Nonequilibrium system

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. 

Constant driving forces cause steady flows, which leads to a stationary state. For example, a constant temperature difference applied to a metal bar will induce a heat flow that will cause a change in all local temperatures. After a while, a constant distribution of temperature will be attained and the heat flow will become steady. The steady state flow and constant distribution of forces characterizing a system form the ultimate state of irreversible systems corresponding to the states of equilibrium in classical thermodynamics. 

In some systems, the distance from equilibrium reaches a critical point, after which the states in the thermodynamic branch become metastable or unstable. This region is the nonlinear region where the linear phenomenological equations are not valid. We observe bifurcations and multiple solutions in this region. 

In nonequilibrium systems, the intensive properties of temperature, pressure, and chemical potential are not uniform. However, they all are defined locally in an elemental volume with a sufficient number of molecules for the principles of thermodynamics to be applicable. For example, in a region A , we can define the densities of thermodynamic properties such as energy and entropy at local temperature. The energy density, the entropy density, and the amount of matter are expressed by uk(T, Nk), s T, Nk), and Nk, respectively. The total energy U, the total entropy S, and the total number of moles N of the system are determined by the following volume integrals  

Since the temperature is not uniform for the whole system, the total entropy is not a function of the other extensive properties of U, V, and N. However, with the local temperature, the entropy of a nonequilibrium system is defined in terms of an entropy density, sk. 

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Nicolis G and Prigogine I 1977 Self-organization in Nonequilibrium Systems (New York Wiley)... 

Nicolis, G., Prigogine, I. Self-organization in nonequilibrium systems. John Willey Sons, New York (1977) 512... 

It is also possible to simulate nonequilibrium systems. For example, a bulk liquid can be simulated with periodic boundary conditions that have shifting boundaries. This results in simulating a flowing liquid with laminar flow. This makes it possible to compute properties not measurable in a static fluid, such as the viscosity. Nonequilibrium simulations give rise to additional technical difficulties. Readers of this book are advised to leave nonequilibrium simulations to researchers specializing in this type of work. 

Polymers are difficult to model due to the large size of microcrystalline domains and the difficulties of simulating nonequilibrium systems. One approach to handling such systems is the use of mesoscale techniques as described in Chapter 35. This has been a successful approach to predicting the formation and structure of microscopic crystalline and amorphous regions. 

P. Meakin, D. J. Scalapino. Simple models for heterogeneous catalysis Phase transitions-like behavior in nonequilibrium systems. J Chem Phys 57 731-741, 1987. 

In the preceding sections, various types of fluctuations and instabilities essential to corrosion were examined. As a result, it was shown that a corrosion system involves various kinds of problems of stability and instability. Unlike thermodynamic equilibrium systems, in nonequilibrium systems like corrosion systems, a drastic change in the reaction state should be defined as a bifurcation phenomenon. 

Nicolis, G. and Prigogine, I. (1977) Self-Organization in Nonequilibrium Systems, Wiley Interscience, New York. 

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 generic case is a subsystem with phase function x(T) that can be exchanged with a reservoir that imposes a thermodynamic force Xr. (The circumflex denoting a function of phase space will usually be dropped, since the argument T distinguishes the function from the macrostate label x.) This case includes the standard equilibrium systems as well as nonequilibrium systems in steady flux. The probability of a state T is the exponential of the associated entropy, which is the total entropy. However, as usual it is assumed (it can be shown) [9] that the... 

An important class of nonequilibrium systems are those in which mechanical work, either steady or varying, is performed on the subsystem while it is in contact with a heat reservoir. Such work is represented by a time-dependent Hamiltonian, t), where p(f) is the work parameter. (For example, this... 

The canonical nonequilibrium system consists of a subsystem sandwiched between two thermal reservoirs of different temperatures, with heat flowing steadily through the subsystem from the hot reservoir to the cold reservoir. Application of the general theory to this canonical problem illustrates the theory and serves to make the analysis more concrete. The first task is to identify explicitly the thermodynamic variables appropriate for this problem. 

The availability of a phase space probability distribution for the steady state means that it is possible to develop a Monte Carlo algorithm for the computer simulation of nonequilibrium systems. The Monte Carlo algorithm that has been developed and applied to heat flow [5] is outlined in this section, following a brief description of the system geometry and atomic potential. 

Perhaps the most common computer simulation method for nonequilibrium systems is the nonequilibrium molecular dynamics (NEMD) method [53, 88]. This typically consists of Hamilton s equations of motion augmented with an artificial force designed to mimic particular nonequilibrium fluxes, and a constraint force or thermostat designed to keep the kinetic energy or temperature constant. Here is given a brief derivation and critique of the main elements of that method. 

