Nonequilibrium theories

Because the focus is on a single, albeit rather general, theory, only a limited historical review of the nonequilibrium field is given (see Section IA). That is not to say that other work is not mentioned in context in other parts of this chapter. An effort has been made to identify where results of the present theory have been obtained by others, and in these cases some discussion of the similarities and differences is made, using the nomenclature and perspective of the present author. In particular, the notion and notation of constraints and exchange with a reservoir that form the basis of the author s approach to equilibrium thermodynamics and statistical mechanics [9] are used as well for the present nonequilibrium theory. 

Perhaps the best starting point in a review of the nonequilibrium field, and certainly the work that most directly influenced the present theory, is Onsager s celebrated 1931 paper on the reciprocal relations [10]. This showed that the symmetry of the linear hydrodynamic transport matrix was a consequence of the time reversibility of Hamilton s equations of motion. This is an early example of the overlap between macroscopic thermodynamics and microscopic statistical mechanics. The consequences of time reversibility play an essential role in the present nonequilibrium theory, and in various fluctuation and work theorems to be discussed shortly. 

Consider an isolated system containing N molecules, and let T = q v. p v be a point in phase space, where the ith molecule has position q, and momentum p . In developing the nonequilibrium theory, it will be important to discuss the behavior of the system under time reversal. Accordingly, define the conjugate... 

Beyond that, despite the numerous instances in high energy physics and in condensed matted physics where (real) time dependance is essential, a nonequilibrium theory has not been fully developed as yet. This difficulty was recognized early as a flaw in the Matsubara equilibrium formalism and has been motivating attempts to construct real-time formalisms at finite temperature (D.N. Zubarev et.al., 1991 R. Floreanini... 

Experiments indicate that the smooth variations of thermodynamic properties (e.g., V, Ky, and the specific heat at constant pressure Cp) with temperature are intermpted by the kinetic process of glass formation, leading to cooling rate dependent kinks in these properties as a function of temperature. In our view, these kinks cannot be described by an equilibrium statistical mechanical theory, but rather are a challenge for a nonequilibrium theory of glass formation. Nonetheless, some insight into the origin of these kinks and the qualitative... 

The above band-broadening mechanism assumes a more concrete form when it is described specifically for chromatography in Section 10.6. When this mechanism is expressed mathematically it becomes the nonequilibrium theory, an important tool describing zone evolution in chromatography (Section 10.6) and field-flow fractionation [2, 3J. 

Clearly, departures from equilibrium—along with the resultant zone spreading—will decrease as means are found to speed up equilibrium between velocity states. One measure of equilibration time is the time defined in Section 9.4 as teq, equivalent to the transfer or exchange time between fast- and slow-velocity states. Time teq must always be minimized this conclusion is seen to follow from either random-walk theory or nonequilibrium theory. These two theories simply represent alternate conceptual approaches to the same band-broadening phenomenon. Thus the plate height from Eqs. 9.12 and 9.17 may be considered to represent simultaneously both nonequilibrium processes and random-walk effects. 

The above phenomena have been incorporated into a theoretical approach which explicitly associates zone spreading with flow velocity and the rates of various equilibration processes. This is the generalized nonequilibrium theory developed by the author [5]. While the theoretical details are too lengthy to develop here, some semi-quantitative reasoning can be used to understand the nature of nonequilibrium-induced zone spreading and the parameters that control it. 

Extensive development of the generalized nonequilibrium theory has led to mathematical expressions for Hs and Hm for many different models, allowing for various microscopic configurations of the stationary phase and flow profiles for the mobile phase. Details, too lengthy to repeat here, can be found in our main reference [5]. 

Application of the Nonequilibrium Theory of Chromatography to a Variable Flow Correlation Model of Complex Flow and Coupling, P. D. Schettler and J. C. Giddings, J. Phys. Chem., 73, 2582 (1969). 

Generalized Nonequilibrium Theory of Plate Height in Large Scale Gas Chromatography, J. C. Giddings, J. Gas Chromatogr., 1, 38 (1963). 

