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The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances; see partition function (mathematics) for
A partition in which no part occurs more than once is called strict, or is said to be a partition into distinct parts. The function q(n) gives the number of these strict partitions of the given sum n. For example, q(3) = 2 because the partitions 3 and 1 + 2 are strict, while the third partition 1 + 1 + 1 of 3 has repeated parts.
The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition of a partition function in statistical mechanics. It is a special case of a normalizing constant in probability theory, for the Boltzmann distribution.
[8] [9] This result was proved by Leonhard Euler in 1748 [10] and later was generalized as Glaisher's theorem. For every type of restricted partition there is a corresponding function for the number of partitions satisfying the given restriction. An important example is q(n) (partitions into distinct parts).
If there is a remainder in solving a partition problem, the parts will end up with unequal sizes. For example, if 52 cards are dealt out to 5 players, then 3 of the players will receive 10 cards each, and 2 of the players will receive 11 cards each, since 52 5 = 10 + 2 5 {\textstyle {\frac {52}{5}}=10+{\frac {2}{5}}} .
A conceptual explanation for Ramanujan's observation was finally discovered in January 2011 [3] by considering the Hausdorff dimension of the following function in the l-adic topology: P ℓ ( b ; z ) := ∑ n = 0 ∞ p ( ℓ b n + 1 24 ) q n / 24 . {\displaystyle P_{\ell }(b;z):=\sum _{n=0}^{\infty }p\left({\frac {\ell ^{b}n+1}{24}}\right)q^{n ...
Generally, a partition is a division of a whole into non-overlapping parts. Among the kinds of partitions considered in mathematics are partition of a set or an ordered partition of a set,
The following notations are used to specify how many partitions have a given rank. Let n, q be a positive integers and m be any integer. The total number of partitions of n is denoted by p(n). The number of partitions of n with rank m is denoted by N(m, n). The number of partitions of n with rank congruent to m modulo q is denoted by N(m, q, n).