The partition function p(n) is a nonmultiplicative function and represents the number of possible partitions of a natural number n, which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers. The partition function is easy to calculate. One way of doing so involves an intermediate function p(k,n) which represents the number of partitions of n using only natural numbers at least as large as k. For any given value of k, partitions counted by p(k,n) fit into exactly one of the following categories:
1. smallest addend is k
2. smallest addend is strictly greater than[?] than k
The number of partitions meeting the first condition is p(k,nk). If the reason for this is not immediately apparent, imagine a list of all the partitions of the number nk into numbers of size at least k, then imagine appending "+k" to each partition in the list. Now what is it a list of?
The number of partitions meeting the second condition is p(k+1,n). Can anyone explain to us why?
Since the two conditions are mutually exclusive[?], the number of partitions meeting either condition is p(k+1,n)+p(k,nk). The base cases of this recursive function are as follows:
This function will mess with one's mind if one lets it. Consider the following:
In statistical mechanics, the partition function Z is used for the statistical description of a system in thermodynamic equilibrium. It depends on the physical system under consideration and is a function of temperature as well as other parameters (such as volume enclosing a gas etc.). The partition function forms the basis for most calculations in statistical mechanics. It is most easily formulated in quantum mechanics:
Given the energy eigenvalues <math>E_j</math> of the system's Hamiltonian operator[?] <math>\hat H</math>, the partition function at temperature <math>T</math> is defined as:
Here the sum runs over all energy eigenstates (counted by the index j) and <math>k_B</math> is Boltzmann's constant.
The partition function has the following meanings:
More formally, the partition function Z of a quantummechanical system may be written as a trace over all states (which may be carried out in any basis, as the trace is basisindependent):
If the Hamiltonian contains a dependence on a parameter <math>\lambda</math>, as in <math>\hat H=\hat H_0 + \lambda \hat A</math> then the statistical average over <math>\hat A</math> may be found from the dependence of the partition function on the parameter, by differentiation:
If one is interested in the average of an operator that does not appear in the Hamiltonian, one often adds it artificially to the Hamiltonian, calculates Z as a function of the extra new parameter and sets the parameter equal to zero after differentiation.
There is also a partition function in game theory[?].
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