Average occupation numbers

We will denote by n, the distribution of particles that consists of a particular set of values n, that determine the occupation of the levels e,. The number of states of the entire system corresponding to a particular distribution , will be denoted as W( n, ). This number is proportional to the volume in phase space occupied by this particular distribution ,. The most probable distribution must correspond to the largest volume in phase space. That is, if we denote the most probable distribution of particles in microscopic states by [f], then W( 7i ) is the maximum value of W. Let us suppose that the degeneracy or multiplicity of the microscopic state labeled i is g,, that is, there are g, individual microscopic states with the same energy ,. With these definitions, and with the restriction of constant N and E, we can now derive the average occupation of level i, which will be the same as /,-. There are three possibilities, which we examine separately. 

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Flere the zero point energy is ignored, which is appropriate at reasonably large temperatures when the average occupation number is large. In such a case one can also replace the sum over by an integral. Each of the triplet n can take the values 0, 1, 2,. . ., co. Thus the sum over can be replaced by an... 

From equation (A2.2.145). the average occupation number of an ideal Bose gas is... 

The second property is that each term of the PF is proportional to the probability of occurrence of the particular state it represents when the system is at equilibrium. We shall use mainly the second property of the PF. The next section is devoted to this aspect of the theory. Once we have the probabilities of all possible events we can compute average quantities pertaining to the system at equilibrium. Of these, the average occupation number, or the binding isotherm, will be the central quantity to be examined and analyzed in this book. 

Since site a can be either empty [with probability 1 - P(a)] or occupied [with probability P(a)], 9 is the average occupation number for the site a. Clearly, 0 S 6 1. When forming the sumn in Eq. (2.1.7) or (2.1.8), we sum over all average quantities 0,. and obtain the average occupation number for the entire molecule. Clearly 0[Pg.28]

In order to device an effective averaging procedure one still has to specify formulas for the average occupation numbers n . 

The conservation of the number of particles introduces a new Fermi level and helps to define the averaged occupation numbers, e g., in the HF case ... 

The averaged occupation numbers h, are a formal ingredient in Strutinsky s averaging method. Combining Eqs (15) and (17) yields their explicit form ... 

Introducing the averaged occupation numbers nf, n°, nj from Eqn (17) into Eqn (39), one obtains the averaged parts of the above ... 

In order to determine the averaged occupation numbers n,. for the considered atoms (3 < Z 30), we have solved numerically the system of highly non-linear equations composed from the particle number conservation condition (17) and the plateau condition (24). It was found that the condition (25) is fulfilled with desired accuracy (with less than 1% discrepancy for 5iEhfr) for values of M > 35, and the roots Yo and X found at M = 35 allow a correct determination of the n. ... 

Next suppose at t = 0 the bath is in thermal equilibrium with the temperature j3 1, while S is in level N0. The average occupation number of the n-th bath oscillator is given by... 

Alternatively, we assume that within each plane I parallel to the solid substrates the occupation number at eacli lattice site can be replaced by an average occupation number for the entire plane. On account of the symmetrybreaking nature of the solid substrate, these average occupation numbers will generally vary between planes that is, they will change with /. Hence, we introduce the total local density... 

Figure 45. The dependence of the pulse size on the average occupancy number of (3-naphthol. 50pM Bromo Cresol Green 500pJW micellar concentration and varying (3-naphthol. Data taken from Table VII.

Specifically consider the average occupation number of the /th state ... 

Stoner and Wohlfarth were able to derive a self-consistent equation for the spontaneous magnetization Mo in the molecular field approximation (MFA) using the model described above. Mo is then written in terms of the average occupation numbers (n ) for electrons of spin cr ... 

It should be noted that the solubilisate will be distributed among the water pools according to the Poisson distribution law [129]. That is, the probability Pk of having/ monomers per water pool such that the average occupancy number is av is given by... 

It would be expected that for a given concentration of water pools, the average occupancy number would increase with an increase in reactant concentration. On the other hand, for a given reactant concentration, an increase in the concentration of the water pools... 

We noted that the energy terms (8.8) and (8.12) of order zero and two do not depend on the separation of particles 1 and 2. Accordingly, we calculated the free energy of interaction from the fourth order energy expression (8.16). Considering this term, it is certainly consistent to calculate the average occupation numbers of the excited states from Fermi and Bose statistics. The error is at worst of order six in the interaction parameter. 

The renormalization of the quasi-photon energies entails a renormalization of the quasi-electron energies. The renormalized quasi-electron Hamiltonian results from the total Hamiltonian (8.31) by replacing the photon occupation numbers + i by coth hcoJ2kT). These are just the average occupation numbers given by the Bose distribution (16). 

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