Liquids classical statistical mechanics

There are two basic approaches to the computer simulation of liquid crystals, the Monte Carlo method and the method known as molecular dynamics. We will first discuss the basis of the Monte Carlo method. As is the case with both these methods, a small number (of the order hundreds) of molecules is considered and the difficulties introduced by this restriction are, at least in part, removed by the use of artful boundary conditions which will be discussed below. This relatively small assembly of molecules is treated by a method based on the canonical partition function approach. That is to say, the energy which appears in the Boltzman factor is the total energy of the assembly and such factors are assumed summed over an ensemble of assemblies. The summation ranges over all the coordinates and momenta which describe the assemblies. As a classical approach is taken to the problem, the summation is replaced by an integration over all these coordinates though, in the final computation, a return to a summation has to be made. If one wishes to find the probable value of some particular physical quantity, A, which is a function of the coordinates just referred to, then statistical mechanics teaches that this quantity is given by... 

The concept of corresponding states was based on kinetic molecular theory, which describes molecules as discrete, rapidly moving particles that together constitute a fluid or solid. Therefore, the theory of corresponding states was a macroscopic concept based on empirical observations. In 1939, the theory of corresponding states was derived from an inverse sixth power molecular potential model (74). Four basic assumptions were made (/) classical statistical mechanics apply, (2) the molecules must be spherical either by actual shape or by virtue of rapid and free rotation, (5) the intramolecular vibrations are considered identical for molecules in either the gas or liquid phases, and (4) the potential energy of a collection of molecules is a function of only the various intermolecular distances. 

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. 

Evans R. Nature of the liquid-vapor interface and other topics in the statistical-mechanics of nonuniform, classical fluids. Adv. Phys., 1979 28(2) 143-200. 

The density functional theory for classical(equilibrium) statistical mechanics is generalized to deal with various dynamical processes associated with density fluctuations in liquids and solutions. This is effected by deriving a Langevin-diffusion equation for the density field. As applications of our theory we consider density fluctuations in both supercooled liquids and molecular liquids and transport coefficients. 

Thermodynamic effects of directional forces in liquid mixtures.— The theory applied to pure liquids in the last two sections can be generalized to liquid mixtures and can be used to discuss the effects of directional forces on the thermodynamic functions of mixing. Classical statistical mechanics leads to a complete expression for the free energy of a multicomponent system in terms of the intermolecular energies Ust for all pairs of components s and t. Each Ust can be expanded in the general manner (2.1), so that it is separated into a spherically symmetric part and various directional terms. 

Classical statistical mechanics views fluids (i.e., gases and liquids) as a collection of N mutually interacting molecules confined to a volume V at a temperature T and specifies the system by a total intermolecular potential energy U, U (xi,X2,..., xn) = U( 1,. , N), where Xi stands for a set of generalized coordinates of molecule i. Not only for convenience and simplicity, but as an utmost necessity if a tractable theory is to be ultimately applied, the assumption of pairwise additivity is made at this stage, and U is simplified to... 

L. Verlet (1967) Computer experiments on classical fluids I. Thermodynamics properties of Lermard Jones molecules. Phys. Rev. 98, p. 159 D. Chandler (1978) Statistical mechanics of isomerization dynamics in liquids and transition-state approximation. J. Chem. Phys. 68, pp. 2959-2970... 

Chemistry s connections to other theories are similarly complex. Sklar s (1993) detailed treatment of statistical mechanics demonstrates the difficulty of translating, without transformation or remainder, phenomenological concepts such as temperature, boiling point, and liquid into the framework of statistical mechanics. Temperature, for instance, is a quantifiable property of an individual system when defined via classical thermodynamic laws. But within a statistical mechanical framework, temperature... 

The discussion in this chapter has focused on the properties of liquids at interfaces. A related area of contemporary research is the study of solid gas interface. The solid surface is quite different in that atomic or molecular components of a solid are relatively motionless compared to those of liquid. For this reason it is easier to define a plane associated with a well-defined solid surface. The approach to studying adsorption on solids has been more molecular with the development of sophisticated statistical mechanical models. On the other hand, the study of liquid I gas and liquid liquid interfaces has been much more macroscopic in approach with a firm connection to classical thermodynamics. As the understanding of liquids has improved at the molecular level using contemporary statistical mechanical tools, these methods are being applied now to fluids at interfaces. 

