Canonical ensemble grand

We consider the special choice X = A. In this case we speak of the grand-canonical 

Insertion of Eq. (5.119) into the Gibbs entropy equation yields 

This equation may be transformed using various thermodynamic equations. Eirst we apply the Gibbs-Duhem equation(2.168), i.e. djx = V/ ]SI))dP T, to obtain 

Here (N) is identical to the thermodynamic particle number N. Using Eqs. (A.2) and (2.6) we find 

In a normal situation, i.e. Kp Ikj is finite, we see immediately that the right side vanishes as N) approaches infinity. In thecase of E) Eq. (5.68) remains 

Comparison of Derivations of the Partition Function Earlier Derivation Canonical Ensemble 

N = total number of particles N = total number of subsystems 

Total energy E = Ej nj8i Total energy = Ei HjEj 

We need three Lagrangean multipliers, a, p, and y. to satisfy the three conditions at maximization of In Q(n). The result is analogous to Equations 5.5 through 5.15. The number of systons in state) with the number of particles N is 

Using of Equations 5.100 and 5.101, we may generalize the equations that we had derived earlier for the microcanonical ensemble. The average energy, for example, is 

---

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... 

In a canonical ensemble, the system is held at fixed (V, T, N). In a grand canonical ensemble the (V, T p) of the system are fixed. The change from to p as an independent variable is made by a Legendre transfomiation in which the dependent variable, the Flelmlioltz free energy, is replaced by the grand potential... 

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... 

Thennodynamics of ideal quantum gases is typically obtained using a grand canonical ensemble. In principle this can also be done using a canonical ensemble partition function, Q =. exp(-p E ). For the photon and... 

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. 

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... 

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... 

Yoon K, Chae D G, Ree T and Ree F H 1981 Computer simulation of a grand canonical ensemble of rodlike molecules J. Chem. Phys. 74 1412-23... 

If a confined fluid is thermodynamically open to a bulk reservoir, its exposure to a shear strain generally gives rise to an apparent multiplicity of microstates all compatible with a unique macrostate of the fluid. To illustrate the associated problem, consider the normal stress which can be computed for various substrate separations in grand canonical ensemble Monte Carlo simulations. A typical curve, plotted in Fig. 16, shows the oscillatory decay discussed in Sec. IV A 2. Suppose that instead... 

M. Schoen. Taylor-expansion Monte Carlo simulations of classical fluids in the canonical and grand canonical ensembles. J Comput Phys 775 159-171, 1995. 

T. Gruhn, M. Schoen. A grand canonical ensemble Monte Carlo study of confined planar and homeotropically anchored Gay-Berne films. Mol Phys 95 681-692, 1998. 

M. Thommes, G. H. Findenegg, M. Schoen. Critical depletion of a pure fluid in controlled-pore glass. Experimental results and grand canonical ensemble Monte Carlo simulation. Langmuir 77 2137-2142, 1995. 

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]. 

Let us consider a simple model of a quenched-annealed system which consists of particles belonging to two species species 0 is quenched (matrix) and species 1 is annealed, i.e., the particles are allowed to equlibrate between themselves in the presence of 0 particles. We assume that the subsystem composed of 0 particles has been a usual fluid before quenching. One can characterize it either by the density or by the value of the chemical potential The interparticle interaction Woo(r) does not need to be specified for the moment. It is just assumed that the fluid with interaction woo(r) has reached an equlibrium at certain temperature Tq, and then the fluid has been quenched at this temperature without structural relaxation. Thus, the distribution of species 0 is any one from a set of equihbrium configurations corresponding to canonical or grand canonical ensemble. We denote the interactions between annealed particles by Un r), and the cross fluid-matrix interactions by Wio(r). 

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... 

A MC study of adsorption of living polymers [28] at hard walls has been carried out in a grand canonical ensemble for semiflexible o- 0 polymer chains and adsorbing interaction e < 0 at the walls of a box of size C. A number of thermodynamic quantities, such as internal energy (per lattice site) U, bulk density (f), surface coverage (the fraction of the wall that is directly covered with segments) 9, specific heat C = C /[k T ]) U ) — U) ), bulk isothermal compressibility... 

---