Department of Physics, North Dakota State University, Fargo, USA
In this tutorial I will discuss the major features of discrete simulations of fluctuating hydrodynamics. The original lattice gases included fluctuations, but there were some draw-backs to these lattice gas methods which lead researchers to develop the lattice Boltzmann method. By going to a Boltzmann equation fluctuations were averaged out, which removed the need for averaging of lattice gas methods to recover standard hydrodynamics results.
Soon lattice Boltzmann methods were applied to a wide variety of applications. Tony Ladd developed his famous method of simulating colloidal suspensions with lattice Boltzmann. During this development he realized that he needed fluctuations to recover the Brownian motion of colloids. So he introduced a fluctuating stress tensor in the collision process so that he could recover fluctuating hydrodynamics.
However, these fluctuating lattice Boltzmann methods showed some disturbing peculiarities, and it was recognized by Adhikari that one not only needs a fluctuating stress tensor but that there needs to be a fluctuating component for all dissipated moments, not only the hydrodynamic modes but also the non-hydrodynamic modes.
In this tutorial we will follow the historical development of the different moments an (some of) their properties. We will conclude this with some very recent results on new fluctuating lattice Boltzmann methods with local collision operators and density- and (maybe) local temperature- dependent fluctuations.