Depending on the point of view, it is either a strength or a weakness of the NEMD method that it gives a uniform structure for the nonequilibrium system... 

In nonequilibrium systems, chemical processes spontaneously alter the composition or phase of the system until equilibrium is attained. Simple systems, such as a mixture of sodium chloride and water, attain equilibrium quickly, whereas complex systems may reach equilibrium only after decades or eons. 

In contrast to a mixture of redox couples that rapidly reach thermodynamic equilibrium because of fast reaction kinetics, e.g., a mixture of Fe2+/Fe3+ and Ce3+/ Ce4+, due to the slow kinetics of the electroless reaction, the two (sometimes more) couples in a standard electroless solution are not in equilibrium. Nonequilibrium systems of the latter kind were known in the past as polyelectrode systems [18, 19]. Electroless solutions are by their nature thermodyamically prone to reaction between the metal ions and reductant, which is facilitated by a heterogeneous catalyst. In properly formulated electroless solutions, metal ions are complexed, a buffer maintains solution pH, and solution stabilizers, which are normally catalytic poisons, are often employed. The latter adsorb on extraneous catalytically active sites, whether particles in solution, or sites on mechanical components of the deposition system/ container, to inhibit deposition reactions. With proper maintenance, electroless solutions may operate for periods of months at elevated temperatures, and exhibit minimal extraneous metal deposition. 

The two main assumptions underlying the derivation of Eq. (5) are (1) thermodynamic equilibrium and (2) conditions of constant temperature and pressure. These assumptions, especially assumption number 1, however, are often violated in food systems. Most foods are nonequilibrium systems. The complex nature of food systems (i.e., multicomponent and multiphase) lends itself readily to conditions of nonequilibrium. Many food systems, such as baked products, are not in equilibrium because they experience various physical, chemical, and microbiological changes over time. Other food products, such as butter (a water-in-oil emulsion) and mayonnaise (an oil-in-water emulsion), are produced as nonequilibrium systems, stabilized by the use of emulsifying agents. Some food products violate the assumption of equilibrium because they exhibit hysteresis (the final c/w value is dependent on the path taken, e.g., desorption or adsorption) or delayed crystallization (i.e., lactose crystallization in ice cream and powdered milk). In the case of hysteresis, the final c/w value should be independent of the path taken and should only be dependent on temperature, pressure, and composition (i.e.,... 

Lerman, A. (1971). Nonequilibrium Systems in Natural Water Chemistry. American Chemical Society, Washington, DC. 

G. Nicolis and I. Prigogine, Self Organization in Nonequilibrium System, John Wiley Sons, New York, 1977. 

LS.12. G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems From Dissipative Structures to Order through Fluctuations, John Wiley Sons [Interscience], New York. Translations in Russian, Japanese, Itahan, and Chinese. 

TNC.30. G. Nicolis and 1. Prigogine, Fluctuations in Nonequilibrium Systems, Proc. Nat Acad. Sciences, 68, 2102-2107 (1971). 

TNC.59. D. K. Kondepudi and II. Prigogine, Sensitivity of nonequilibrium systems, Physica, 107A, 1-24, 1981. 

TNC.68. 1. Prigogine and G. Nicolis, Self-organization in nonequilibrium systems Towards a dynamics of complexity, in Bifurcation Analysis, M. Hatzewinkel, ed., Reidel, Dordrecht, 1985, pp. 3-12. 

MSN. 107. 1. Prigogine, Irreversibility and space-time structure, in Proceedings, International Conference on Fluctuations and Sensitivity in Nonequilibrium Systems, Austin, 1984, W. Horsthemke and K. D. K. Kondepudi, eds., Springer, Berlin, 1984, pp. 2-9. 

GEN. 128. 1. Prigogine, The message of entropy, in Workshop, Patterns, Defects and Microstructures in Nonequilibrium Systems, Austin, Texas 1986. 

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

We here have a principle showing that the second law of thermodynamics plays a constmctive role in nonequilibrium systems and thus does not have the need to exceed some threshold in the nonequilibrium constraints as was the case for the Glansdorff-Prigogine dissipative stmctures [67]. The principle of temporal ordering is valid as soon as the system is out of equilibrium and holds arbitrarily far from equilibrium. 

The nonequilibrium effect is due to the different velocities at which the components of the analyte are carried down the channel. The different velocities, in turn, originate from the laminar nature of the flow since the constituents of the analyte are dispersed in these laminae, they undergo differential migration velocities in the axial direction. The expression of the nonequilibrium system dispersion takes the following form ... 

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