Statistical mechanics nonequilibrium

In many systems found in nature, there is a continuous flux of matter and energy so that the system cannot reach equilibrium. Equilibrium statistical mechanics says nothing about the rate of a process. Chemical reaction rates will be discussed in Chapter 8. The nonequilibrium processes to be discussed here are transport processes like diffusion, heat transfer, or conductivity, where the statistics are expressed as time-evolving probability distributions. Transport processes are due to random motion of molecules and are therefore called stochastic. The equations are partial differential equations describing the time evolution of a probability function rather than properties of equilibrium. 

How fast do molecules move around in the gas phase The velocity distribution, f(v), depends on tenperature, of course. If the molecular mass is M, the kinetic energy is 

E = MvV2. The number of molecules with velocity in a range dv, df(v), is proportional to the Boltzmann factor multiplied by the volume of a shell with thickness dv (k = kg)  

This expression is known as Maxwell s velocity distribution. 

In this equation, we multiply Avogadro s number, in the numerator and denominator to obtain the molecular mass M = 1000 m u (the unit u corresponds to gram/mole) in the numerator and the gas constant R = N k in the denominator (thus we use here the notation M for the molecular mass and m (kilogram) for the mass of the molecule), v is obtained from Equation 5.133  

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In this review we put less emphasis on the physics and chemistry of surface processes, for which we refer the reader to recent reviews of adsorption-desorption kinetics which are contained in two books [2,3] with chapters by the present authors where further references to earher work can be found. These articles also discuss relevant experimental techniques employed in the study of surface kinetics and appropriate methods of data analysis. Here we give details of how to set up models under basically two different kinetic conditions, namely (/) when the adsorbate remains in quasi-equihbrium during the relevant processes, in which case nonequilibrium thermodynamics provides the needed framework, and (n) when surface nonequilibrium effects become important and nonequilibrium statistical mechanics becomes the appropriate vehicle. For both approaches we will restrict ourselves to systems for which appropriate lattice gas models can be set up. Further associated theoretical reviews are by Lombardo and Bell [4] with emphasis on Monte Carlo simulations, by Brivio and Grimley [5] on dynamics, and by Persson [6] on the lattice gas model. 

A method is outlined by which it is possible to calculate exactly the behavior of several hundred interacting classical particles. The study of this many-body problem is carried out by an electronic computer which solves numerically the simultaneous equations of motion. The limitations of this numerical scheme are enumerated and the important steps in making the program efficient on the computer are indicated. The applicability of this method to the solution of many problems in both equilibrium and nonequilibrium statistical mechanics is discussed. 

This nonequilibrium Second Law provides a basis for a theory for nonequilibrium thermodynamics. The physical identification of the second entropy in terms of molecular configurations allows the development of the nonequilibrium probability distribution, which in turn is the centerpiece for nonequilibrium statistical mechanics. The two theories span the very large and the very small. The aim of this chapter is to present a coherent and self-contained account of these theories, which have been developed by the author and presented in a series of papers [1-7]. The theory up to the fifth paper has been reviewed previously [8], and the present chapter consolidates some of this material and adds the more recent developments. 

The present theory can be placed in some sort of perspective by dividing the nonequilibrium field into thermodynamics and statistical mechanics. As will become clearer later, the division between the two is fuzzy, but for the present purposes nonequilibrium thermodynamics will be considered that phenomenological theory that takes the existence of the transport coefficients and laws as axiomatic. Nonequilibrium statistical mechanics will be taken to be that field that deals with molecular-level (i.e., phase space) quantities such as probabilities and time correlation functions. The probability, fluctuations, and evolution of macrostates belong to the overlap of the two fields. 

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. 

