Indiscernible objects

Microscopic modeUng is based on the statistics of collections of discernible or indiscernible objects, the introduction of canonical and molecular partition functions and the use of data about the molecule obtained from spectroscopy. Chapters 3, 4, 5 and 6 will develop these concepts and follow onto microscopic modeling. 

Fermi-Dirac quantum statistics applies for non-localized quantum objects, meaning indiscernible objects with fractional spin (some molecules and ions, and electrons). The distribution of the objects obeys the expression ... 

The second parameter refers to the indifference threshold (Qk), which defines the range of variation of each criterion for which it is not possible for the decision-maker to favor the criterion of one solution over the corresponding criterion of another solution. It therefore represents the range of values over which two objective functions are indiscernible. 

In this section we will consider collections of indiscernible, and hence non-localized, objects. 

In this case, we can only distribule N objects irrlo the groups tiy, 2, tij in one way since they are indiscernible. Thert, the identical objects are distributed between g, energy states and represented byg, cases (Figure 4.2) and are therefore separated by g, - 1 bars. We can swap objects and g, -lbars, i.e. , +g, -1 objects, and corrsequently ertsme ( + g, -l) permutations. 

The indiscemibility of objects and bars remains to be discussed. If a permutation of objects is carried out when the bars and the population of cases are fixed, nothing changes as they are indiscernible. As there are k, ... 

Having counted the objects, as they are indiscernible, we can swap them in any form without changing anything. We must therefore divide the previous number by k, , i.e. for the distribution of n, objects in g, cases ... 

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