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On thermalization in many-body classical Floquet systems

Anton Kapustin

Abstract

It is expected that a generic closed many-body system prepared in a well-behaved initial state and subjected to a periodic drive will eventually thermalize, i.e. approach the state of maximal entropy. This property, while compatible with and even demanded by the physical intuition, is much stronger than ergodicity or mixing and is difficult to justify mathematically. We describe an infinite set of classical many-body Floquet systems of algebraic origin for which thermalization of very general initial states can be proved. For example, we show that a Gibbs state of any sufficiently uniform local differentiable Hamiltonian heats up to infinite temperature at long times. We show that in agreement with the physical intuition, the only obstruction to thermalization is the existence of local observables which are periodic in time.

On thermalization in many-body classical Floquet systems

Abstract

It is expected that a generic closed many-body system prepared in a well-behaved initial state and subjected to a periodic drive will eventually thermalize, i.e. approach the state of maximal entropy. This property, while compatible with and even demanded by the physical intuition, is much stronger than ergodicity or mixing and is difficult to justify mathematically. We describe an infinite set of classical many-body Floquet systems of algebraic origin for which thermalization of very general initial states can be proved. For example, we show that a Gibbs state of any sufficiently uniform local differentiable Hamiltonian heats up to infinite temperature at long times. We show that in agreement with the physical intuition, the only obstruction to thermalization is the existence of local observables which are periodic in time.
Paper Structure (6 sections, 8 theorems, 80 equations)

This paper contains 6 sections, 8 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Algebraic Floquet systems
  3. Results
  4. The frequency blowup property and thermalization
  5. Regularity and the frequency blowup property
  6. Quantum versus classical Floquet spin chains

Key Result

Theorem 3.1

Let ${\mathcal{F}}$ be a symplectomorphism of the form (eq:F). If ${\mathcal{F}}$ has the FB property and $\mu$ is a probability measure satisfying the URL condition, then for $f\in C({\mathsf M})$ we have

Theorems & Definitions (21)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.1
  • Definition 3.4
  • Theorem 3.2
  • Definition 3.5
  • Theorem 3.3
  • proof : Proof of Theorem \ref{['thm:FBimpliestherm']}
  • proof : Proof of Theorem \ref{['thm:locHimpliesFB']}
  • ...and 11 more