The cluster expansion and super-position approximation

To solve eqn. (294) for the doublet density, the hierarchy of the equation must be broken in a manner analogous to the super-position approximation of Kirkwood or that of Felderhof and Deutch [25], which was presented in Chap. 9, Sect. 5. Furthermore, it is not unreasonable to assume that the system is quite near to thermal equilibrium. Were the system at thermal equilibrium, then collisions would not change the velocity distribution of the particles and the equilibrium distribution would be of the usual Maxwellian form, 0 (v,), etc. These are the solutions of the psuedo-Liouville equation 

X can be regarded as the local non-equilibrium distribution of the species a at a position rj. As the system approaches equilibrium, x becomes -small. In a like manner, the doublet density fjf3 can be decomposed into an equilibrium term, two contributions from the local non-equilibrium departures of the particle a and (3 separately and then their mutual effect 

The doublet correlation function x 3 is not xTXi, else fjf would be equal to ff and the equations would reduce to the Boltzmann equation. 

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