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Microscopic thermal 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 Fokker-Planck and Boltzmann equations that by proxy solve thestochastic Navier-Stokes-Fourier equations and a non-ideal, cubic van der Waalsequation of state. We present the derivation of the kineticlattice-Fokker-Planck-Boltzmann equations facilitated by Gauss-Hermitequadrature, and show by multi-scale asymptotic analysis that thenon-equilibrium dynamics in velocity space inherent to the Fokker-Planckequation manifest as stresses. The resulting coarse-grainedlattice-Fokker-Planck-Boltzmann method (LFPBM) is attractive as its dynamicsare hypothesized to continually evolve thermal fluctuations introduced into thethermo-hydrodynamic variables by initial conditions in a manner that obeys thefundamental fluctuation-dissipation balances. Simulations will be showcased infuture publications.

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