  Molecular 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 stochastic Fokker-Planck and Boltzmann equations that by proxy solve
the compressible fluctuating Navier-Stokes-Fourier equations. We present the
derivation of the kinetic lattice-Fokker-Planck-Boltzmann equations facilitated
by means of 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 fluctuating stresses analogous to the
conventional viscous stress tensor. The key implication of this analysis is a
direct correlation between the thermal white noise of the Fokker-Planck
equation and the effective viscosities $\mu, \zeta$ that in concert with a
cubic equation of state enable simulations of phase-changing fluids perturbed
by stochastic thermal fluctuations.
