Equations for Macroscopic LG Behavior

4 Equations for Macroscopic LG Behavior The generalized form of equation 9.71, 

In the first case (i.e. in the absence of all collisions), we know that all particles at position X with velocity cp at time T move to the site at position X + cp at time T-t-1  

Taking the lowest order terms 6 and 5t, we have the basic collisionless form of the discrete Boltzman equation  

Including collisions yields the full Boltzman equation  

On a hexagonal lattice, for example, the two-particle distribution function, is 

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