Classical equilibrium statistical mechanics

Marcus uses the Born-Oppenheimer approximation to separate electronic and nuclear motions, the only exception being at S in the case of nonadiabatic reactions. Classical equilibrium statistical mechanics is used to calculate the probability of arriving at the activated complex only vibrational quantum effects are treated approximately. The result is... 

III. Irrelevance of Classical Equilibrium Statistical Mechanics for the Gravitational Problem... 

III. IRRELEVANCE OF CLASSICAL EQUILIBRIUM STATISTICAL MECHANICS FOR THE GRAVITATIONAL PROBLEM... 

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. 

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... 

This section summarizes the classical, equilibrium, statistical mechanics of many-particle systems, where the particles are described by their positions, q, and momenta, p. The section begins with a review of the definition of entropy and a derivation of the Boltzmann distribution and discusses the effects of fluctuations about the most probable state of a system. Some worked examples are presented to illustrate the thermodynamics of the nearly ideal gas and the Gaussian probability distribution for fluctuations. 

Thompson, C.J. (1988) Classical Equilibrium Statistical Mechanics, Clarendon Press, Oxford. 

If we interpret Gibbs entropy in the same spirit as the missing information of information theory, it can be viewed as a measure of statistical uncertainty. Adopting this point of view, it seems natural to treat the following principle as a basic postulate of classical equilibrium statistical mechanics. 

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

In equilibrium statistical mechanics involving quantum effects, we need to know the density matrix in order to calculate averages of the quantities of interest. This density matrix is the quantum analog of the classical Boltzmann factor. It can be obtained by solving a differential equation very similar to the time-dependent Schrodinger equation... 

Raymond Kapral carried out his doctoral studies on molecular quantum mechanics at Princeton University and received his Ph.D. in 1967. He pursued his postdoctoral work at the Massachusetts Institute of Technology where his interests switched to non-equilibrium statistical mechanics. He then took a faculty position at the University of Toronto in 1969 where he is currently Professor of Chemistry. He is a Fellow of the Royal Society of Canada and received the Palladium Medal of the Chemical Institute of Canada in 2003. European collaborations have not only provided an opportunity to mix quantum and classical mechanics but also to mix science with good food and wine. 

In describing thermodynamic and equilibrium statistical-mechanical behaviors of a classical fluid, we often make use of a radial distribution function g r). The latter for a fluid of N particles in volume V expresses a local number density of particles situated at distance r from a fixed particle divided by an average number density p = NjV), when the order of IjN is negligible in comparison with 1. Various thermodynamic quantities are related to g(r). For a single-component monatomic system of particles interacting with a pairwise additive potential 0(r), the relationship connecting the pressure P to g(r) is the virial theorem, ... 

Monte Carlo techniques are methods of estimating the values of manydimensional integrals by sampling with the help of random numbers/ It is obvious that this makes them methods appropriate to equilibrium statistical mechanics. Among the integrals of interest in classical statistical mechanics are ensemble averages of any mechanical quantity M( ),... 

In the classical free statistics, the number of the functional groups on the surface of a tree-like cluster is of the same order of that of the groups inside the cluster, so that a simple thermodynamic limit without surface term is impossible to take. The equilibrium statistical mechanics for the polycondensation was refined by Yan [14] to treat surface correction in such finite systems. He found the same result as Ziff and Stell. Thus the treatment of the postgel regime is not unique. The rigorous treatment of the problem requires at least one additional parameter defining relative probability of occurrence of infra- and intermolecular reactions in the gel. 

Interlude 4.2 Getting Microscopic Information from Macroscopic Observations The Inverse Problem When one thinks of equilibrium statistical mechanics, what usually comes to mind is its classical mission, namely, predicting macroscopic structure and properties from microscopic, pairwise interaction forces. However, more often than not, we are faced with a need to deduce information on the (conservative) pair interaction forces [equivalently, the pair potential observed macroscopic properties. The information thus obtained can then be used to predict other properties of interest. The problem of extracting microscopic information from macroscopic observations is known as the inverse problem. 

The most fundamental description of processes, in the present context, would be based on molecular considerations. A molecular description is distinguished by the fact that it treats an arbitrary system as if it were composed of individual entities, each of which obeys certain rules. Consequently, the properties and state variables of the system are obtained by summing over all of the entities. Quantum mechanics, equihbrium and non-equilibrium statistical mechanics, and classical mechanics are typical methods of analysis, by which the properties and responses of the system can be calculated. 

We now give a brief introduction to equilibrium statistical mechanics based on classical mechanics. Classical mechanics is a good approximation for the translational motion of atoms and molecules near room temperature, and appears to be a usable approximation for the rotational motion of most molecules near room temperature. It works very poorly for vibrations, and fails completely for electronic motion. However, we have seen that in many systems the vibrational and electronic energies are numerically unimportant, and classical statistical mechanics can be used with good results in these systems. 

In going from statics to dynamics we need new experimental tools and also theoretical machinery that allows for the dependence on time. This means that the stationary states that are usually the subject of an introductory quantum mechanics course have to be extended to non-stationary ones. Fairly often, classical dynamics is sufficient to describe the time evolution but there are a number of interesting exceptions. Non-equilibrium statistical mechanics is necessary to describe systems with many degrees of freedom and their far-from-equilibrium pattern formation. 

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. 

There is considerable interest in the use of discretized path-integral simulations to calculate free energy differences or potentials of mean force using quantum statistical mechanics for many-body systems [140], The reader has already become familiar with this approach to simulating with classical systems in Chap. 7. The theoretical basis of such methods is the Feynmann path-integral representation [141], from which is derived the isomorphism between the equilibrium canonical ensemble of a... 

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