Thermodynamic and Stochastic Theory of Transport Processes

In this chapter we present the thermodynamic and stochastic theory of simple transport processes, linear and non-linear diffusion, thermal conduction and viscous flow. We refer only briefly to more advanced work on hydrodjmamic equations and some interesting experiments, and on hght scattering from a fluid in a temperature gradient. A suitable introduction to the chapter may well be a review of the some of the main results obtained in earlier chapters on chemical reactions, and we begin with that review. 

For a model chemical reaction such as the Schlogl model 

The function provides necessary and sufficient conditions of global stability it has an extremum at each stationary state 

The function is the driving force (frequently so-called, in fact a potential) toward stable stationary states. is lower bounded and is a Liapunov function with the derivative of with respect to time 

is a component of the total dissipation and is zero at stationary states. 

---

So far we have considered only homogeneous reaction systems in which concentrations are functions of time only. Now we turn to inhomogeneous reaction systems in which concentrations are functions of time and space. There may be concentration gradients in space and therefore diffusion will occur. We shall formulate a thermodynamic and stochastic theory for such systems [1] first we analyze one-variable systems and then two- and multi-variable systems, with two or more stable stationary states, and then apply the theory to study relative stabihty of such multiple stable stationary states. The thermodynamic and stochastic theory of diffusion and other transport processes is given in Chap. 8. 

In Chap. 2 9 we presented a thermodynamic and stochastic theory of chemical reactions and transport processes in non-equilibrium stationary and transient states approaching non-equilibrium stationary states. We established a state function systems approaching equilibrimn reduces to AG. Since Gibbs free energy changes can be determined by macroscopic electrochemical measurements, we seek a parallel development for the determination of by macroscopic electrochemical and other measurements. 

The roots of QSM theory lie in Mori s statistical mechanical theory of transport processes and Kubo s theory of thermal disturbances. The version of QSM theory given here with its refinements and modem embellishments was used by the author to develop an irreversible thermodynamic theory for photophysical phenomena and a quantum stochastic Fokker-Planck theory for adiabatic and nona-diabatic processes in condensed phases. 

Somewhat closer to the designation of a microscopic model are those diffusion theories which model the transport processes by stochastic rate equations. In the most simple of these models an unique transition rate of penetrant molecules between smaller cells of the same energy is determined as function of gross thermodynamic properties and molecular structure characteristics of the penetrant polymer system. Unfortunately, until now the diffusion models developed on this basis also require a number of adjustable parameters without precise physical meaning. Moreover, the problem of these later models is that in order to predict the absolute value of the diffusion coefficient at least a most probable average length of the elementary diffusion jump must be known. But in the framework of this type of microscopic model, it is not possible to determine this parameter from first principles . 

---