Liouville equation steady-state solutions

Now, following the same procedures as in Sec. 4.1, show by separation of variables that a particular steady-state solution to the extended Liouville equation can be written as... 

In the foregoing discussion of the Brownian motion method, the ensemble averages are all constructed from an ensemble of replica systems of the subset of h molecules, the behavior of each replica having been time-smoothed over the interval r. However, in a steady-state transport process dfiN)/dt = 0 at every point in phase space, where f N) is the instantaneous phase density of the N molecules. In principle, at least, it should thus be possible to express the steady-state pressure tensor and the mass and heat currents in terms of ensemble averages constructed without preliminary time-averaging in the replica systems. Thus it is desirable to examine the possibility of obtaining solutions to the reduced Liouville equation, Eq. 8, without preliminary timeaveraging. 

We are inclined to view the steady states (equilibrium solutions) of the Liouville equation as solutions of... 

A word of caution is in order, however, to the development given here. In the preceding sections, we have obtained a particular solution to the steady-state Liouville equation, Eq. (4.1), under the equilibrium restrictions given by Eqs. (4.14) to (4.16). The question arises, however, as to whether or not any system beginning in some arbitrary initial state, if left to itself, would evolve to the time-independent equilibrium solution (existence question). The answer to this question is a definite no as there are many examples of interacting particle systems whose dynamics continually depend on its initial state. Furthermore, there may be... 

If dFJdpa 0, then / = 1 is not a solution to the Liouville equation (9). Consequently the principle of equal a priori probabilities is not valid, in the system phase space. This is not surprising equal volumes in the phase space for the universe do not project onto equal volumes in a subspace. In the past there have been some attempts to describe the non-equilibrium steady state by a procedure similar to that used for equilibrium, that is, by treating the steady state as a state of maximum probability... 

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