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A large deviation principle for the multispecies stirring process

Francesco Casini, Frank Redig, Hidde van Wiechen

Abstract

In this paper we consider the multispecies stirring process on the discrete torus. We prove a large deviation principle for the trajectory of the vector of densities of the different species. The technique of proof consists in extending the method of the foundational paper [1] based on the superexponential estimate to the multispecies setting. This requires a careful choice of the corresponding weakly asymmetric dynamics, which is parametrized by fields depending on the various species. We also prove the hydrodynamic limit of this weakly asymmetric dynamics, which is similar to but different from the ABC model in [2]. Using the appropriate asymmetric dynamics, we also obtain that the mobility matrix relating the drift currents to the fields coincides with the covariance matrix of the reversible multinomial distribution, which then further leads to the Einstein relation

A large deviation principle for the multispecies stirring process

Abstract

In this paper we consider the multispecies stirring process on the discrete torus. We prove a large deviation principle for the trajectory of the vector of densities of the different species. The technique of proof consists in extending the method of the foundational paper [1] based on the superexponential estimate to the multispecies setting. This requires a careful choice of the corresponding weakly asymmetric dynamics, which is parametrized by fields depending on the various species. We also prove the hydrodynamic limit of this weakly asymmetric dynamics, which is similar to but different from the ABC model in [2]. Using the appropriate asymmetric dynamics, we also obtain that the mobility matrix relating the drift currents to the fields coincides with the covariance matrix of the reversible multinomial distribution, which then further leads to the Einstein relation

Paper Structure

This paper contains 24 sections, 19 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Organization of the paper and main results
  4. Acknowledgments.
  5. The multispecies stirring process
  6. The symmetric case
  7. The weakly asymmetric stirring process
  8. Superexponential estimate
  9. Hydrodynamic limit of the weakly asymmetric model
  10. Potentials for large deviations
  11. Currents and the Einstein relation
  12. Macroscopic currents.
  13. Einstein's relation.
  14. Large deviations
  15. Radon-Nikodym derivative and the exponential martingale
  16. ...and 9 more sections

Key Result

Theorem 1

For any $\delta>0$, for all $T>0$ and $\phi \in C(\Omega)$

Theorems & Definitions (44)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Corollary 1
  • Remark 3
  • Theorem 2
  • proof
  • Remark 4
  • Remark 5
  • Remark 6
  • ...and 34 more