Deterministic collision operator

We begin our analysis with a general description of the dynamics of the LB equation, based on a Chapman-Enskog expansion (Sect. 3.2). Then we consider the equilibrium disUibution for the D3Q19 model (Sect. 3.3), followed by deterministic (Sect. 3.4) and stochastic (Sect 3.5) collision operators. Finally, we consider the connection of the fluctuating LB model to statistical mechanics (Sect. 3.6) and the effects of external forces (Sect 3.7). 

In a deterministic model, the collision operator 4, is a nnique function of the distrib-ntion n. Therefore, we can obtain the Chapman-Enskog ordering of 4 via a Taylor expansion with respect to n ... 

Although a deterministic GLG collision operator would be difficult to constract, we can nevertheless determine the distribution in a homogeneous equilibrium state from the conservation laws alone. First we note that there is an entropy associated with each v,-. 

For simplicity, we only consider the deterministic case the analysis of the fluctuating part (A/) remains the same. Application of the new collision operator should leave the mass density unchanged, but increase the momentum density by M. This implies the following conditions on the moments of A- ... 

Molecules move from node to node by deterministic propagation dynamics which models the free streaming real molecules undergo between the effective (strong) collisions (cf below). Thus at each time step molecules move one lattice unit in the directions specified by their velocities. This propagation step will be denoted by the operator P. 

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