Nonequilibrium Ensemble Method for Dense Fluids

Abstract

The Boltzmann kinetic theory for dilute gases which we have used to develop the nonequilibrium ensemble method in the previous chapters has taught us a lesson about the kinetic theory of irreversible processes and provided us with a paradigm for the treatment of such processes in dense and complex fluids. The Boltzmann kinetic theory is based on a time reversal symmetry breaking irreversible kinetic equation. It is an evolution equation for singlet distribution functions for a system which consists of statistically uncorrelated subunits made up of a single particle and evolves through binary encounters of the subunits in phase space. The crucial point of the theory is that the kinetic equation is irreversible, in sharp contrast to the Newtonian dynamics or the quantum dynamics on which the underlying particulate evolution of the subunits is based. We consider this feature essential to any kinetic theory aspiring to transcend the level of density to which the Boltzmann kinetic theory is limited. There are two basic premises that have to do with the density limitation of the Boltzmann kinetic theory. They are the statistically uncorrelated subunits of a single particle and binary encounters of subunits. Of these two basic premises the former has a deeper physical significance than the latter from the statistical viewpoint; it suggests that a subunit consisting of a single particle with which the subunit—the particle—in question is interacting via a binary collision can serve as a model for the rest of the system which may double as a heat reservoir. This picture, when the second premise regarding the binary encounters is removed so as to include encounters involving many subunits, can be readily generalized to a situation where the subunits no longer consist of a single particle, but may contain many particles. We have also learned that when a formal theory of irreversible processes is the principal aim the details of the collision term in the Boltzmann kinetic equation are not all that crucial to have at the stage of structuring such a formal theory. All that is necessary for the purpose is that the collision term satisfies a set of conditions which guarantee the conservation laws of mass, momentum, and energy and the H theorem for the Boltzmann entropy. In the case of dilute gases, with a kinetic equation satisfying such conditions and with the help of the phenomenological theory of irreversible processes constructed on the basis of the thermodynamic laws, it was possible to develop a nonequilibrium ensemble method in parallel with the Gibbs equilibrium ensemble method [1–4]. The details of the collision term in the kinetic equation become important when the transport processes in the nonequilibrium system are studied and compared with experiments. Transport processes are those on which experimental measurements are made and hence the testing ground for the quality of the collision terms representing dynamical mechanisms in the system at the molecular level. In this chapter the paradigm acquired from the aforementioned lessons is shown to remain valid for dense fluids since a formal theory of irreversible processes and an accompanying nonequilibrium ensemble method can be formulated in the same form as for the case of dilute gases. For the sake of simplicity we will confine the discussion to the case of simple dense fluids consisting of monatomic molecules. The generalization of the results presented in this chapter can be straightforwardly made to more complex fluids with internal degrees of freedom, if the techniques described for simple dense fluids can be made use of in treating the internal degrees of freedom. Such generalizations therefore are left to the reader as exercises.