For nonequilibrium statistical mechanics, the present development of a phase space probability distribution that properly accounts for exchange with a reservoir, thermal or otherwise, is a significant advance. In the linear limit the probability distribution yielded the Green-Kubo theory. From the computational point of view, the nonequilibrium phase space probability distribution provided the basis for the first nonequilibrium Monte Carlo algorithm, and this proved to be not just feasible but actually efficient. Monte Carlo procedures are inherently more mathematically flexible than molecular dynamics, and the development of such a nonequilibrium algorithm opens up many, previously intractable, systems for study. The transition probabilities that form part of the theory likewise include the influence of the reservoir, and they should provide a fecund basis for future theoretical research. The application of the theory to molecular-level problems answers one of the two questions posed in the first paragraph of this conclusion the nonequilibrium Second Law does indeed provide a quantitative basis for the detailed analysis of nonequilibrium problems. 

B. C. Eu, Nonequilibrium Statistical Mechanics, Kluwer Academic Publishers, Dordrecht, 1998. 

R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, London, 2001. 

Monte Carlo heat flow simulation, 69-70 nonequilibrium statistical mechanics, microstate transitions, 44 46 nonequilibrium thermodynamics, 7 time-dependent mechanical work, 52-53 transition probability, 53-57 Angular momentum, one- vs. three-photon... 

Monte Carlo heat flow simulation, 68—70 steady-state probability distribution, nonequilibrium statistical mechanics, 40-43... 

Microstate transitions, nonequilibrium statistical mechanics, 44—51 adiabatic evolution, 44—46 forward and reverse transitions, 47-51 stationary steady-steat probability, 47 stochastic transition, 46—47... 

Nonequilibrium statistical mechanics Green-Kubo theory, 43-44 microstate transitions, 44-51 adiabatic evolution, 44—46 forward and reverse transitions, 47-51 stationary steady-state probability, 47 stochastic transition, 464-7 steady-state probability distribution, 39—43 Nonequilibrium thermodynamics second law of basic principles, 2-3 future research issues, 81-84 heat flow ... 

Although the collision and transition state theories represent two important methods of attacking the theoretical calculation of reaction rates, they are not the only approaches available. Alternative methods include theories based on nonequilibrium statistical mechanics, stochastic theories, and Monte Carlo simulations of chemical dynamics. Consult the texts by Johnson (62), Laidler (60), and Benson (59) and the review by Wayne (63) for a further introduction to the theoretical aspects of reaction kinetics. 

Hillery, M., R.O. Connell, M.O. Scully and E.P. Wigner. Phys. Rep., 106 121, 1984. Zwanzig, R. Nonequilibrium Statistical Mechanics. Oxford University Press, Oxford, 2001. 

The vision of irreversibility that appeared in this first group of works, which formed the object of the first monograph on nonequilibrium statistical mechanics by Prigogine (1962, LS.9), was the following. The necessary conditions for an irreversible evolution were ... 

LS.9. I. Prigogine, Nonequilibrium Statistical Mechanics, John Wiley Sons [Interscience Division], New York. 

MSN.64. I. Prigogine, G. Nicolis, and P. Allen, Eyring s theory of viscosity of dense media and nonequilibrium statistical mechanics, in Chemical Dynamics, Papers in Honor of H. Eyring, Hirshfelder, ed., Wiley, New York, 1971. 

MSN.93. I. Prigogine and A. P. Grecos, Topics in nonequilibrium statistical mechanics, in Problems in the Foundations of Physics, LXXII Corso, Soc. Ital. Fisica. 

MSN. 102. I. Prigogine, Entropy, time and kinetic description, in Order and Fluctuations in Equilibrium and Nonequilibrium Statistical Mechanics, 17th International Solvay Conference on Physics, G. Nicolis, G. Dewel, and J. W. Turner, eds., Wiley, 1980, pp. 35-75. 

MSN. 120. 1. Prigogine and T. Petrosky, Nonequilibrium statistical mechanics Beyond the Van Hove limit, in Festschrift in Honor of Leon Van Hove, World Scientihc, Singapore, 1989. 

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