Equilibrium statistical mechanics ensembles

B.C. Eu, Non-Equilibrium Statistical Mechanics, Ensemble Method, Academic Kluwer Publishers, Dordrecht, 1998. 

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... 

Fig.12. Computation by Monte Carlo methods of the first four order parameters of an ensemble of 1000 chromophores (of dipole moment 13 Debye) existing in a medium of uniform dielectric constant. At the beginning of the calculation, the chromophores are randomly ordered thus, ==O. During the first 400 Monte Carlo steps, an electric poling field (600 V/micron) is on but the chromophore number density (=10 7 molecules/cc) is so small that intermolecular electrostatic interactions are unimportant. The order parameters quickly evolve to well-known equilibrium values obtained analytically from statistical mechanics (black dots in figure also see text). During steps 400-800 the chromophore number density is increased to 5xl020 and intermolecular electrostatic interactions act to decrease order parameters consistent with the results of equilibrium statistical mechanical calculations discussed in the text. Although Monte Carlo and equilibrium statistical mechanical approaches described in the text are based on different approximations and mathematical methods, they lead to the same result (i.e., are in quantitative agreement)...

The general mathematical formulation of the equilibrium statistical mechanics based on the generalized statistical entropy for the first and second thermodynamic potentials was given. The Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles were investigated as an example. It was shown that the statistical mechanics based on the Tsallis statistical entropy satisfies the requirements of equilibrium thermodynamics in the thermodynamic limit if the entropic index z=l/(q-l) is an extensive variable of state of the system. 

The organization of this document is as follows The basis of all of these methods, equilibrium statistical mechanics, will be reviewed in Sect. 2. Section 3 will discuss the use of non-Hamiltonian systems to generate important ensembles. Novel non-Hamiltonian method, such as variable transformation techniques and adiabatic free energy dynamics will be discussed in Sect. 4. Finally, some conclusions and remarks will be provided in Sect. 5. 

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( ),... 

The maximum information entropy procedure is the derivation of the Gibbs ensemble in equilibrium statistical mechanics, but the information entropy is not defined by a probability measure on phase space, but rather on path space. The path a information entropy is... 

Progress on the theoretical description at the properties of moderately dense and dense fluids systems has traditionally been limited by the lack of an equation of state for dense fluid phases. The past decade has seen rapid progress on this problem using electronic computers to obtain ensemble average properties for fluid systems modelled with simple particles. The equilibrium and non-equilibrium statistical mechanical and kinetic theory relationships admit the direct calculation of the fluid properties without identifying the mathematical form of the equation of state. This application of computers has revolutionized the fluid properties studies in ways as profound as their application to process design problems. 

Before we introduce our restricted partition function formalism, we need to review some salient aspects of the equilibrium statistical mechanics formalism. This will also help with the clarity and continuity of presentation. We will restrict our discussion to the canonical ensemble, but the extension to other ensembles is straightforward and trivial. 

This chapter introduces tools from statistical mechanics that are useful for analyzing the behavior of static and slowly driven granular media. (For fast dynamics, refer to the previous chapter on Kinetic Theory by Jenkins.) These tools encompass techniques used to predict emergent properties from microscopic laws, which are the analogs of calculations in equilibrium statistical mechanics based on the concept of statistical ensembles and stochastic dynamics. Included, for example, are the Edwards approach to static granular media, and coarse-grained models of... 

For static granular packings, which are end states of some protocol, statistical techniques are needed for analyzing properties such as stress transmission and porosity. I first introduce the concept and utility of statistical ensembles in the context of equilibrium statistical mechanics and then generalize the approach to static granular assemblies. 

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