Nonequilibrium Kinetic Theory

The theory which we have developed for the kinetic behavior of nonequilibrium systems suffers at two important points. The most serious 

A more basic difficulty and one not yet adequately resolved is that encountered in the use of artificial models to represent molecules. From a rigorous point of view the entire behavior of a molecular encounter is determined by the force field surrounding each molecule. By representing molecular force fields by artificial models we avoid the impossible mathematical problem involved in the rigorous approach. The result, however, is to introduce an entirely new set of molecular parameters which remain as yet unpredictable from simpler molecular properties. In the case of the hard sphere model we have introduced the molecular diameter r and the definition of a collision as two new parameters. But there exist two additional parameters which were somewhat concealed in the discussion, namely, the two accommodation coefficients, one for velocity transfers between molecules in collision and the other for collision between molecules and surfaces. 

For the more complicated molecular models such as, for example, those that assume central forces, we replace the above set of parameters by a new set involved in defining the force field. If we add to this the problem of complex molecules (i.c., those with internal structure), then there is the additional set of parameters needed to define the interactions between the internal molecular motions and the external force fields. From the point of view of the hard sphere model this would involve the definition of still more accommodation coefficients to describe the efficiency of transfer of internal energy between colliding molecules. 

Despite these difficulties, the kinetic theory in its simple equilibrium approximation and in its more accurate nonequilibrium representation is capable of reproducing physical behavior in a form which is mathematically simple, qualitatively correct in so far as it represents the interdependence of physical variables, and quantitatively correct to within better than an order of magnitude. As such it presents a valuable direct insight into the relations between molecular processes and macroscopic properties and, as we shall see, provides a valuable guide to understanding kinetic behavior. 

However, before we proceed to an application of these findings to systems undergoing chemical change, we shall undertake in the next chapter the discussion of still more powerful methods for the analysis of the molecular behavior of equilibrium systems. 

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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. 

Basically, two fundamental approaches are used (I) continuum or field dynamics and (2) kinetic theory and nonequilibrium statistical mechanics. The study of fluids tends to be quite complex. 

Kinetic Theory. In the kinetic theory and nonequilibrium statistical mechanics, fluid properties are associated with averages of pruperlies of microscopic entities. Density, for example, is the average number of molecules per unit volume, times the mass per molecule. While much of the molecular theory in fluid dynamics aims to interpret processes already adequately described by the continuum approach, additional properties and processes are presented. The distribution of molecular velocities (i.e., how many molecules have each particular velocity), time-dependent adjustments of internal molecular motions, and momentum and energy transfer processes at boundaries are examples. 

When Bernie Shizgal arrived at UBC in 1970, his research interests were in applications of kinetic theory to nonequilibrium effects in reactive systems. He subsequently applied kinetic theory methods to the study of electron relaxation in atomic and molecular moderators,46 hot atom chemistry, nucleation,47 rarefied gas dynamics,48 gaseous electronics, and other physical systems. An important area of research has been the kinetic theory description of the high altitude portion of planetary atmospheres, and the escape of atmospheric species.49 An outgrowth of these kinetic theory applications was the development of a spectral method for the solution of differential and integral equations referred to as the quadrature discretization method (QDM), which has been used with considerable success in statistical, quantum, and fluid dynamics.50... 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

Comparison of MEIS capabilities (equilibrium descriptions) with capabilities of kinetics, theory of dynamic systems, nonequilibrium thermodynamics, synergetics, thermodynamic finite time, and thermodynamic analysis of motion equations. 

William Russel May I follow up on that and sharpen the issue a bit In the complex fluids that we have talked about, three types of nonequilibrium phenomena are important. First, phase transitions may have dynamics on the time scale of the process, as mentioned by Matt Tirrell. Second, a fluid may be at equilibrium at rest but is displaced from equilibrium by flow, which is the origin of non-Newtonian behavior in polymeric and colloidal fluids. And third, the resting state itself may be far from equilibrium, as for a glass or a gel. At present, computer simulations can address all three, but only partially. Statistical mechanical or kinetic theories have something to say about the first two, but the dynamics and the structure and transport properties of the nonequilibrium states remain poorly understood, except for the polymeric fluids. 

Medium behavior can be visualized in two ways. First, it can be considered to be a large collection of small spheres (molecules) that are in random motion with each other but may be in ordered motion overall. A general treatment of matter from a molecular point of view is called statistical mechanics, and the nonequilibrium gaseous portion is referred to as kinetic theory. 

There is a close connection between molecular mass, momentum, and energy transport, which can be explained in terms of a molecular theory for low-density monatomic gases. Equations of continuity, motion, and energy can all be derived from the Boltzmann equation, producing expressions for the flows and transport properties. Similar kinetic theories are also available for polyatomic gases, monatomic liquids, and polymeric liquids. In this chapter, we briefly summarize nonequilibrium systems, the kinetic theory, transport phenomena, and chemical reactions. 

