The grand canonical ensemble

By analogy with Eqs. (2.117) and (2.118), it follows that the probability density in the grand canonical ensemble is given by 

A random change of the number of molecules accommodated by the system, that is, Nn N  

As we also point out in Appendix E.1.2, the Chapman-KolmogorofF equation is derived under the assumption of small changes in the random processes represented by y. Hence, the Metropolis algorithm proceeds in two consecutive steps, namely. 

Henceforth, the sequence of N displacement attempts followed by creation/destruction attempts in grand canonical ensemble MC (GCEMC) simulations will be referred to as a GCEMC cycle.  

For step 1 of this cycle, we notice that iV i = Nn — N remains constant between members n — 1 and n in the Markov chain. Hence, we compute (see Eq. (E.20)  

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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... 

The coimection between the grand canonical ensemble and thennodynamics of fixed (V, T, p) systems is provided by the identification... 

In the grand canonical ensemble, the number of particles flucPiates. By differentiating log E, equation (A2.2.121) with respect to Pp at fixed V and p, one obtains... 

The grand canonical ensemble is a collection of open systems of given chemical potential p, volume V and temperature T, in which the number of particles or the density in each system can fluctuate. It leads to an important expression for the compressibility Kj, of a one-component fluid ... 

It was shown in section A2.3.3.2 that the grand canonical ensemble (GCE) PF is a generating fiinction for the canonical ensemble PF, from which it follows that correlation fiinctions in the GCF are just averages of the fluctuating numbers N and N - 1... 

Orkoulas G and Panagiotopoulos A Z 1999 Phase behavior of the restricted primitive model and square-well fluids from Monte Carlo simulations in the grand canonical ensemble J. Chem. Phys. 110 1581... 

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 ... 

The grand canonical ensemble corresponds to a system whose number of particles and energy can fluctuate, in exchange with its surroundings at specified p VT. The relevant themiodynamic quantity is the grand potential n = A - p A. The configurational distribution is conveniently written... 

The definition of the distribution function given above is valid in the canonical ensemble. This means that N is finite. Of course, N will, in general, be very large. Hence, g(ri,..., r/,) approaches 1 when aU the molecules are far apart but there is a term of order X/N that sometimes must be considered. This problem can be avoided by using the grand canonical ensemble. We will not pursue this point here but do wish to point it out. 

To test the results of the chemical potential evaluation, the grand canonical ensemble Monte Carlo simulation of the bulk associating fluid has also been performed. The algorithm of this simulation was identical to that described in Ref. 172. All the calculations have been performed for states far from the liquid-gas coexistence curve [173]. 

However, if one focuses on the adsorption of a fluid in heterogenous matrices [32,33] and/or on the fluctuations in an adsorbed fluid, it is inevitable to perform developments similar to those above in the grand canonical ensemble. Moreover, this derivation is of importance for the formulation of the virial route to thermodynamics of partially quenched systems. For this purpose, we include only some basic relations of this approach. 

By definition, an observable property of a partly quenched system,/pg, in the grand canonical ensemble is obtained as... 

The theory presented in this section is based on the grand canonical ensemble formulation, which is perfectly well-suited for the description of confined systems. Undoubtedly, in the case of attractive-repulsive interparticle forces unexpected structural and thermodynamic behavior in partly... 

The main idea of a lattice model is to assume that atomic or molecular entities constituting the system occupy well-defined lattice sites in space. This method is sometimes employed in simulations with the grand canonical ensemble for the simulation of surface electrochemical proceses. The Hamiltonians H of the lattice gas for one and two adsorbed species from which the ttansition probabilities 11 can be calculated have been discussed by Brown et al. (1999). We discuss in some detail MC lattice model simulations applied to the electrochemical double layer and electrochemical formation and growth two-dimensional phases not addressed in the latter review. MC lattice models have also been applied recently to the study the electrox-idation of CO on metals and alloys (Koper et al., 1999), but for reasons of space we do not discuss this topic here. 

With applications to protein solution thermodynamics in mind, we now present an alternative derivation of the potential distribution theorem. Consider a macroscopic solution consisting of the solute of interest and the solvent. We describe a macroscopic subsystem of this solution based on the grand canonical ensemble of statistical thermodynamics, accordingly specified by a temperature, a volume, and chemical potentials for all solution species including the solute of interest, which is identified with a subscript index 1. The average number of solute molecules in this subsystem is... 

Let us illustrate this procedure with the grand-canonical ensemble, and take the scenario in which we desire to achieve a uniform distribution in particle number N at a given temperature. In the weights formalism, we introduce the weighting factor r/(/V) into the microstate probabilities from (3.31) so that... 

Here the notation [GC y indicates that the system to be treated is only the inner-shell volume, and the material enclosed is described by an ensemble of fluctuating composition - as with the grand canonical ensemble - under the influence of the molecular-field p. With longer-ranged interactions, a correction for those... 

In the canonical example, we could estimate the free energy difference between two runs by examining the overlap in their probability distributions. Similarly, in the grand canonical ensemble, we can estimate the pressure difference between the two runs. If the conditions for run I arc f//1. V. > ) and for run 2 (po, VjK), then... 

We can, therefore, let /cx be the subject of our calculations (which we approximate via an array in the computer). Post-simulation, we desire to examine the joint probability distribution p(N, U) at normal thermodynamic conditions. The reweighting ensemble which is appropriate to fluctuations in N and U is the grand-canonical ensemble consequently, we must specify a chemical potential and temperature to determine p. Assuming -7CX has converged upon the true function In f2ex, the state probabilities are given by... 

Chapters 10 and 11 cover methods that apply to systems different from those discussed so far. First, the techniques for calculating chemical potentials in the grand canonical ensemble are discussed. Even though much of this chapter is focused on phase equilibria, the reader will discover that most of the methodology introduced in Chap. 3 can be easily adapted to these systems. Next, we will provide a brief presentation of the methods devised for calculating free energies in quantum systems. Again, it will be shown that many techniques described previously for classical systems, such as PDT, FEP and TI, can be profitably applied when quantum effects are taken into account explicitly. 

The starting point is the definition of the partition function, S, in the grand canonical ensemble ... 

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