Thermodynamics, nonequilibrium

Flux equations derived from irreversible thermodynamics give a real description of transport through membranes. In this description the membrane is considered as a black box and no information is obtained or is required about the structure of the membrane. Thus, no physico-chemical view is obtained how the molecules or particles fiermeate through the membrane. Because of the limitations of this approach with respect to the 

The flows do not only refer to the transport of mass but also to the transfer of heat and of electric current. The fluxes are expressed relative to the fixed membrane as reference frame with constant boundaries. 

Close to equilibrium it can be assumed that each force is linearly related to the fluxes (eq. V - 12) or each flux is linearly related to the forces (eq. V - 13). This latterapproach is often used in membrane transport. 

Considering eq. V - 13 then for single component transport a very simple relation is obtained with only one proportionality coefficient. If the driving force is the gradient in the chemical potential then 

In the case of the transport of two components 1 and 2 there are two flux equadons with four coefficients (L[, L22 J-12 J-2i)- (J e case of transport of three components 

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Katchalsky A and Curran P F 1965 Nonequilibrium Thermodynamics in Biophysics (Cambridge, MA ITarvard University Press)... 

A. Katachalsky and P. F. Curran, Nonequilibrium Thermodynamics in Biophysics, Harvard University Press, Cambridge, Mass., 1967. 

H. J. Kme2er, Nonequilibrium Thermodynamics and its Statistical Foundations, Clarendon Press, Oxford, 1981. 

In this review we put less emphasis on the physics and chemistry of surface processes, for which we refer the reader to recent reviews of adsorption-desorption kinetics which are contained in two books [2,3] with chapters by the present authors where further references to earher work can be found. These articles also discuss relevant experimental techniques employed in the study of surface kinetics and appropriate methods of data analysis. Here we give details of how to set up models under basically two different kinetic conditions, namely (/) when the adsorbate remains in quasi-equihbrium during the relevant processes, in which case nonequilibrium thermodynamics provides the needed framework, and (n) when surface nonequilibrium effects become important and nonequilibrium statistical mechanics becomes the appropriate vehicle. For both approaches we will restrict ourselves to systems for which appropriate lattice gas models can be set up. Further associated theoretical reviews are by Lombardo and Bell [4] with emphasis on Monte Carlo simulations, by Brivio and Grimley [5] on dynamics, and by Persson [6] on the lattice gas model. 

The current density j is, according to linear nonequilibrium thermodynamics, proportional to the gradient of a chemical potential difference... 

A generalized model of transport allowing for component interactions is provided by nonequilibrium thermodynamics where the flux of component i through the membrane /, [gmol/(cm -s)] is written as a first-order perturbation of the chemical potential dp,/dx [cal/(gmohcm)] ... 

He is the author of two other books. Nonequilibrium Thermodynamics (1962) and Vector Analysis in Chemistry (1974), and has published research articles on the theory of optical rotation, statistical mechanical theory of transport processes, nonequilibrium thermodynamics, molecular quantum mechanics, theory of liquids, intermolecular forces, and surface phenomena. 

This begs the question of whether a comparable law exists for nonequilibrium systems. This chapter presents a theory for nonequilibrium thermodynamics and statistical mechanics based on such a law written in a form analogous to the equilibrium version ... 

This nonequilibrium Second Law provides a basis for a theory for nonequilibrium thermodynamics. The physical identification of the second entropy in terms of molecular configurations allows the development of the nonequilibrium probability distribution, which in turn is the centerpiece for nonequilibrium statistical mechanics. The two theories span the very large and the very small. The aim of this chapter is to present a coherent and self-contained account of these theories, which have been developed by the author and presented in a series of papers [1-7]. The theory up to the fifth paper has been reviewed previously [8], and the present chapter consolidates some of this material and adds the more recent developments. 

The present theory can be placed in some sort of perspective by dividing the nonequilibrium field into thermodynamics and statistical mechanics. As will become clearer later, the division between the two is fuzzy, but for the present purposes nonequilibrium thermodynamics will be considered that phenomenological theory that takes the existence of the transport coefficients and laws as axiomatic. Nonequilibrium statistical mechanics will be taken to be that field that deals with molecular-level (i.e., phase space) quantities such as probabilities and time correlation functions. The probability, fluctuations, and evolution of macrostates belong to the overlap of the two fields. 

On a related point, there have been other variational principles enunciated as a basis for nonequilibrium thermodynamics. Hashitsume [47], Gyarmati [48, 49], and Bochkov and Kuzovlev [50] all assert that in the steady state the rate of first entropy production is an extremum, and all invoke a function identical to that underlying the Onsager-Machlup functional [32]. As mentioned earlier, Prigogine [11] (and workers in the broader sciences) [13-18] variously asserts that the rate of first entropy production is a maximum or a minimum and invokes the same two functions for the optimum rate of first entropy production that were used by Onsager and Machlup [32] (see Section HE). 

The aim of this section is to give the steady-state probability distribution in phase space. This then provides a basis for nonequilibrium statistical mechanics, just as the Boltzmann distribution is the basis for equilibrium statistical mechanics. The connection with the preceding theory for nonequilibrium thermodynamics will also be given. 

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