Langevin theory of fluctuations in the discrete Boltzmann equation

The discrete Boltzmann equation for both the ideal and a non-ideal fluid is extended by adding Langevin noise terms in order to incorporate the effects of thermal fluctuations. After casting the fluctuating discrete Boltzmann equation in a form appropriate to the Onsager-Machlup theory of linear fluctuations, the statistical properties of the noise are determined by invoking a fluctuation-dissipation theorem at the kinetic level. By integrating the fluctuating discrete Boltzmann equation, the fluctuating lattice Boltzmann equation is obtained, which provides an efficient way to solve the equations of fluctuating hydrodynamics for ideal and non-ideal fluids. Application of the framework to a generic force-based non-ideal fluid model leads to ideal gas-type thermal noise. Simulation results indicate proper thermalization of all degrees of freedom.