Equilibrium Statistical Mechanics Using Ensembles

However, a statistical treatment of the elections in a metal using the methods that we have used so far would certainly fail. All electrons wonld be in the lowest state when T = 0, but if this is the case, the fundamental Pauli principle stating that electrons are fermions and form an antisymmetric wave function would not be satisfied. As we have seen, the Pauli principle implies that each spin orbital can be occupied by only one electron. 

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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... 

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. 

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. 

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... 

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... 

The above realization of the abstract mesoscopic equilibrium thermodynamics is called a Canonical-Ensemble Statistical Mechanics. We shall now briefly present also another realization, called a Microcanonical-Ensemble Statistical Mechanics since it offers a useful physical interpretation of entropy. 

Next, we wish to find the relation between A (P ) and the experimental Helmholtz energy of solvation of the molecule 5, irrespective of its conformation. We carry out the derivation in two steps, and for convenience we use the T, L, N ensemble. Suppose first that s can attain only two conformations A and B, say the cis and trans conformations of a given molecule at equilibrium. The PCP is the change in the Helmholtz energy for placing an A molecule at a fixed position /. The corresponding statistical-mechanical expression is... 

Linear response theory [152] is perfectly suited to the study of fluid structures when weak fields are involved, which turns out to be the case of the elastic scattering experiments alluded to earlier. A mechanism for the relaxation of the field effect on the fluid is just the spontaneous fluctuations in the fluid, which are characterized by the equilibrium (zero field) correlation functions. Apart from the standard technique used to derive the instantaneous response, based on Fermi s golden rule (or on the first Bom approximation) [148], the functional differentiation of the partition function [153, 154] with respect to a continuous (or thermalized) external field is also utilized within this quantum context. In this regard, note that a proper ensemble to carry out functional derivatives is the grand ensemble. All of this allows one to gain deep insight into the equilibrium structures of quantum fluids, as shown in the works by Chandler and Wolynes [25], by Ceperley [28], and by the present author [35, 36]. In doing so, one can bypass the dynamics of the quantum fluid to obtain the static responses in k-space and also make unexpected and powerful connections with classical statistical mechanics [36]. 

The present article presents an introduction to the path integral formulation of quantum dynamics and quantum statistical mechanics along with numerical procedures useful in these areas and in electronic structure theory. Section 2 describes the path integral formulation of the quantum mechanical propagator and its relation to the more conventional Schrddinger description. That section also derives the classical limit and discusses the connection with equilibrium properties in the canonical ensemble, Numerical techniques are described in Section 3. Selective chemical applications of the path integral approach are presented in Section 4 and Section 5 concludes. 

Let be a function of coordinates only, so a point in is specified by [3(A(ab - 1) - 1] coordinates and 3(A(ab - ) momenta. Identify the missing coordinate as the reaction coordinate s (so s becomes a coordinate normal to the hypersurface), and identify the momentum conjugate to s as p. Let C denote the [6(A(ab — 1) - 2]-dimensional hyperface in in which ps = 0. Assume that the % region of phase space is populated according to a Boltzmann equilibrium distribution then Liouville s theorem of classical statistical mechanics shows it will evolve into a Boltzmann equilibrium distribution at and hence also at C. Consider the one-way flux of this equilibrium ensemble of phase points through in the 5 —> P direction. This flux may be calculated quite generally, and using this calculation plus equation (2) yields... 

In order to theoretically understand the Monte Carlo method, we need to use the concepts of statistical mechanics. These concepts are developed in detail in books of fundamental statistical mechanics [16]. This formalism guides us in the task of calculating various quantities and/or properties that describe the behavior of the system. This is accomplished from the partition function that specifies the statistical properties of the system in thermodynamic equilibrium. There are different classes of partition functions, each corresponding to a determined statistical ensemble. The canonical ensemble is applied for a system of N particles in a volume V and temperature T. These magnitudes have fixed values and heat exchange is allowed. For this ensemble the corresponding partition function is ... 

The energy and entropy, which are related in this equation to the molecular partition function, are statistical entities, defined by particles which do not interact except to maintain the equilibrium conditions. To obtain similar relationships for real systems, it is necessary to apply statistical mechanics to the calculation of the thermodynamic entities, which correspond to the molar quantities of particles, or that is N approaches 1 this treatment it is convenient to use the canonical ensemble already discussed and presented in Figure n.l. This ensemble consists of a very large number of systems, N, each containing 1 mol of molecules and separated from the others by diathermic walls, which allow heat conduction but do not allow particles to pass. The set of all the systans is isolated from the outside and has a fixed energy E, which is the energy of the canonical ensemble. 

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