# Publications

## 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)

Abstract

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;

DOI: doi:10.1088/1742-5468/2011/03/P03030

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