A Program for Fluctuating Kinetic Theory

Abstract

A program for fluctuating kinetic theory is proposed in this paper to study nonlinear L\'evy-type fluctuation corrections to kinetic PDEs in a unified way. The Vlasov--Fokker--Planck (VFP), Landau, and Boltzmann equations can each be decomposed into a kinetic transport term and a generalized gradient structure generated by a nonlinear dual pair of dissipation potentials. Going beyond the linear Onsager structure and the classical fluctuation--dissipation relation, we propose a new nonlinear L\'evy-type fluctuation--dissipation relation that identifies the thermal fluctuations associated with this generalized gradient structure. On a regular tilted class, the canonical action of the generalized gradient structure agrees with the Freidlin--Wentzell-type action predicted by the resulting SPDE. This consistency gives thermodynamic support for the new nonlinear fluctuation--dissipation relation and leads to a master fluctuating kinetic equation. We further establish a sufficient condition on the nonlinear dual dissipation potential under which its centered cumulant admits a L\'evy--Khintchine representation, and show that the resulting L\'evy current can be realized through Gaussian white noise and a compensated Poisson random measure. As an application of this framework, an \(\alpha\)-stable L\'evy-type Dean-Kawasaki equation is derived. Furthermore, by rescaling the fluctuating Boltzmann equation under a distinguished relation between the noise intensity and the hydrodynamic scaling parameter, we prove that its Hamiltonian converges to the Hamiltonian of the Landau--Lifshitz--Navier--Stokes equations. This reveals a scaling relation between kinetic fluctuations and fluctuating hydrodynamics.