GENERIC formalism of a Vlasov-Fokker-Planck equation and connection to large-deviation principles (Q2864560)

A GENERIC equation (general equation for non-equilibrium reversible-irrevesible coupling, see [\textit{H. C. Öttinger}, Beyond equilibrium thermodynamics. Hoboken, NJ: John Wiley \& Sons (2005)]) for an unknown \(\mathsf{z}\) in a state space \(\mathsf{Z}\) is a mixture of both reversible and dissipative dynamics: \(\partial_t\mathsf{z}=\mathsf{L}\mathsf{d E} + \mathsf{M}\mathsf{d S}\). Here, \(\mathsf{E,S}:\mathsf{Z}\to \mathbb{R}\) are interpreted as energy and entropy functionals, \(\mathsf{d E},\mathsf{d S}\) are ``appropriate derivatives'' of \(\mathsf{E}\) and \(\mathsf{S}\), \(\mathsf{L}=\mathsf{L}(\mathsf{z})\) is (for each \(\mathsf{z}\)) an antisymmetric operator satisfying the Jacobi identity, \(\mathsf{M}=\mathsf{M}(\mathsf{z})\) is symmetric and positive semidefinite. The building blocks \(\mathsf{L,M,E,S}\) are required to fulfill the degeneracy conditions: \(\mathsf{L}\mathsf{d S}=0\) and \(\mathsf{M}\mathsf{d E}=0\). As a consequence, energy is conserved and the entropy is non-decreasing. A GENERIC system is fully characterized by \(\mathsf{Z,E,S,L,M}\). The goal of this paper is twofold: to connect the GENERIC structure with large deviations of stochastic processes, and to construct a useful variational principle for an abstract GENERIC equation. As a guiding example the Vlasov-Fokker-Plank (VFP) equation is considered. The authors construct a large-deviation principle for the SDE associated with the VFP equation. After that they construct a GENERIC structure for the VFP equation and reformulate the large-deviation rate function in this context. Finally, a variational formulation for the VFP equation (and for any GENERIC system) is deduced from the large-deviation result. A generalized VFP equation is considered as well.