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The permutation group of acts on this space by permuting the entries. By definition the expectation values for an observable a {\displaystyle a} of n {\displaystyle n} indistinguishable particles should be invariant under these permutation.
According to the first meaning of permutation, each of the six rows is a different permutation of three distinct balls. In mathematics, a permutation of a set can mean one of two different things: an arrangement of its members in a sequence or linear order, or; the act or process of changing the linear order of an ordered set. [1]
In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number.
Combinations and permutations in the mathematical sense are described in several articles. Described together, in-depth: Twelvefold way; Explained separately in a more accessible way: Combination; Permutation; For meanings outside of mathematics, please see both words’ disambiguation pages: Combination (disambiguation) Permutation ...
It can be used to solve many simple counting problems, such as how many ways there are to put n indistinguishable balls into k distinguishable bins. [4] The solution to this particular problem is given by the binomial coefficient ( n + k − 1 k − 1 ) {\displaystyle {\tbinom {n+k-1}{k-1}}} , which is the number of subsets of size k − 1 that ...
In classical physics, Maxwell–Boltzmann statistics is used to describe particles that are identical and treated as distinguishable. For both Bose–Einstein and Maxwell–Boltzmann statistics, more than one particle can occupy the same state, unlike Fermi–Dirac statistics.
If the gas particles are distinguishable, closing the doors will not return the system to its original state – many of the particles will have switched containers. There is a freedom in defining what is "ordered", and it would be a mistake to conclude that the entropy has not increased.
This is just the multinomial coefficient, the number of ways of arranging N items into k boxes, the l-th box holding N l items, ignoring the permutation of items in each box. Now, consider the case where there is more than one way to put N i {\displaystyle N_{i}} particles in the box i {\displaystyle i} (i.e. taking the degeneracy problem into ...