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Classical dynamics of infinite particle systems in an operator algebraic framework

T. D. H. van Nuland, C. J. F. van de Ven

TL;DR

The paper develops a rigorous operator-algebraic framework for the classical dynamics of infinite particle systems with harmonic pinning and general, summable interactions. By employing the commutative resolvent algebra $C_ extnormal{R}(oldsymbol{ abla})$, it proves that the algebra is invariant under time evolution for a broad class of Hamiltonians and identifies a nontrivial time-stable subalgebra that implements the dynamics as a C*-dynamical system. The key technical advance is a finite-dimensional Dyson-series approach combined with controllable approximations and a thermodynamic-limit argument that yields a well-defined one-parameter group of automorphisms on the inductive limit algebra. The work highlights the advantages of an algebraic viewpoint for studying equilibrium properties, phase structure, and the thermodynamic limit in classical lattice-like systems, while laying groundwork for potential quantum analogs and extensions to more abstract geometries or Coulomb interactions.

Abstract

We construct C*-dynamical systems for the dynamics of classical infinite particle systems describing harmonic oscillators interacting with arbitrarily many neighbors on lattices, as well on more general structures. Our approach allows particles with varying masses, varying frequencies, irregularly placed lattice sites and varying interactions subject to a simple summability constraint. A key role is played by the commutative resolvent algebra, which is a C*-algebra of bounded continuous functions on an infinite dimensional vector space, and in a strong sense the classical limit of the Buchholz--Grundling resolvent algebra, which suggests that quantum analogs of our results are likely to exist. For a general class of Hamiltonians, we show that the commutative resolvent algebra is time-stable, and admits a time-stable sub-algebra on which the dynamics is strongly continuous, therefore obtaining a C*-dynamical system.

Classical dynamics of infinite particle systems in an operator algebraic framework

TL;DR

The paper develops a rigorous operator-algebraic framework for the classical dynamics of infinite particle systems with harmonic pinning and general, summable interactions. By employing the commutative resolvent algebra , it proves that the algebra is invariant under time evolution for a broad class of Hamiltonians and identifies a nontrivial time-stable subalgebra that implements the dynamics as a C*-dynamical system. The key technical advance is a finite-dimensional Dyson-series approach combined with controllable approximations and a thermodynamic-limit argument that yields a well-defined one-parameter group of automorphisms on the inductive limit algebra. The work highlights the advantages of an algebraic viewpoint for studying equilibrium properties, phase structure, and the thermodynamic limit in classical lattice-like systems, while laying groundwork for potential quantum analogs and extensions to more abstract geometries or Coulomb interactions.

Abstract

We construct C*-dynamical systems for the dynamics of classical infinite particle systems describing harmonic oscillators interacting with arbitrarily many neighbors on lattices, as well on more general structures. Our approach allows particles with varying masses, varying frequencies, irregularly placed lattice sites and varying interactions subject to a simple summability constraint. A key role is played by the commutative resolvent algebra, which is a C*-algebra of bounded continuous functions on an infinite dimensional vector space, and in a strong sense the classical limit of the Buchholz--Grundling resolvent algebra, which suggests that quantum analogs of our results are likely to exist. For a general class of Hamiltonians, we show that the commutative resolvent algebra is time-stable, and admits a time-stable sub-algebra on which the dynamics is strongly continuous, therefore obtaining a C*-dynamical system.
Paper Structure (20 sections, 15 theorems, 162 equations)

This paper contains 20 sections, 15 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The index set and its thermodynamic limit
  4. The phase space
  5. The commutative resolvent algebra
  6. A dense Poisson algebra
  7. Local Hamiltonians
  8. Definition of the dynamics on $C_\mathcal{R}(\Omega_\Lambda)$
  9. Finite dimensional dynamical systems
  10. Noninteracting time evolution
  11. Time evolution for compactly supported potentials
  12. Proof of 1.
  13. Proof of 2.
  14. Time evolution for general potentials
  15. Infinite dimensional dynamical systems
  16. ...and 5 more sections

Key Result

Lemma 4

The time evolution of the simple harmonic oscillator $H_{\Lambda}^{0}$ of the finite particle system $\Lambda\Subset\Gamma$ preserves the commutative resolvent algebra, i.e., for each $t\in{\mathbb R}$. Moreover, $(\Phi_{H_\Lambda^{0}}^t)^*(\mathcal{S}_\mathcal{R}(\Omega_\Lambda))=\mathcal{S}_\mathcal{R}(\Omega_\Lambda).$

Theorems & Definitions (35)

  • Remark 1
  • Remark 2
  • Definition 3: Time evolution in algebraic sense
  • Lemma 4
  • proof
  • Proposition 5
  • Definition 6
  • Lemma 7
  • proof
  • Lemma 8: Gronwall's inequality
  • ...and 25 more