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Dissipative dynamics for infinite lattice systems

Shreya Mehta, Boguslaw Zegarlinski

Abstract

We study dissipative dynamics constructed by means of non-commutative Dirichlet forms for various lattice systems with multiparticle interactions associated to CCR algebras. We give a number of explicit examples of such models. Using an idea of quasi-invariance of a state, we show how one can construct unitary representations of various groups. Moreover in models with locally conserved quantities associated to an infinite lattice we show that there is no spectral gap and the corresponding dissipative dynamics decay to equilibrium polynomially in time.

Dissipative dynamics for infinite lattice systems

Abstract

We study dissipative dynamics constructed by means of non-commutative Dirichlet forms for various lattice systems with multiparticle interactions associated to CCR algebras. We give a number of explicit examples of such models. Using an idea of quasi-invariance of a state, we show how one can construct unitary representations of various groups. Moreover in models with locally conserved quantities associated to an infinite lattice we show that there is no spectral gap and the corresponding dissipative dynamics decay to equilibrium polynomially in time.
Paper Structure (15 sections, 24 theorems, 291 equations)

This paper contains 15 sections, 24 theorems, 291 equations.

Table of Contents

  1. Introduction
  2. The Infinite Quantum System
  3. Domain Issues
  4. Adjoint Operators
  5. Quasi Invariance and Unitary Group Representations
  6. Modular Dynamics and Finite Speed of Propagation of Information
  7. Convergence of $\mathbb{L}_p$ Norms
  8. Noncommutative Dirichlet forms and Markov Generators
  9. Examples of models
  10. Mean Field Models
  11. Non-diagonal Dirichlet Forms
  12. No Spectral Gap Property
  13. Algebra of Invariant Derivations
  14. Polynomial Decay to Equilibrium
  15. Appendix: $\Gamma_1$

Key Result

Proposition 2.1

If $X^\ast X, X X^\ast\in\mathbb{L}_2(\omega)$, then the derivation is well defined on a domain In particular for any $X\in \mathcal{D}\cup \mathcal{A}$ The above condition holds if $X,\alpha_{\pm i/4}(X)\in \mathbb{L}_4(\omega)$ and then we have $D(\delta_X)\supset \{B ,\alpha_{\pm i/4}(B)\in \mathbb{L}_4(\omega)\}$.

Theorems & Definitions (46)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Theorem 3.1
  • Theorem 4.1
  • proof
  • Definition 6.1
  • Proposition 6.1
  • Example 7.1
  • Proposition 7.1
  • ...and 36 more