Statistical mechanics canonical ensemble

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. 

We can determine the thermodynamic properties of an ideal gas using canonical ensemble statistical mechanical calculations. We follow the same steps as in the W E ideal gas. The Hamiltonian of an W T ideal... 

The grand canonical ensemble is a set of systems each with the same volume V, the same temperature T and the same chemical potential p (or if there is more than one substance present, the same set of p. s). This corresponds to a set of systems separated by diathennic and penneable walls and allowed to equilibrate. In classical thennodynamics, the appropriate fimction for fixed p, V, and Tis the productpV(see equation (A2.1.3 7)1 and statistical mechanics relates pV directly to the grand canonical partition function... 

No system is exactly unifomi even a crystal lattice will have fluctuations in density, and even the Ising model must pemiit fluctuations in the configuration of spins around a given spin. Moreover, even the classical treatment allows for fluctuations the statistical mechanics of the grand canonical ensemble yields an exact relation between the isothemial compressibility K j,and the number of molecules Ain volume V ... 

Statistical mechanics may be used to derive practical microscopic fomuilae for themiodynamic quantities. A well-known example is tire virial expression for the pressure, easily derived by scaling the atomic coordinates in the canonical ensemble partition fiinction... 

Chesnut D A and Salsburg Z W 1963 Monte Carlo procedure for statistical mechanical calculation in a grand canonical ensemble of lattice systems J. Chem. Phys. 38 2861-75... 

The canonical ensemble is the name given to an ensemble for constant temperature, number of particles and volume. For our purposes Jf can be considered the same as the total energy, (p r ), which equals the sum of the kinetic energy (jT(p )) of the system, which depends upon the momenta of the particles, and the potential energy (T (r )), which depends upon tlie positions. The factor N arises from the indistinguishability of the particles and the factor is required to ensure that the partition function is equal to the quantum mechanical result for a particle in a box. A short discussion of some of the key results of statistical mechanics is provided in Appendix 6.1 and further details can be found in standard textbooks. 

In statistical mechanics the properties of a system in equilibrium are calculated from the partition function, which depending on the choice for the ensemble considered involves a sum over different states of the system. In the very popular canonical ensemble, that implies a constant number of particles N, volume V, and temperature T conditions, the quasiclassical partition function Q is... 

This equation forms the fundamental connection between thermodynamics and statistical mechanics in the canonical ensemble, from which it follows that calculating A is equivalent to estimating the value of Q. In general, evaluating Q is a very difficult undertaking. In both experiments and calculations, however, we are interested in free energy differences, AA, between two systems or states of a system, say 0 and 1, described by the partition functions Qo and (), respectively - the arguments N, V., T have been dropped to simplify the notation ... 

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... 

In quantum statistical mechanics where a density operator replaces the classical phase density the statistics of the grand canonical ensemble becomes feasible. The problem with the classical formulation is not entirely unexpected in view of the fact that even the classical canonical ensemble that predicts equipartitioning of molecular energies, is not supported by observation. 

Limitation to ensembles that allow exchange of energy, but not of matter, with their environment is unnecessarily restrictive and unrealistic. What is required is an ensemble for which the particle numbers, Nj also appear as random variables. As pointed out before, the probability that a system has variable particle numbers N and occurs in a mechanical state (p, q) can not be interpreted as a classical phase density. In quantum statistics the situation is different. Because of second quantization the grand canonical ensemble, like the microcanonical and canonical ensembles, can be represented by means of a density operator in Hilbert space. 

Monte Carlo Methods. Although several statistical mechanical ensembles may be studied using MC methods (2,12,14), the canonical ensemble has been the most frequently used ensemble for studies of interfacial systems. In the canonical ensemble, the number of molecules (N), cell volume (V) and temperature (T) are fixed. Hence, the canonical ensemble is denoted by the symbols NVT. The choice of ensemble determines which thermodynamic properties can be computed. 

In MC methods the ultimate objective is to evaluate macroscopic properties from information about molecular positions generated over phase space. To evaluate average macroscopic properties, p, in the canonical ensemble from statistical mechanics, the following expression is used ... 

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is... 

The JE (Eq. (40)) indicates a way to recover free energy differences by measuring the work along all possible paths that start from an equihbrium state. Its mathematical form reminds one of the partition function in the canonical ensemble used to compute free energies in statistical mechanics. The formulas for the two cases are... 

In order to deal with collections of molecules in statistical mechanics, one typically requires that certain macroscopic conditions be held constant by external influence. The enumeration of these conditions defines an ensemble . We will confine ourselves in this chapter to the so-called canonical ensemble , where the constants are the total number of particles N (molecules, and, for our purposes, identical molecules), the volume V, and the temperature... 

Just as there is a fundamental function that characterizes the microscopic system in quantum mechanics, i.e., the wave function, so too in statistical mechanics there is a fundamental function having equivalent status, and this is called the partition function. For the canonical ensemble, it is written as... 

Obviously Eq. (11) can also be derived directly from equilibrium thermodynamic considerations involving equilibrium constants. A statistical-mechanical derivation4 of Eq. (11), utilizing an equilibrium (actually grand canonical) ensemble, points up the analogy between this transition and a special case of helix-coil transition, the latter being most usually treated with equilibrium ensembles. 

We take a rather simple approach to the treatment of statistical thermodynamics and the derivations that follow. A rigorous discussion requires introduction of the canonical ensemble and is beyond the scope of our immediate interest. The details can be found, for example, in textbooks on statistical mechanics [269]. The approach taken here yields all of the needed results, and is a compact introduction to the theory. 

In the following, we first describe (Section 13.3.1) a statistical mechanical formulation of Mayer and co-workers that anticipated certain features of thermodynamic geometry. We then outline (Section 13.3.2) the standard quantum statistical thermodynamic treatment of chemical equilibrium in the Gibbs canonical ensemble in order to trace the statistical origins of metric geometry in Boltzmann s probabilistic assumptions. In the concluding two sections, we illustrate how modem ab initio molecular calculations can be enlisted to predict thermodynamic properties of chemical reaction (Sections 13.3.3) and cluster equilibrium mixtures (Section 13.3.4), thereby relating chemical and phase thermodynamics to a modem ab initio electronic stmcture picture of molecular and supramolecular interactions. 

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 statistical mechanics of solutions at nonvamshing concentration is best-formulated within the grand canonical ensemble, where the number of solute molecules contained in volume Si is allowed to fluctuate. We only control the average concentration cp via the chemical potential fi.p This has great technical advantages, allowing for a very simple, analysis of the thermodynamic limit. In contrast, in the canonical ensemble, where the particle number M = Qc.p is taken as basic variable, an analysis of the thermodynamic limit is more tricky. A short discussion is given in the appendix. 

Three of the eight thermodynamic potentials for a system with one species are frequently used in statistical mechanics (McQuarrie, 2000), and there are generally accepted symbols for the corresponding partition functions V[T = A = — RTlnQ, where Q is the canonical ensemble partition function ... 

We have derived a formula for the molecular partition function by considering a system containing many molecules at equilibrium with a heat bath. We can generalize our statistical mechanics by a gedanken experiment of considering a large number of identical systems, each with volume V and number of particles N at equilibrium with the heat bath at temperature T. Such a supersystem is called a canonical ensemble. Our derivation is the same the fraction of systems that are in a state with energy Et is... 

---