Distribution Functions in the Grand Canonical Ensemble

MOLECULAR DISTRIBUTION FUNCTIONS IN THE GRAND CANONICAL ENSEMBLE 

We recall that the probability of finding a system in the T, F, fi ensemble with exactly N particles is  

The conditional th-order MDF of finding the configuration X , given that the system has N particles, is 

The bar over denotes average in the T, V, p ensemble. Recalling that 

The normalization condition for p X ) is obtained from (5.7.3) by integrating over all the configurations X  

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Molecular distribution functions in the grand canonical ensemble... 

Introduction 88. 2. Distribution Functions in the Canonical Ensemble 88. 3, Distribution Functions in the Grand Canonical Ensemble 90. 4. Idulticomponent... 

The distribution functions (ri... r ) defined in 2 refer to systems with well defined volume V, temperature T and total number of molecules iV. It is appropriate to define also distribution functions in open systems corresponding to given values of V, T and of the chemical potential /h. As we have seen, open sjretems are described by the grand canonical ensemble (Ch. II, 3). Let us call />< > (ri... r ) the distribution function in the grand canonical ensemble. The relation between (n. .. r ) and (ri... r ) is immediately deduced from the definition of the grand ensemble (we have introduced the index N in to remember that it refers to a well defined total number N of molecules.)... 

This statistical mechanical expression for surface tension depends explicitly on the potentials of inteimolecular force and molecular distribution functions. Upon recognition that the two-phase system under consideration is thermodynamically open, it follows that the distribution functions must be represented in the grand canonical ensemble. Thus, the dependence of y on temperature, T, and chemical potentials, enters through the implicit dependence of the distribution func-... 

MOLECULAR DISTRIBUTION FUNCTIONS (MDF) IN THE GRAND CANONICAL ENSEMBLE... 

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. 

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... 

The Kirkwood—Buff (KB) theory of solution (often called fluctuation theory) employs the grand canonical ensemble to relate macroscopic properties, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volnmes, to microscopic properties in the form of spatial integrals involving the radial distribution function. This theory allows one to obtain information regarding some microscopic characteristics of mnlti-component mixtures from measurable macroscopic thermodynamic quantities. However, despite its attractiveness, the KB theory was rarely used in the first three decades after its publication for two main reasons (1) the lack of precise data (in particular regarding the composition dependence of the chemical potentials) and (2) the difficulty to interpret the results obtained. Only after Ben-Naim indicated how to calculate numerically the Kirkwood—Buff integrals (KBIs) for binary systems was this theory used more frequently. 

The KB theory of solution [15] (often called fluctuation theory of solution) employed the grand canonical ensemble to relate macroscopic properties, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility and the partial molar volumes to microscopic properties in the form of spatial integrals involving the radial distribution function. 

One can prove that in thermal equilibrium, in a grand canonical ensemble (i.e., volume, chemical potential, and temperature are fixed), the grand canonical free energy Q(p(r)) of a system can be written as a functional of the one-body density p(r) alone, which will depend on the position r in inhomogeneous systems. The density distribution peq(r) which minimizes the grand potential functional is the equilibrium density distribution. This statement is the basis of the equilibrium density functional theory (DFT) for classical fluids which has been used with great... 

We begin our discussion with a bit of notation. The n body distribution function in an open system described by the grand canonical ensemble is,... 

Computationally, polydispersity is best handled within a grand canonical (GCE) or semi-grand canonical ensemble in which the density distribution p(a) is controlled by a conjugate chemical potential distribution p(cr). Use of such an ensemble is attractive because it allows p(a) to fluctuate as a whole, thereby sampling many different realizations of the disorder and hence reducing finite-size effects. Within such a framework, the case of variable polydispersity is considerably easier to tackle than fixed polydispersity The phase behavior is simply obtained as a function of the width of the prescribed p(cr) distribution. Perhaps for this reason, most simulation studies of phase behavior in polydisperse systems have focused on the variable case [90, 101-103]. 

During World War II, little or no work was done on solution theory, but after the war, activity began again. Now, the emphasis of many theories began to fall on the properties and usefulness of molecular distribution functions, in particular the pair correlation function. This was due, in part, I believe, to the thesis of Jan de Boer (De Boer 1940,1949). As an aside, I once asked J. E. Mayer why he used the canonical ensemble in his early work on statistical mechanics and the grand ensemble in his later works. He replied, Oh, I switched after I read de Boer s thesis and saw how easy the grand ensemble made things. De Boer s work was for pure fluids, not solutions, and other authors, in particular John G. Kirkwood (Kirkwood 1935), also developed the correlation function method. 

A third approach is to inject particles based on a grand canonical ensemble distribution. At each predetermined molecular dynamics time step, the probability to create or destroy a particle is calculated and a random number is used to determine whether the update is accepted (the probability for both the creation and the destruction of a particle must be equal to ensure reversibility). The probability function depends on the excess chemical potential and must be calculated in a way that is consistent with the microscopic model used to describe the system. In the work of Im et al., a primitive water model is used, and the chemical potential is determined through an analytic solution to the Ornstein-Zernike equation using the hypemetted chain as a closure relation. This method is very accurate from the physical viewpoint, but it has a poorer CPU performance compared with simpler schemes based on... 

We performed Monte Carlo in the constant-pressure, semi-grand-canonical ensemble of the type described in [52]. In such a simulation it is not possible to impose the size distribution of the particles directly, but the size distribution can be controlled through variation of the imposed activity-ratio distribution function. In our simulations we imposed a Gaussian activity distribution and a typical particle size distribution function is shown in Fig. 14. 

Fig. 14. Typical particle size distribution functions from Monte Carlo simulations in the constant-pressure, semi-grand-canonical ensemble in a bulk liquid and solid. At that pressure the volume fraction of the liquid is = 0.5775 at a polydispersity of 10%. The volume fraction of the solid in coexistence with the liquid is = 0.6196 and has a polydispersity of 4.2%...

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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