Grand canonical distribution

The T-P ensemble distribution is obtained in a maimer similar to the grand canonical distribution as (quantum mechanically)... 

The grand canonical distribution of an ideal mixture of chemical compounds is... 

Yet the grand canonical distribution (3.1) does not correctly describe the fluctuations of the rij in a closed vessel. This is obvious because it assigns a... 

At relatively high temperatures typical for CsHg oxidation, oxygen diffusion is rapid compared to other steps (for the Arrhenius parameters for this process, see [21]). In this case, the relation between and 6q is given by the grand canonical distribution, i.e., one should have... 

The Ursell cluster functions are usually defined by their relationship with the grand canonical distribution functions pi l.K)... 

Statistical theories of thermodynamics yield many correct and practical results. For example, they yield the canonical and grand canonical distributions for petit and grand systems, respectively these distributions, which were proposed by Gibbs, have been shown by innumerable comparisons with experiments to describe accurately the properties of quasistable states. Again, they predict the equality of temperatures of systems in mutual stable equilibrium, the Maxwell relations, and the Gibbs equation. 

The only equilibrium states that are stable are those for which the density operator yields the canonical distribution if the system is a petit system, and the grand canonical distribution if the system is a grand system. 

A set of grand canonical distribution functions can then be defined for species... 

Some historical remarks. The physical assumption adopted by van Kampen (1976) is that the grand-canonical distribution of the particle number of an ideal mixture is Poissonian. Based on this — strongly restrictive — assumption, and utilising the conservation of the total number of atoms the stationary distribution can be obtained. This stationary distribution can be identified with the stationary solution of the master equation, and it is not Poissonian in general, even for large systems. 

In the books of classical statistical mechanics, the properties of the ideal gas have been evaluated, so we can focus on the configurational contribution. With the Monte Carlo method [17-19], we explore the configuration phase subspace according to the canonical or grand canonical distributions. The corresponding trajectory is essentially the projection of the phase space onto the subspace of coordinates, becoming independent of the subspace of moments. 

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. 

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). 

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... 

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... 

Abbreviations MD, molecular dynamics TST, transition state theory EM, energy minimization MSD, mean square displacement PFG-NMR, pulsed field gradient nuclear magnetic resonance VAF, velocity autocorrelation function RDF, radial distribution function MEP, minimum energy path MC, Monte Carlo GC-MC, grand canonical Monte Carlo CB-MC, configurational-bias Monte Carlo MM, molecular mechanics QM, quantum mechanics FLF, Hartree-Fock DFT, density functional theory BSSE, basis set superposition error DME, dimethyl ether MTG, methanol to gasoline. 

As a convenient starting point for the model, the grand canonical partition function is developed from the canonical partition function, to incorporate the above assumptions. The canonical partition function is written as the product of three factors the water lattice, the guest distribution within the cages, and the states of the guest molecules themselves assuming that they behave as ideal gas molecules, as follows ... 

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