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Molecular fluctuations are known to affect the macroscopic andspatio-temporal evolution of a host of physical phenomena central to the studyof biological systems, turbulence, and reactive mixtures, among others. Inphase-changing fluids metastability and nucleation rates of embryos are knownto be non-trivially affected by thermal noise stemming from molecules randomvelocity fluctuations, which ultimately determine the long-term growth,morphology, and decay of macroscopic bubbles in cavitation and boiling. Weherein present the mathematical groundwork for a lattice-based solution of thecombined stochastic Fokker-Planck and Boltzmann equations that by proxy solvethe compressible fluctuating Navier-Stokes-Fourier equations. We present thederivation of the kinetic lattice-Fokker-Planck-Boltzmann equations facilitatedby means of Gauss-Hermite quadrature, and show by multi-scale asymptoticanalysis that the non-equilibrium dynamics in velocity space inherent to theFokker-Planck equation manifest as fluctuating stresses analogous to theconventional viscous stress tensor. The key implication of this analysis is adirect correlation between the thermal white noise of the Fokker-Planckequation and the effective viscosities $\mu, \zeta$ that in concert with acubic equation of state enable simulations of phase-changing fluids perturbedby stochastic thermal fluctuations.

Towards a lattice-Fokker-Planck-Boltzmann method for simulating thermal noise in compressible phase-changing fluids