Statistical mechanics can provide phenomenological descriptions of nonequilibrium processes. An alternative approach based on kinetic theory is favorable especially in describing the transport and rate phenomena. A kinetic theory of nonequilibrium systems has been developed for dilute monatomic gases at low pressure. Substantial progress has also been achieved in extending the theory to dense gases, real gases, and liquids. 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

In classical physics we are familiar with another kind of stationary states, so-called steady states, for which observables are still constant in time however fluxes do exist. A system can asymptotically reach such a state when the boundary conditions are not compatible with equilibrium, for example, when it is put in contact with two heat reservoirs at different temperatures or matter reservoirs with different chemical potentials. Classical kinetic theory and nonequilibrium statistical mechanics deal with the relationships between given boundary conditions and the resulting steady-state fluxes. The time-independent formulation of scattering theory is in fact a quantum theory of a similar nature (see Section 2.10). 

Since Eyring s work, much effort has been devoted to obtain a derivation of transport properties based on nonequilibrium statistical mechanics, which is the more general version of the classical kinetic theory. It is very interesting that this development has led to ideas which largely validate Eyring s simple physical picture. 

Although the kinetic theory has been successfully applied to predict the thermal conductivity, it cannot be used under nonequilibrium conditions. For such cases, the Boltzmann transport theory is required. 

Often it is not the rate kernel itself that is of interest, but rather the decay of the system from some initial nonequilibrium state. A case in point is the study of atom recombination following a photodissociation event. The initial state is presumed to correspond to a specific phase-space configuration of the pair of atomic radicals in an equilibrium solvent. The decay of this initial state is then monitored in the experiment. This is discussed in more detail in Section XII here we simply show how the kinetic theory can be formulated to accommodate this situation. 

The current frontiers for the subject of non-equilibrium thermodynamics are rich and active. Two areas dominate interest non-linear effects and molecular bioenergetics. The linearization step used in the near equilibrium regime is inappropriate far from equilibrium. Progress with a microscopic kinetic theory [38] for non-linear fluctuation phenomena has been made. Careful experiments [39] confirm this theory. Nonequilibrium long range correlations play an important role in some of the light scattering effects in fluids in far from equilibrium states [38, 39]. 

In this section, we give a brief review of important selected theories for surfactant and block copolymer micelles. First, the classical thermodynamic theories covering both mean-field and scaling approaches are briefly reviewed before discussing kinetics. Classical theories for equilibrium and near-equilibrium surfactant and block copolymer micelle kinetics will be briefly reviewed before covering nonequilibrium kinetics in the final part. 

Various flow problems involving evaporation and condensation phenomena are quite common in ordinary circumstances and have aroused an interest of scientists not only in the field of fluid dynamics but also of kinetic theory. The reason for this is that the ordinary continuum-based fluid dynamics cannot describe qualitatively correctly the process of evaporation and condensation occurring at the interface even in ihe continuum limit because of the existence of a nonequilibrium region, the thickness of which is of the order of the molecular mean free path, in the close vicinity of the interface between the condensed phase and the gas phase. Such a nonequilibrium region is called the Knudsen layer, in which collisions between molecules are not so frequent that the momentum and energy exchanges between the molecules leaving the interface... 

Statistical mechanics can provide phenomenological descriptions of nonequilibrium processes. An alternative approach based on kinetic theory is favorable especially in describing the transport and rate phenomena. 

The kinetic theory of gases attempts to explain the macroscopic nonequilibrium properties of gases in terms of the microscopic properties of the individual gas molecules and the forces between them. A central aim of this theory is to provide a microscopic explanation for the fact that a wide variety of gas flows can be described by the Navier-Stokes hydrodynamic equations and to provide expressions for the transport coefficients appearing in these equations, such as the coefficients of shear viscosity and thermal conductivity, in terms of the microscopic prop>erties of the molecules. We devote most of our attention in this article to this problem. 

The next important advance in the theory, and the one that provided the foundation for all later work in this field, was made by Boltzmann, who in 1872 derived an equation for the time rate of change of the distribution function for a dilute gas that is not in equilibrium—the Boltzmann transport equation. (See Boltzmann and also Klein. " ) Boltzmann s equation gives a microscopic description of nonequilibrium processes in the dilute gas, and of the approach of the gas to an equilibrium state. Using the Boltzmann equation. Chapman and Enskog derived the Navier-Stokes equations and obtained expressions for the transport coefficients for a dilute gas of particles that interact with pairwise, short-range forces. Even now, more than 100 years after the derivation of the Boltzmann equation, the kinetic theory of dilute gases is largely a study of special solutions of that equation for various initial and boundary conditions and various compositions of the gas.t... 

Nagnibeda, E. Kustova, E. (2009). Nonequilibrium Reacting Gas Flows. Kinetic Theory of Transport and Relaxation Processes, Springer-Verlag, Berlin, Heidelberg. 

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