Liouville equation equilibrium solution

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

One such systematic generalization was obtained by Cohen,8 whose method is now given the point of departure was the expansion in clusters of the non-equilibrium distribution functions. This procedure is formally analogous to the series expansion in the activity where the integrals of the Ursell cluster functions at equilibrium appear in the coefficients. Cohen then obtained two expressions in which the distribution functions of one and two particles are given in terms of the solution of the Liouville equation for one particle. The elimination of this quantity between these two expressions is a problem which presents a very full formal analogy with the elimination (at equilibrium) of the activity between the Mayer equation for the concentration and the series... 

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... 

J. Ma, D. Hsu, J.E. Straub, Approximate solution of the classical Liouville equation using Gaussian phase packet dynamics application to enhanced equilibrium averaging and global optimization, J. Chem. Phys. 99 (1993), 4024. 

We wish to examine equilibrium solutions, f r) or df/dt = 0. Clearly, any function consisting of conserved quantities of the dynamics, /(C i...C m) will satisfy the Liouville Equation where dCk/dt = 0. We wish to consider a distribution function that will allow us to visit all points in phase space with equal a priori probability subject to the constraints embodied by the conserved quantity, H. Clearly,... 

There are a number of different formulations of the time correlation function method, all of which lead to the same results for the linearized hydrodynamic equations. One way is to generalize the Chapman-Enskog normal solution method so as to apply it to the Liouville equations, and obtain the N-particle distribution function for a system near a local equilibrium state. " Expressions for the heat current and pressure tensor for a general fluid system can be obtained, which have the form of the macroscopic linear laws, with explicit expressions for the various transport coefficients. These expressions for the transport coefficients have the form of time integrals of equilibrium correlation functions of microscopic currents, viz., a transport coefficient t is given by... 

As will be shown in subsequent chapters, the solution to the Liouville equation for the N -particle density function p r is the basis for determination of the equilibrium and nonequilibrium properties of matter. However, because of the large number of dimensions (6N), solutions to the Liouville equation represent formidable problems Fortunately, as will be shown in Chaps. 4 and 5, the equilibrium and nonequilibrium properties of matter can usually be expressed in terms of lower-ordered or reduced density functions. For example, the thermodynamic and transport properties of dilute gases can be expressed in terms of the two-molecule density function /02(ri,r2, pi,p2,f). In Chap. 3 we will examine the particular forms of the reduced Liouville equation. 

It suffices to say that extensions of the reduced Liouville equation to higher orders, necessary for dense gases and liquids, become extremely cumbersome. Approximate solutions to the reduced Liouville equation for nonequilibrium dense gas and liquid systems will be considered in more detail in Chap. 6. In the next chapter, however, we will show that exact analytical solutions to the Liouville equation, and its reduced forms, are indeed possible for systems at equilibrium. 

Equilibrium Solution to the Liouville Equation and the Thermodynamic Properties of Matter... 

Summarizing, from Eqs. (4.23) and (4.28) through (4.30), the equilibrium solution to the Liouville equation in terms of the configurational integral is... 

Under the pairwise additivity assumption, the configurational part of the equilibrium solution to the Liouville equation, Eq. (4.24), can be easily reduced to the following integral equation (Prob. 4.12)... 

This completes our description of the thermodynamic basis functions in terms of the configurational and momenta density functions obtained directly from the equilibrium solution to the Liouville equation. As will be shown in the next chapter, the nonequilibrium counterparts (local in space and time) of the thermodynamic basis functions can also be obtained directly from the Liouville equation, thus, providing a unified molecular view of equilibrium thermodynamics and chemical transport phenomena. Before moving on, however, we conclude this chapter by noting some important aspects of the equilibrium solution to the Liouville equation. 

Using the same type of arguments, it is possible to extend all thermodynamic functions in this manner. Of course, we have the added complexities of obtaining n ivap) in the multicomponent swarm of molecules, which is no easy task. A more extensive discussion can be found in the book by McQuarrie listed at the end of this chapter. (In particular, see pages 292-295.) Multiphase S3rstems are even more complex, since more than one single equilibrium solution to the Liouville equation must exist. For more details, see the Further Reading section at the end of this chapter. 

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... 

It is, of course also, possible to obtain a description of nonequilibrium processes by examining the behavior of an isolated system subject to an initial condition imposed at a finite time. This is the procedure adopted by Mori, who chooses local equilibrium as the initial condition. The resulting solution to the Liouville equation can be put into the form (51), with D given by... 

One aspect of irreversibility, namely an as5mimetry between past and future, is introduced by a choice of a solution to the Liouville equation, that is, by our choice of the initial condition (47). There are of course other solutions for example, one can impose the final condition that the system be in equilibrium at = - - 00 and obtain thereby an "advanced solution for which the time integration of Eq. (54) is altered according to... 

A second aspect of irreversibility concerns the approach to equilibrium. Since this question arises only for an isolated system, we will suppose the external forces to be turned off at some time, say < = 0, the system then being isolated for later times. Now the Navier-Stokes equations imply an approach to equilibrium, with monotonically increasing entropy. However, for times t > 0 the ensemble (51) is an exact solution to the Liouville equation for an isolated system, and as is well known the quantity S of Eq. (85) must then be time-independent and cannot approach its equilibrium value. The resolution of this paradox lies in the approximate nature of the Navier-Stokes equations, which hold only for slowly varying processes. The exact transport relations, taking into account processes of arbitrarily rapid variation, are evidently such as to maintain S constant. One may say that the low frequency contributions to S increase at the expense of the high frequency contributions, and measurements of a sufficiently coarse nature will show an apparent approach to equilibrium. Thus the approach to equilibrium is obtained as a natural consequence of an approximation method suited to slowly varying processes. 

In Sections I.C and YD it was shown that the basic results from equilibrium and nonequilibrium thermodynamics can be established from statistical mechanics by starting from the maximum entropy principle. The success of this approach to the formulation of equilibrium and nonequilibrium thermodynamics suggests that the maximum entropy principle can also be used to formulate a general theory of nonequihbrium processes that automatically includes the thermodynamic description of nonequilibrium systems. In this section, we formulate a theory possessing this character by making use of a time-dependent projection operator P(t) that projects the thermodynamic description p,) of a system out of the global description pt) given by the solution of the Liouville equation. We shall refer to this theory as the maximitm entropy approach to nonequilibrium processes. 

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