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Asymptotic analysis for the Generalized Relativistic Langevin Equation

Ethan Baker, Manh Hong Duong, Hung Dang Nguyen

Abstract

In this paper, we study a non-Markovian generalized relativistic Langevin equation (GRLE). We show that when the memory kernel is a sum of exponentials, the GRLE is equivalent to a Markovian system with added variables. We establish the well-posedness and polynomial ergodicity, obtaining an algebraic rate of convergence to the unique Gibbs distribution. From the Markovian GRLE, we recover the relativistic underdamped Langevin dynamics in a small-noise limit, as well as the classical (non-relativistic) generalized Langevin dynamics in the Newtonian limit.

Asymptotic analysis for the Generalized Relativistic Langevin Equation

Abstract

In this paper, we study a non-Markovian generalized relativistic Langevin equation (GRLE). We show that when the memory kernel is a sum of exponentials, the GRLE is equivalent to a Markovian system with added variables. We establish the well-posedness and polynomial ergodicity, obtaining an algebraic rate of convergence to the unique Gibbs distribution. From the Markovian GRLE, we recover the relativistic underdamped Langevin dynamics in a small-noise limit, as well as the classical (non-relativistic) generalized Langevin dynamics in the Newtonian limit.
Paper Structure (21 sections, 16 theorems, 194 equations, 1 figure)

This paper contains 21 sections, 16 theorems, 194 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The generalized relativistic Langevin equation
  3. Summary of the main results
  4. Markovian formulation
  5. Polynomial ergodicity
  6. White-noise limit
  7. Newtonian limit
  8. Future works
  9. Organization of the paper
  10. Well-posdeness of the Markovian GRLE
  11. Ergodicity
  12. Invariant measure
  13. Construction of Lyapunov functions
  14. Hörmander's and Solvability Conditions
  15. Proof of Theorem \ref{['polynomialerogicity']}
  16. ...and 6 more sections

Key Result

Proposition 1.2

Let $\Lambda\in\mathbb{R}^{d\times k}$, and ${\bf{A}}\in\mathbb{R}^{k\times k}$ be symmetric positive definite. Then, given the memory kernel can be written as, $(q,p)$ given by GRLE is equivalent to the process $(q,p)$ solving, for some $k$-dimensional Brownian motion $\widetilde{W}$, where $z:[0,\infty)\to \mathbb{R}^{{k}}$ with the initial distribution given by, and the matrix $\Sigma\in \ma

Figures (1)

  • Figure 1: Diagram of the relationships between Langevin equations.

Theorems & Definitions (33)

  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.6: The Relativistic White Noise Limit
  • Remark 1.7
  • Theorem 1.8: The Newtonian Limit for the GRLE
  • proof : Proof of Proposition \ref{['prop: Markovian formulation']}
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['existenceuniquenessmarkovtheorem']}
  • ...and 23 more