Towards a lattice-Fokker-Planck-Boltzmann model of thermal fluctuations in non-ideal fluids

Microscopic thermal fluctuations are known to affect the macroscopic and spatio-temporal evolution of a host of physical phenomena central to the study of biological systems, turbulence, and reactive mixtures, among others. In phase-changing fluids metastability and nucleation rates of embryos are known to be non-trivially affected by thermal noise stemming from molecules random velocity fluctuations, which ultimately determine the long-term growth, morphology, and decay of macroscopic bubbles in cavitation and boiling. We herein present the mathematical groundwork for a lattice-based solution of the combined Fokker-Planck and Boltzmann equations that by proxy solve the stochastic Navier-Stokes-Fourier equations and a non-ideal, cubic van der Waals equation of state. We present the derivation of the kinetic lattice-Fokker-Planck-Boltzmann equations facilitated by Gauss-Hermite quadrature, and show by multi-scale asymptotic analysis that the non-equilibrium dynamics in velocity space inherent to the Fokker-Planck equation manifest as stresses. The resulting coarse-grained lattice-Fokker-Planck-Boltzmann method (LFPBM) is attractive as its dynamics are hypothesized to continually evolve thermal fluctuations introduced into the thermo-hydrodynamic variables by initial conditions in a manner that obeys the fundamental fluctuation-dissipation balances. Simulations will be showcased in future publications.