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On the absence of time-translation symmetry breaking in some non-reversible interacting particle systems

Jonas Köppl

Abstract

The conditions under which stochastic systems of infinitely many interacting particles can maintain sufficient spatial order to move coherently along a time-periodic orbit, thereby breaking the time-translation invariance of the underlying dynamical equation, have been an elusive issue. Via a free energy technique, we prove that if a non-reversible interacting particle system on $\mathbb{Z}^d$, $d=1,2$, with strictly positive rates admits a product measure as a stationary measure, then it cannot exhibit time-periodic behaviour. This provides a first step towards a general conjecture that time-periodic behaviour cannot occur in one- and two-dimensional systems with short-range interactions and constitutes the first result for non-reversible dynamics in dimension two.

On the absence of time-translation symmetry breaking in some non-reversible interacting particle systems

Abstract

The conditions under which stochastic systems of infinitely many interacting particles can maintain sufficient spatial order to move coherently along a time-periodic orbit, thereby breaking the time-translation invariance of the underlying dynamical equation, have been an elusive issue. Via a free energy technique, we prove that if a non-reversible interacting particle system on , , with strictly positive rates admits a product measure as a stationary measure, then it cannot exhibit time-periodic behaviour. This provides a first step towards a general conjecture that time-periodic behaviour cannot occur in one- and two-dimensional systems with short-range interactions and constitutes the first result for non-reversible dynamics in dimension two.
Paper Structure (18 sections, 13 theorems, 65 equations)

This paper contains 18 sections, 13 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Interacting particle systems
  4. Relative entropy loss
  5. Time-stationary measures, orbits, and the attractor
  6. Main results
  7. Outlook
  8. Towards a no-go theorem in $d=1,2$
  9. Towards a minimal example
  10. Proof strategy
  11. Finite-volume relative entropy loss and stationary orbits
  12. The positive-mass property
  13. Proof of the time-averaged relative entropy loss principle
  14. Rewriting the finite-volume relative entropy loss
  15. The time-averaged zero-loss estimate
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $d \in \{1,2\}$ and assume that $\mathscr{L}$ is the generator of an interacting particle system that satisfies assumptions $\mathbf{(R1)-(R4)}$ and that it admits a product measure $\mu$ as time-stationary measure.Then we have that

Theorems & Definitions (22)

  • Theorem 2.1
  • Corollary 2.2
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Proposition 4.1: Time-averaged entropy loss principle
  • Proposition 4.2: Positive-mass property
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • ...and 12 more