Linearized Hydrodynamics and Green-Kubo Relations

The GK relations for SRD differ from the well-known continuous versions due to the discrete-time dynamics, the underlying lattice structure, and the multi-particle interactions. In the following, we briefly outline this approach for determining the transport coefficients. More details can be found in [20,27]. 

The starting point of this theory are microscopic definitions of local hydrodynamic densities A. These slow variables are the local number, momentum, and 

The equilibrium correlation functions for the conserved variables are defined by 5Ap r,t)5Ay r, t )), where (5A) = A - (A), and the brackets denote an average over the equilibrium distribution. In a stationary, translationally invariant system, the correlation functions depend only on the differences r - r and t — t, and the Fourier transform of the matrix of correlation functions is 

The primary reason is that in SRD, the collisional contribution corresponds to a nonlocal (on the scale of the cell size) force which acts only at discrete time intervals. 

Bi has a number of important properties which simplify the calculation of the transport coefficients. In particular, it is shown in [28,51] that stress-stress correlation functions involving one in the GK relations for the transport coefficients are zero, so that, for example, Aap (k) = with 

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