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Abstract

We propose a fluctuating discrete Boltzmann equation (FDBE) for both the ideal and nonideal fluids [M. Gross, M.E. Cates, F. Varnik, R. Adhikari, arXiv:1012.3632v1 [physics.compph] ]. After casting the FDBE in a form appropriate to the Onsager-Machlup theory of linear fluctuations, the statistical properties of the noise are determined by invoking a fluctuationdissipation theorem. The thus obtained fluctuating discrete Boltzmann equation provides a sound theoretical framework to construct fluctuating lattice Boltzmann equations. The framework is applied to generic force-based non-ideal fluid models with the result that ideal gas-type thermal noise is sufficient for a proper thermalization of all relevant degrees of freedom. It is shown that the results obtained hold in principle for any model that employs an interaction force  F^int that can be written as the divergence of a tensor, F^int = −∇·P. Moreover, details regarding how this interaction force is incorporated into the model (i.e., via an explicit forcingterm or by redefining the macroscopic velocity) is not relevant concerning the fluctuation dissipation theorem. The utility of the fluctuating discrete Boltzmann route is illustrated further by re-deriving the fluctuating LBEs of the ideal gas [R. Adhikari, K. Stratford, M.E. Cates, and A.J. Wagner, Europhys. Lett. 71, 473 (2005)] and the modified-equilibrium model [M. Gross, M.E. Cates, R. Adhikari, F. Varnik, Phys. Rev. E 82, 056714 (2010)]. Moreover, the thus obtained fluctuating non-ideal LB model is used to study the spreading dynamics of nano-droplets on flat substrates. It is found that the well-known Tanner’s law (which states that the droplet’s base radius scales as R ∼ t^1/10) is replaced by a faster spreading, R ∼ t^1/6. This result is in agreement with previous works using scaling analysis of a stochastic lubrication equation [B. Davidovitch, E. Moro, H.A. Stone, Phys. Rev. Lett. 95, 244505 (2005)]. Further perspectives of the approach are also discussed.