Ensembles and Thermodynamic Functions

In Chapter 4, we saw statistical methods to process collections of quantum or non-quantum objects. Now, we will assume that not only the entropy as the Boltzmaim theorem reveals, but all the thermodynamic properties of a constituent can be accurately obtained by considering that this constituent is in fact a collection of many identical qnantnm objects. Similarly, a mixture with several constituents will be considered as many collections of quantum objects. So, we thus move onto the concept of nonquantum ensembles with an infinite number of system reproductions under well-defined conditions a canonical ensemble. 

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Use of the Grand Ensemble has the disadvantage that all the calculated thermodynamic functions are dependent on z, V, and T. However, Mayer s programme can be adapted to the canonical ensemble (N, V, T). 

The organization of this chapter is as follows. In Sect. 5.1 we present the basic formalism and work out the Feynman rules for the grand canonical ensemble. Diagrammatic representations valid in the thermodynamic limit are derived for both thermodjmamic quantities and correlation functions. The proof of the Linked Cluster Theorem is given in Appendix A 5.1. Section 5.2... 

At first glance, the averaged model would appear to serve most researchers who are looking for a molecular model to help them explain the function of the molecule and rationalize other chemical, spectroscopic, thermodynamic, and kinetic data. On the other hand, you might think that the ensemble and distance-restraint files are of most use to those working to improve structure determination techniques. There are good reasons however, for all researchers to look carefully at the ensemble, as discussed in the next section. 

If a system contains two types of species, but the membrane is permeable only to species number 1, the natural variables for the system are T, K //, and N2, where N2 is the number of molecules of type 2 in the system. The thermodynamic potential for this system containing two species is represented by U[T, //,]. The corresponding ensemble is referred to as a semigrand ensemble, and the semigrand partition function can be represented by P(71 K /q, N2). The thermodynamic potential of the system is related to the partition function by... 

In real situations surface and volume changes are often made with systems that are at equilibrium with their environment, characterized by a set of chemical potentials p, rather than keeping In ] fixed, as in [2.2.7 and 8j. In other words, area changes in open systems are considered. In statistical thermodynamics the conversion from closed to open implies the transition from the canonical to the grand canonical ensemble. The characteristic function of the latter is nothing other than the sum of the bulk and surface mechemical work terms (see [1.3.3.12] and [I.A6.23D which are the quantities of interest ... 

A system that is constrained to have a constant number of molecules, volume, and temperature constitutes a canonical ensemble. The thermodynamic properties of the system can be calculated from the corresponding partition function, Q N, V, T) [24]. For the adsorbed phase the partition function can be written as... 

Since the goal of a simulation is to calculate properties of a particular macroscopic thermodynamic system from the configurations of many microscopic states, the final consideration in model selection is the choice of the thermodynamic state of the system. This can be done by fixing three thermodynamic variables (such as pressure, P, temperature T, internal energy, E) and designing the simulation so that these functions remain constant throughout the calculation. The choice of the thermodynamic state defines the ensemble and since the ensemble is chosen based upon the properties of interest, more details about the different ensembles will be given in Section 4.2.4.3. 

V, N, and T are held constant. In this way, the configurational integral Z is calculated. This immediately allows one to calculate the partition function Q for the ensemble (equation (2.2.31)) and thus the thermodynamic functions for the system. In addition, the pair correlation function for the fiuid is obtained in this calculation. 

An important partition function can be derived by starting from Q (T, V, N) and replacing the constant variable AT by fi. To do that, we start with the canonical ensemble and replace the impermeable boundaries by permeable boundaries. The new ensemble is referred to as the grand ensemble or the T, V, fi ensemble. Note that the volume of each system is still constant. However, by removing the constraint on constant N, we permit fluctuations in the number of particles. We know from thermodynamics that a pair of systems between which there exists a free exchange of particles at equilibrium with respect to material flow is characterized by a constant chemical potential fi. The variable N can now attain any value with the probability distribution... 

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