Introduction The Scope of Transport Phenomena

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  

Nonequilibrium basis functions Corresponding conservation equation 

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The Liouville equation, Eq. (2.6), describes the behavior of the collection of phase points as they move through a multidimensional space, or phase space, representing the position and momentum coordinates of all molecules in the system. The phase points tend to be concentrated in regions of phase space where it is most likely to find the N molecules with a certain momentum and position. Thus, the density function pN can be interpreted (aside from a normalization constant) as a probability density function, i.e., pjvdr- dp is proportional to the probability of finding a phase point in a multidimensional region between (r, p ) and (r - - d r, + d p ) at any time t. 

Just as one defines the mean, variance, and other moments of probability density functions, we can also examine these quantities with respect to the density function More specifically, as shown originally by Irving and Kirkwood, this averaging can be performed directly with the Liouville equation leading to the so-called transport equations. Since the Liouville equation is a conservation equation, the transport equations also represent conservation equations for the various moments of the density function p The moments will be defined more specifically in this chapter. 

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