In this paper we discuss the connections between a Vlasov–Fokker–Planck equation and an underlying microscopic particle system, and we interpret those connections in the context of the GENERIC framework (Öttinger 2005 Beyond Equilibrium Thermodynamics (New York: Wiley-Interscience)). This interpretation provides (a) a variational formulation for GENERIC systems, (b) insight into the origin of this variational formulation, and (c) an explanation of the origins of the conditions that GENERIC places on its constitutive elements, notably the so-called degeneracy or non-interaction conditions. This work shows how the general connection between large-deviation principles on one hand and gradient-flow structures on the other hand extends to non-reversible particle systems.
Duong, M. H., Peletier, M. A., & Zimmer, J. (2013). GENERIC formalism of a Vlasov-Fokker-Planck equation and connection to large-deviation principles. Nonlinearity, 26(11), 2951-2971. https://doi.org/10.1088/0951-7715/26/11/2951