Exact function, Maxwell equation

Classical physics is based on the concept that things happen deterministically From A follows B andfrom B follows C and each of these successive outcomes can be described with some exact functional relationship. Such are Newtons s laws and Maxwell s equations, to mention just two examples. Of course, scientists like Newton, Maxwell, and others were well aware of the seemingly irregular, unpredictable (random) nature of certain phenomena. The movement of the smoke from a chimney and the flow of the water in a river look rather irregular. Yet, these scientists were convinced that if only we were able to break down the description of the system to the smallest possible level (e.g., the molecules), the motion would turn out to be deterministic again. 

Equation (A 1.25) is known as the Maxwell relation. If this relationship is found to hold for M and A in a differential expression of the form of equation (A 1.22), then 6Q — dQ is exact, and some state function exists for which dQ is the total differential. We will consider a more general form of the Maxwell relationship for differentials in three dimensions later. 

Equations 5.1.6 represent a set of n - 1 coupled partial differential equations. Since the Fick matrix [ )] is a strong function of composition it is not always possible to obtain exact solutions to Eqs. 5.1.6 without recourse to numerical techniques. The basis of the method put forward by Toor and by Stewart and Prober is the assumption that c and [D] can be considered constant. (Actually, Toor worked with the generalized Fick s law formulation, whereas Stewart and Prober worked with the Maxwell-Stefan formulation. Toor et al. (1965) subsequently showed the two approaches to be equivalent.) With this assumption Eqs. 5.1.6 reduce to... 

The Boltzmann equation is a nonlinear, integrodifferential equation. As such it is extremely difficult to solve and, in fact, almost no exact solutions are known, apart from the Maxwell-Boltzmann equilibrium solution. Furthermore, only a few existence theorems are known notable are the theorems of Carleman, later extended by Wild and by Morgenstern, proving the existence of a solution of the nonlinear Boltzmann equation for special intermolecular potentials in the case that the system is spatially uniform, i.e., that the distribution function does not depend on r. However, there are a number of circumstances where the system is close enough to equilibrium that the distribution function may be written... 

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