Bose-Einstein statistics, permutational

As mentioned above, we assume that the molecular energy does not depend on the nuclear spin state For the initial rovibronic state nuclear spin functions available, for which the product function 4 i) in equation (2) is an allowed complete internal state for the molecule in question, because it obeys Fermi-Dirac statistics by permutations of identical fermion nuclei, and Bose-Einstein statistics by permutations of identical boson nuclei (see Chapter 8 in Ref. [3]). By necessity [3], the same nuclear spin functions can be combined with the final rovibronic state form allowed complete... 

The development of quantum theory, particularly of quantum mechanics, forced certain changes in statistical mechanics. In the development of the resulting quantum statistics, the phase space is divided into cells of volume hf. where h is the Planck constant and / is the number of degrees of freedom. In considering the permutations of the molecules, it is recognized that the interchange of two identical particles does not lead to a new state. With these two new ideas, one arrives at the Bose-Einstein statistics. These statistics must be further modified for particles, such as electrons, to which the Pauli exclusion principle applies, and the Fermi-Dirac statistics follow. 

The properties of an ideal Bose gas are entirely controlled by permutation symmetry, and the resulting Bose-Einstein statistics are obeyed by the particles. All complicating effects of interparticle interactions, which play a dominant role in determining the properties of bulk liquid He and of ( He)jy clusters, are... 

Distributions which can be derived from one another by mere permutation of the cells among themselves, or of the particles among themselves, do not, however, represent different states, but one and the same state the number of these permutations is gs nj, We thus obtain for the number of distinguishable arrangements in the sheet which is characterized by the index s, in the case of the Bose-Einstein statistics,... 

Permutations of this type have to be considered in PIMC simulations if a full account of the quantum statistics is intended in the study and required by the physical effect under consideration, which means that additional permutation moves have to be done in the simulation. In this way quantum statistics has been included in a few PIMC simulations, in particular for the study of superfluidity in He [287] and in adsorbed H2 layers [92], for the Bose-Einstein condensation of hard spheres [269], and for the analysis of... 

Systems containing more than one identical particles are invariant under the interchange of these particles. The permutations form a symmetry group. If these particles have several degrees of freedom, the group theoretical analysis is essential to extract symmetry properties of the permissible physical states. Examples include Bose-Einstein, Fermi-Dirac, Maxwell-Boltzmann statistics, Pauli exclusion principle, etc. 

Explicitly, the Pauli principle states that the wave function of an ensemble of particles with half-valued spin, which obey Fermi-Dirac quantum statistics and are therefore called/emions, must change its sign upon exchange of any two coordinates in the wave function. A fermionic many-particle wave function is then said to be antisymmetric under pair exchange of coordinates. By contrast, the wave function of a set of particles with integer spin, which obey Bose-Einstein quantum statistics and are therefore called bosons, does not change its sign upon pair permutation of any two coordinates in the function. 

If particles are indistinguishable and there are no restrictions on how many particles can occupy a cell, which is the case for most atoms and molecules, these particles are said to obey Bose-Einstein (B-E) statistics. There are (gi + N ways of arranging the sequence of numbers and letters. Since particles are indistinguishable, only the number of particles in each cell, not which particles, is important. Therefore, we must divide by redundancies. Cells are distinguishable (each corresponds to a particular position and momenta), but the sequence in which they are taken is not important. That is, the above sequence could have been written as 3bd, Ic, 2, 4ae,... without changing anything physically. Since there are cells, the sequence can be permuted gi ways that are redundant. 

---