Partition function

Partition function in number theory

The partition function p(n) is a non-multiplicative 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,n-k). If the reason for this is not immediately apparent, imagine a list of all the partitions of the number n-k 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,n-k). The base cases of this recursive function are as follows:

• p(k,n) = 0 if k > n

• p(k,n) = 1 if k = n

This function will mess with one's mind if one lets it. Consider the following:

p(1,4)=5

p(2,8)=7

p(3,12)=9

p(4,16)=11

p(5,20)=13

p(6,24)=16

Partition function in statistical mechanics

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:

<math>Z\equiv \sum_j e^{-{E_j \over k_B T}}</math>

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:

• It is needed as the normalization denominator for Boltzmann's probability distribution[?] which gives the probability to find the system in state j when it is in thermal equilibrium at temperature T (the sum over probabilities has to be equal to one):

<math>P(j)={e^{- {E_j \over k_B T}} \over Z}</math>

• Qualitatively, Z grows when the temperature rises, because then the exponential weights increase for states of larger energy. Roughly, Z is a measure of how many different energy states are populated appreciably in thermal equilibrium (at least when we suppose the ground state energy to be zero).

• Given Z as a function of temperature, we may calculate the average energy as

<math>E=\sum_j P(j) E_j=k_B T^2 {d \over dT} \ln Z</math>

• The free energy of the system is basically the logarithm of Z:

<math>F=E-TS=-k_B T \ln Z</math>

• From these two relations, the entropy S may be obtained as

<math>S=k_B \sum_j P(j) \ln P(j)=(E-F)/T=k_B T^2 {d \over dT} {\ln Z \over T}</math>

• Alternatively, with <math>\beta\equiv 1/(k_B T)</math>, we have <math>E=-{d \over d \beta} \ln Z</math> and <math>F=-\beta^{-1} \ln Z</math>, as well as <math>S=-k_B \beta^2 {d \over d \beta} (\beta^{-1} \ln Z)</math>.

More formally, the partition function Z of a quantum-mechanical system may be written as a trace over all states (which may be carried out in any basis, as the trace is basis-independent):

<math>Z=tr e^{-\beta \hat H}</math>

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:

<math><\hat A>= -\beta^{-1} {d \over d\lambda} Z(\beta,\lambda)</math>

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[?].