Langevin theory of fluctuations in the discrete Boltzmann equation
M. Gross, M. E. Cates, F. Varnik, R. Adhikari.
Journal of Statistical Mechanics: Theory and Experiment, 03, 1742-5468, (2011)
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, a 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.
Keyword(s): kinetic theory of gases and liquids; lattice Boltzmann methods; computational ﬂuid dynamics;