Fluctuation symmetry leads to GENERIC equations with non-quadratic dissipation

Abstract

We develop a formalism to discuss the properties of GENERIC systems in terms of corresponding Hamiltonians that appear in the characterization of large-deviation limits. We demonstrate how the GENERIC structure naturally arises from a certain symmetry in the Hamiltonian, which extends earlier work that has connected the large-deviation behaviour of reversible stochastic processes to the gradient-flow structure of their deterministic limit. Natural examples of application include particle systems with inertia.

Gradient and GENERIC Systems in the Space of Fluxes, Applied to Reacting Particle Systems

“…The quadruple satisfying this condition falls within the class of what is recently coined pre-(Generalised) GENERIC (pGGEN) [ 9 , 10 ].…”

“…Although this implies the existence of a GGEN system induced by ℒ , one can not uniquely decide on the basis of ℒ what the “correct” Hamiltonian structure should be. Additional physical or mathematical arguments needed to uniquely fix the Hamiltonian structure are beyond the scope of this paper; for possible constructions of Poisson structures L and energies , we refer the reader to (Section 4, [ 10 ]).…”

“…([ 10 ]) . A Generalised pre-GENERIC system (pGGEN) is a quadruple , where is a differentiable manifold, Ψ is a dissipation potential, , and is a vector field such that: …”

“…One is thus lead to study generalised GENERIC structures (GGEN) [ 8 ]. In a recent work, necessary and sufficient conditions were found under which microscopic fluctuations induce such a GGEN structure [ 9 , 10 ].…”

On the Combined Use of Friction Matrices and Dissipation Potentials in Thermodynamic Modeling

“…(4) cannot make sense for all quantities A with varying physical dimensions. Such dimensional considerations are of no concern in the mathematical literature where dimensionless formulations are assumed from the outset (see, for example, Section 2.5 of [19] or Definition 2.5 of [20]). We should try to find out whether there are any deeper reasons why we need to consider dimensional issues in the context of dissipation potentials, but not in the context of friction matrices.…”