Equilibrium/nonequilibrium theory

Figure 38. Classification of nonequilibrium fluctuations. (Reprinted from M. Asanuma and R. Aogaki, Non-equilibrium fluctuation theory on pitting dissolution. I. Derivation of dissolution current equations." J. Chem. Phys. 106,9938,1997. Copyright 1997, American Institute of Physics.)...

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. 

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... 

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. 

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

States away from global equilibrium are called the thermodynamic branch (Figure 2.2). Systems not far from global equilibrium may be extrapolated around equilibrium state. For systems near equilibrium, linear phenomenological equations may represent the transport and rate processes. The linear nonequilibrium thermodynamics theory determines the dissipation function or the rate of entropy production to describe such systems in the vicinity of equilibrium. This theory is particularly useful to describe coupled phenomena, and quantify the level of coupling in physical, chemical, and biological systems without detailed process mechanisms. 

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. 

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... 

The new concept is derived from a more general nonequilibrium thermodynamical description of heterogeneous polymer systems which has been worked out on the basis of experiments mainly with conductive systems, plus some orienting and critical examples with non-conductive systems [72d]. The principal ideas of the new general non-equilibrium thermodynamic theory of multiphased polymer systems can be outlined as described in Section 9. 

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. 

The power of the nonequilibrium theory, from a simulator s point of view, is that transport properties can be obtained from equilibrium simulations. However, one must pay particular attention to statistical uncertainties. The best way to show this is with a simple example. 

For the examples in Table 7, VDEs computed with the equilibrium PCM are 1.0-1.3 eV smaller than the nonequilibrium values, the latter of which lie closer to experimental results obtained from liquid microjet photoelectron spectroscopy. The 0.5 eV discrepancies that remain between the nonequilibrium theory and the experimental VDEs probably have myriad origins, not the least of which is likely the inherent limitations of a continuum description of the solvent. This comparison between theory and experiment serves as evidence of the complexity of solution-phase VDE calculations. 

The third approach is called the thermodynamic theory of passive systems. It is based on the following postulates (1) The introduction of the notion of entropy is avoided for nonequilibrium states and the principle of local state is not assumed, (2) The inequality is replaced by an inequality expressing the fundamental property of passivity. This inequality follows from the second law of thermodynamics and the condition of thermodynamic stability. Further the inequality is known to have sense only for states of equilibrium, (3) The temperature is assumed to exist for non-equilibrium states, (4) As a consequence of the fundamental inequality the class of processes under consideration is limited to processes in which deviations from the equilibrium conditions are small. This enables full linearization of the constitutive equations. An important feature of this approach is the clear physical interpretation of all the quantities introduced. 

Introduction.—Statistical physics deals with the relation between the macroscopic laws that describe the internal state of a system and the dynamics of the interactions of its microscopic constituents. The derivation of the nonequilibrium macroscopic laws, such as those of hydrodynamics, from the microscopic laws has not been developed as generally as in the equilibrium case (the derivation of thermodynamic relations by equilibrium statistical mechanics). The microscopic analysis of nonequilibrium phenomena, however, has achieved a considerable degree of success for the particular case of dilute gases. In this case, the kinetic theory, or transport theory, allows one to relate the transport of matter or of energy, for example (as in diffusion, or heat flow, respectively), to the mechanics of the molecules that make up the system. 

The second chapter is by Aogaki and includes a review of nonequilibrium fluctuations in corrosion processes. Aogaki begins by stating that metal corrosion is not a single electrode reaction, but a complex reaction composed of the oxidation of metal atoms and the reduction of oxidants. He provides an example in the dissolution of iron in an acidic solution. He follows this with a discussion of electrochemical theories on corrosion and the different techniques involved in these theories. He proceeds to discuss nonequilibrium fluctuations and concludes that we can again point out that the reactivity in corrosion is determined, not by its distance from the reaction equilibrium but by the growth processes of the nonequilibrium fluctuations. ... 

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 ... 

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. 

---