Dynamics of Mass Transfer and the Generalized Nonequilibrium Theory of Chromatography, J. C. Giddings, Ber. Bunsenges. Phys. Chem., 69, 773 (1965). Theory A Practical Tool in Gas Chromatography, J. C. Giddings, Methodes Phys. Anal. (GAMS), 1, 13 (1966). 

Dense-Gas Chromatography of Nonvolatile Substances of High Molecular Weight, L. McLaren, M. N. Myers, and J. C. Giddings, Science, 159, 197 (1968). Nonequilibrium Theory of Field-Flow Fractionation, J. C. Giddings, J. Chem. Phys., 49, 81 (1968). 

Simplified Nonequilibrium Theory of Secondary Relaxation Effects in Programmed Field-Flow Fractionation, J. C. Giddings, Anal. Chem., 58, 735 (1986). Feasibility Study of Dielectrical Field-Flow Fractionation, J. M. Davis and J. C. 

We shall give two answers in equilibrium theories and another answer in nonequilibrium theories (discussed in Sections 3 and 4). 

In his nonequilibrium theory of chromatography, Giddings [67] attempted to derive a general relationship between the broadening of a chromatographic zone due to the mass transfer kinetics and the experimental parameters. Central to his approach, however, is the recognition that the two phases of the chromatographic column are always near equilibrium. 

As discussed already in Chapter 2 (Section 2.2.6), Giddings [10] has developed a nonequilibrium theory of chromatography and showed that the influence of the kinetics of mass transfers can be treated as a contribution to axial dispersion. As illustrated in Chapter 6, this approximation is excellent in linear chromatography, as long as the column efficiency exceeds 20 to 30 theoretical plates. 

From statistical mechanics the second law as a general statement of the inevitable approach to equilibrium in an isolated system appears next to impossible to obtain. There are so many different kinds of systems one might imagine, and each one needs to be treated differently by an extremely complicated nonequilibrium theory. The final equilibrium relations however involving the entropy are straightforward to obtain. This is not done from the microcanonical ensemble, which is virtually impossible to work with. Instead, the system is placed in thermal equilibrium with a heat bath at temperature T and represented by a canonical ensemble. The presence of the heat bath introduces the property of temperature, which is tricky in a microscopic discipline, and relaxes the restriction that all quantum states the system could be in must have the same energy. Fluctuations in energy become possible when a heat bath is connected to the... 

We also find that a simple estimate of Tq within the model also produces remarkable agreement with estimates from the experimental observations. The connection between the low-temperature properties of the glass and its behavior at the transition region follows as a natural consequence of our nonequilibrium theory. The estimate of the number of tunneling levels is very close to the usual predictions. The remarkable agreement of these predictions with the experimental results, of course, cannot be taken too seriously, but it does support our belief that the model correctly describes many of the essential features of the transition. 

Chemical reactions and their rates are central to chemical research. The quantum mechanical theory of reaction rates invokes quantum mechanical scattering theory and statistical mechanics. Thus, one considers the propagation of a system from an initial situation to a different one. Expressions for such processes are developed by means of quasi stationary nonequilibrium theory. 

In this section we introduce these concepts in a very broad way, valid generally for any thermodynamics including nonequilibrium theories, to justify their application in our methodology. For this goal, only several primitives well-known from common life are sufficient. We use the SiUiavy s method [59, 60, 94-97], following mostly the papers of Kratochvfl and Silhavy [98, 99] (see Sects. 1.3, 1.4), because it is appropriate for (at least some) nonequilibrium situations. Unfortunately, this procedure has been demonstrated for pure materials only (for discussion of mixtures see below). 

Three limiting theories are outlined next. The first is an equilibrium theory of the eutectic phase diagram of copolymers as developed by Hory which has been widely used, even for systems not in equilibrium. The second is the corresponding theory for the formation of solid solutions. The third is the application of cold crystallization to copolymers as a limiting, nonequilibrium theory of melting and crystallization. 

Giddings, J.C. Nonequilibrium theory of field-flow fractionation. J. Chem. Phys. 1968, 49, 81. 

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