Liouville equation thermodynamic equilibrium

Section II deals with the general formalism of Prigogine and his co-workers. Starting from the Liouville equation, we derive an exact transport equation for the one-particle distribution function of an arbitrary fluid subject to a weak external field. This equation is valid in the so-called "thermodynamic limit , i.e. when the number of particles N —> oo, the volume of the system 2-> oo, with Nj 2 = C finite. As a by-product, we obtain very easily a formulation for the equilibrium pair distribution function of the fluid as well as a general expression for the conductivity tensor. 

Solve the Liouville equation for the density matrix p(0 at time t, given that the system is initially in thermodynamic equilibrium. 

In an NMR experiment, the energy of the lattice is practically constant (the lattice has a large heat capacity). It is therefore assumed that the lattice is always in a state of thermodynamic equilibrium. Thus, it is possible to use a semi-classical description of its interactions with the spin system. Within this approach, the Liouville-von Neumann equation of motion for a spin system is given by ... 

These results were obtained by using the time-dependent quantum mechanical evolution of a state vector. We have generalized these to non-equilibrium situations [16] with the given initial state in a thermodynamic equilibrium state. This theory employs the density matrix which obeys the von Neumann equation. To incorporate the thermodynamic initial condition along with the von Neumann equation, it is advantageous to go to Liouville (L) space instead of the Hilbert (H) space in which DFT is formulated. This L-space quantum theory was developed by Umezawa over the last 25 years. We have adopted this theory to set up a new action principle which leads to the von Neumann equation. Appropriate variants of the theorems above are deduced in this framework. 

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. 

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

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. 

In the previous chapter, we studied equilibrium expressions for the thermodynamic basis functions V, P,U, and > . For equilibrium systems, these functions are spatially and temporally independent. In nonequilibrium systems, on the other hand, these functions can depend on both space and time. Furthermore, as will be shown in this chapter, their nonequilibrium behavior is described by the so-called transport equations or conservation equations that can be obtained directly from the Liouville equation. Specifically, we have the following relationships that will be established ... 

Chapter 1 introduces basic elements of polymer physics (interactions and force fields for describing polymer systems, conformational statistics of polymer chains, Flory mixing thermodynamics. Rouse, Zimm, and reptation dynamics, glass transition, and crystallization). It provides a brief overview of equilibrium and nonequilibrium statistical mechanics (quantum and classical descriptions of material systems, dynamics, ergodicity, Liouville equation, equilibrium statistical ensembles and connections between them, calculation of pressure and chemical potential, fluctuation... 

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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