Nonequilibrium theories macroscopic

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. 

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 world surrounding us is mostly out of equihbrium, equilibrium being just an idealization that requires specific conditions to be met in the laboratory. Even today we do not have a general theory about nonequilibrium macroscopic systems as we have for equilibrium ones. Onsager theory is probably the most successful attempt, albeit its domain of validity is restricted to the linear response regime. In small systems the situation seems to be the opposite. Over the past years, a set of theoretical results that go under the name of fluctuation theorems have been unveiled. These theorems make specific predictions about energy processes in small systems that can be scrutinized in the laboratory. 

In 4.4 the theory of 4.2 will be applied to study electro-diffusion of ions through a unipolar ion-exchange membrane, separating two electrolyte solutions. This will include the classical treatment of concentration polarization in a solution layer adjacent to an ion-exchange membrane under an electric current. The validity limits of this theory, set by the violations of local electro-neutrality and caused by the development of a macroscopic nonequilibrium space charge, will be indicated. (The effects of the nonequilibrium space charge are to be discussed at some length in Chapter 5.)... 

In 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

The conference was divided into four parts to each of which a full day was devoted the first one treated Equilibrium Statistical Mechanics, with special regard to The Theory of Critical Phenomena the second part regarded Nonequilibrium Statistical Mechanics. Cooperative Phenomena the third one, The Macroscopic Approach to Coherent Behavior in Far Equilibrium Conditions and the fourth and last, Fluctuation Theory and Nonequilibrium Phase Transitions. ... 

The fourth part included a paper by Graham on the onset of cooperative behavior in nonequilibrium states. Suzuki talked about the theory of instability, with special regard to nonlinear Brownian motion and the formation of macroscopic order, and P. W. Anderson developed a series of interesting considerations of very general nature around the question Can broken symmetry occur in driven systems ... 

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. 

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. 

We have thus demonstrated that Newton s law of viscosity, an inherently macroscopic result, can be obtained via linear response theory as the nonequilibrium average in the steady state. Furthermore, the distribution function for the steady state average is determined by microscopic equations of motion. Hence, the SLLOD equations, in the linear regime, reduce to the linear phenomenological law proposed by Newton. Moreover, all the quantities that are needed to compute the shear viscosity can be obtained from a molecular dynamics simulation. 

The second fact that is implicit in macroscopic or continuum laws is the idea of local thermodynamic equilibrium. For example, when we write the Fourier law of heat conduction, it is inherently assumed that one can define a temperature at any point in space. This is a rather severe assumption since temperature can be defined only under thermodynamic equilibrium. The question that we might ask is the following. If there is thermodynamic equilibrium in a system, then why should there be any net transport of energy Thus, we implicitly resort to the concept of local thermodynamic equilibrium, where we assume that thermodynamic equilibrium can be defined over a volume which is much smaller than the overall size of the system. What happens when the size of the object becomes on the order of this volume Obviously, the macroscopic or continuum theories break down and new laws based on nonequilibrium thermodynamics need to be formulated. This chapter focuses on developing more generalized theories of transport which can be used for nonequilibrium conditions. This involves going to the root of the macroscopic or continuum theories. 

In this chapter the formalism of nonequilibrium thermodynamics, is reviewed. This formalism is then applied to the theory of isothermal diffusion and electrophoresis. It is shown that this theory is important in determining the relations between the transport coefficients measured by light scattering and those measured by classical macroscopic techniques. Since much of this material is covered in other chapters, this chapter is very brief. Our presentation closely follows that of Katchalsky and Curran (1965). Other books that can be consulted are those of DeGroot and Mazur (1962) and Prigogine (1955). 

It is well-known that implicit solvent models use both discrete and continuum representations of molecular systems to reduce the number of degrees of freedom this philosophy and methodology of implicit solvent models can be extended to more general multiscale formulations. A variety of DG-based multiscale models have been introduced in an earlier paper of Wei [74]. Theory for the differential geometry of surfaces provides a natural means to separate the microscopic solute domain from the macroscopic solvent domain so that appropriate physical laws are applied to applicable domains. This portion of the chapter focuses specifically on the extension of the equilibrium electrostatics models described above to nonequilibrium transport problems that are relevant to a variety of chemical and biological S5 ems, such as molecular motors, ion channels, fuel cells, and nanofluidics, with chemically or biologically relevant behavior that occurs far from equilibrium [74-76]. 

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. 

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