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Stationary fluctuations of run-and-tumble particles

Frank Redig, Hidde van Wiechen

Abstract

We study the stationary fluctuations of independent run-and-tumble particles. We prove that the joint densities of particles with given internal state converges to an infinite dimensional Ornstein-Uhlenbeck process. We also consider an interacting case, where the particles are subjected to exclusion. We then study the fluctuations of the total density, which is a non-Markovian Gaussian process, and obtain its covariance in closed form. By considering small noise limits of this non-Markovian Gaussian process, we obtain in a concrete example a large deviation rate function containing memory terms.

Stationary fluctuations of run-and-tumble particles

Abstract

We study the stationary fluctuations of independent run-and-tumble particles. We prove that the joint densities of particles with given internal state converges to an infinite dimensional Ornstein-Uhlenbeck process. We also consider an interacting case, where the particles are subjected to exclusion. We then study the fluctuations of the total density, which is a non-Markovian Gaussian process, and obtain its covariance in closed form. By considering small noise limits of this non-Markovian Gaussian process, we obtain in a concrete example a large deviation rate function containing memory terms.
Paper Structure (24 sections, 15 theorems, 153 equations)

This paper contains 24 sections, 15 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Basic notations and definitions
  3. Scaling limit of the single particle dynamics
  4. Basic properties of independent run-and-tumble particles
  5. Stationary ergodic product measures
  6. Duality
  7. Hydrodynamic limit
  8. Fluctuation fields
  9. Stationary fluctuations
  10. Stationary covariance of the fluctuation fields via duality
  11. Interacting case: the multi-layer SEP
  12. Scaling limits of the total density
  13. Hydrodynamic equation for the total density
  14. Fluctuations of the total density
  15. Covariance of the total density
  16. ...and 9 more sections

Key Result

Theorem 2.1

For every $t\geq0$, $\varepsilon>0$ and $\phi \in C_{c,S}^\infty$, we have that where $\rho_t(x,\sigma)$ solves the PDE $\dot{\rho}_t = A^*\rho_t$ with initial condition $\rho_0(x,\sigma) = \rho(x,\sigma)$.

Theorems & Definitions (32)

  • Definition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.2
  • Remark 2.1
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Remark 3.1
  • Proposition 5.1
  • ...and 22 more