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