The fundamental problem in classical equilibrium statistical mechanics is to evaluate the partition function. Once this is done, we can calculate all the thermodynamic quantities, as these are typically first and second partial derivatives of the partition function. Except for very simple model systems, this is an unsolved problem. In the theory of gases and liquids, the partition function is rarely mentioned. The reason for this is that the evaluation of the partition function can be replaced by the evaluation of the grand canonical correlation functions. Using this approach, and the assumption that the potential energy of the system can be written as a sum of pair potentials, the evaluation of the partition function is equivalent to the calculation of... 

One of the most active areas of research in the statistical mechanics of interfacial systems in recent years has been the problem of freezing. The principal source of progress in this field has been the application of the classical density-functional theories (for a review of the fundamentals in these methods, see, for example, Evans ). For atomic fluids, such apphcations were pioneered by Ramakrishnan and Yussouff and subsequently by Haymet and Oxtoby and others (see, for example, Baret et al. ). Of course, such theories can also be applied to the vapor-liquid interface as well as to problems such as phase transitions in liquid crystals. Density-functional theories for these latter systems have not so far involved use of interaction site models for the intermolecular forces. 

All three areas will be addressed here. The application of classical density functional theory has led to some of the most important recent theoretical advances in SFE and these have been the subject of several authoritative review articles [10-16]. On the other hand, we know of no recent comprehensive review addressing theoretical approaches other than density functional theories (DFT) and the other two subject areas, particularly the last one, and it was this that motivated us to write this chapter. We hope that the somewhat broader coverage of molecular modeling research in SFE given in this chapter will be of benefit to researchers new to the field. We should mention that this Chapter is written from a perspective that is more strongly influenced by liquid-state statistical mechanics than by solid-state theory. The interests of the authors in the problem at hand are an outgrowth of their previous work on phase equilibrium in fluids and fluid mixtures. 

The properties of the metal phase have been successfully described by rather simple models, most notably the jelliiun model. In many theoretical treatments of the liquid/metal interface, the hquid electrolyte in contact with the metal has been described, to first order, as an external field, acting on the jellium model (see Ref. 13 and references therein). In many simulation studies, the reverse approach is taken. The focus is on the description of the liquid phase and the effect of the metal on the aqueous phase is approximated, to first order, by an external potential acting on the ions and molecules in the liquid phase. This is done within the framework of classical mechanics and classical statistical mechanics. The models for the interparticle interactions will consist of distributed point charges in combination with soft interatomic repulsions and dispersive attractions. Some of the models can also be considered chemical models they can be regarded as a first step towards electrochemical modeling, very much in the spirit of molecular modeling . 

Chapter 11 reviews the statistical mechanical basis of hydrodynamics and discusses theories that may be used to extend hydrodynamics beyond the classical equations discussed in Chapter 10. Chapter 12 applies the statistical mechanical theory to the calculation of depolarized light-scattering spectra from dense liquids where interactions between anisotropic molecules are important. 

The mathematical and physical theory of equilibrium cooperative phenomena in crystals has been reviewed by Newell and Montroll, and Domb, and the basic statistical mechanics is reviewed in Hill s monograph. Rowlinson has given a very thorough discussion of the classical thermodynamics of the coexistence curve and the critical region, and has also appraised much of the better data on equUibrium properties (of liquids and hquid mixtures). Rice > has several times reviewed the field of critical phenomena. 

Every serious student of fluids will own a copy of Rowlinson s book on liquids and liquid mixtures, and there is no warrant for a repetition of his scholarly and lucid exposition of the classical thermodynamics of the critical state. But a few of the important points must be brought to mind. Consider the classical isotherm portrayed in Fig. 1. The solid fine represents the observed pressure of a system in its most stable state of volume V. Between points 1 and 4 the compressibility of the fluid is infinite, although approximate statistical-mechanical theories, when based on the canonical ensemble, give a loop between points 1 and 4 and... 

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