WIAS Preprint No. 3108, (2024)

Deriving a GENERIC system from a Hamiltonian system


Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Peletier, Mark A.
    ORCID: 0000-0001-9663-3694
  • Zimmer, Johannes

2020 Mathematics Subject Classification

  • 37K06 37L05 37D35 80A05 82C35

Keywords

  • Hamiltonian systems, gradient systems, GENERIC systems, coarse-graining, heat bath, temperature, Gaussian measures, multivariate Ornstein--Uhlenbeck process, dilations, compressions, energy, entropy, Poisson operator, Onsager operator, Caldeira--Leggett

DOI

10.20347/WIAS.PREPRINT.3108

Abstract

We reconsider the fundamental problem of coarse-graining infinite-dimensional Hamiltonian dynamics to obtain a macroscopic system which includes dissipative mechanisms. In particular, we study the thermodynamical implications concerning Hamiltonians, energy, and entropy and the induced geometric structures such as Poisson and Onsager brackets (symplectic and dissipative brackets). We start from a general finite-dimensional Hamiltonian system that is coupled linearly to an infinite-dimensional heat bath with linear dynamics. The latter is assumed to admit a compression to a finite-dimensional dissipative semigroup (i.e., the heat bath is a dilation of the semigroup) describing the dissipative evolution of new macroscopic variables. Already in the finite-energy case (zero-temperature heat bath) we obtain the so-called GENERIC structure (General Equations for Non-Equilibrium Reversible Irreversibe Coupling), with conserved energy, nondecreasing entropy, a new Poisson structure, and an Onsager operator describing the dissipation. However, their origin is not obvious at this stage. After extending the system in a natural way to the case of positive temperature, giving a heat bath with infinite energy, the compression property leads to an exact multivariate Ornstein--Uhlenbeck process that drives the rest of the system. Thus, we are able to identify a conserved energy, an entropy, and an Onsager operator (involving the Green-Kubo formalism) which indeed provide a GENERIC structure for the macroscopic system.

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