish the ergodicity, obtaining an algebraic mixing rate of any order to the Maxwell-Jüttner distribution, and the Newtonian limit (that is when the speed of light tends to infinity), approximating a system of underdamped Langevin dynamics. The proofs rely on the construction of Lyapunov functions that account for irregular potentials and multiplicative noises.
Paper Structure (20 sections, 23 theorems, 373 equations)

This paper contains 20 sections, 23 theorems, 373 equations.

Table of Contents

  1. Introduction
  2. Langevin interacting systems with singular forces and multiplicative noise
  3. Main results
  4. Methodology of the proofs
  5. Novelties
  6. Relevant literature
  7. Organization of the paper
  8. Preliminary
  9. Assumptions
  10. Preliminary results
  11. Langevin dynamics with state-dependent friction
  12. Ergodicity
  13. Small-mass limit
  14. Relativistic Langevin Equation with multiplicative noise
  15. Ergodicity
  16. ...and 5 more sections

Key Result

Theorem 1.1

Suppose Assumptions Assumption U, A:G and Assumption D respectively on the external potential $U$, the interaction potential $G$, and the diffusion matrix $D$ hold. Then the Gibbs-Boltzmann distribution $\pi^m_{\textup{GB}}$ given by form:pi^m_GB is the unique invariant measure of the classical Lang for all probability measure $\mu$ such that

Theorems & Definitions (47)

  • Theorem 1.1: Ergodicity of the classical model
  • Theorem 1.2: Small-mass limit of the classical model
  • Theorem 1.3: Ergodicity of the relativistic model
  • Theorem 1.4: Newtonian limit
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.5: Lyapunov function
  • Definition 2.6: Minorization condition
  • Definition 2.7: Hormander's condition
  • ...and 37 more