Deriving a GENERIC system from a Hamiltonian system
arxiv(2024)
摘要
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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