Function grand probability

To calculate the stochastic time evolution of the system, the key element is the grand probability function... 

For simple cases with populations of small sizes, one can express the Kolmogorov equations in a matrix form. The elements of the grand probability function Pn 1,...,nm (t) can be considered in a vector form ... 

From the very mode of construction of the grand partition function the probability that a set of r sites congruent with an y-mer is occupied by an y-mer molecule is obtained by considering in the summation (3.6.4) only those terms which correspond to the occupation of the considered y sites by an y-mer molecule and dividing by the total grand partition function. Therefore... 

The reader who is less familiar with the theory of grand partition functions may directly proceed to Eqs. 12a and 13. The physical basis of these formulas and the significance of the quantities CK% will then become apparent in the subsequent paragraph is the vapor pressure (or fugacity) of solute K and y i is the probability of finding a K molecule in a cavity of type i. 

The same approach applies to alternate partition functions the WL output is used directly in the macroscopic probability scheme. For example, in the previous grand canonical scenario, the WL simulation would yield which would be substituted directly into the macrostate probabilities as Q exp(J ) for subsequent results generation. 

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... 

A grand jury s function is to determine whether there is probable cause to believe that a certain person(s) or company(ies) have committed a federal offense.24 Prosecutors are permitted to appear before the grand jury and, in practice, conduct the grand jury proceedings. In general, the prosecutor is the one who makes the decision... 

E. A. Guggenheim. Grand partition functions and so-called thermodynamic probability . J. Chem. Phys., 7 103-107,1939. 

The shape of a zeolite sorption uptake isotherm, a quantitation of the amount of a given sorbate taken up as a function of its partial pressure in the gas phase in equilibiitun with the zeolite sorbent, depends both on the zeolite sorbate interaction and on the sorbate - sorbate interactions. Simulation of such isotherms for one or more sorbates is accomplished by the Grand Canonical Monte Carlo method. Additional to the molecular reorientation and movement attempts is a particle creation or annihilation, the probability of which scales with the partial pressure [100,101]. This procedure thus simulates the eqmlibrium between the sorbed phase in the zeolite and an infinite gas / vapor bath. Reasonable reproduction of uptake isotherms for simple gases has been achieved for a small number of systems (e.g. [100,101]), and the molecular simulations have, for example, explained at a molecular level the discontinuity observed in the Ar - VPI-5 isotherm. 

In statistical mechanics we do not measure observables directly. Instead we observe an average over all possible values. The averaging is done by means of a probability distribution function, which in classical mechanics is averaged over all of phase space. Let us compare an equilibrium ensemble with grand potential Q, and an arbitrary nearby ensemble prepared by a small perturbation, AQ. Let the equilibrium probability distribution function be f and that for the nearby... 

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... 

Here S is the grand partition function and the equilibrium distribution is characterized by the probabilities PN.i. Let us consider the probabilities Pn, i defining the nonequilibrium distributions of the same system, at temperature T, generated through small perturbation of the bath parameters, viz. p and v(f). The average over any such nonequilibrium ensemble may be written as... 

The potential for reduced size and complexity is important if BN models are to be applied on a grand scale, modelling the complete set of loss events faced by a financial institution. Because each node in the network needs to be assigned a (conditional) density or probability mass function a simple structure may provide more efficient modelling as the required input is reduced. Hence the model suggested in this paper may potentially reduce the effort needed in constructing the BN model. 

W probability of state (I chemical potential e energy levels p vibration frequency S grand partition function a symmetry factor... 

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... 

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