Partition function

Partition function in number theory

The partition function described here is part of number theory. See the next section for the partition function of statistical mechanics.

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 $E_{j}$ of the system's Hamiltonian operator ${\hat {H}}$, the partition function at temperature $T$ is defined as:

$Z\equiv \sum _{j}e^{-{E_{j} \over k_{B}T}}$

Here the sum runs over all energy eigenstates (counted by the index j) and $k_{B}$ 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):

$P(j)={e^{-{E_{j} \over k_{B}T}} \over Z}$

• 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

$E=\sum _{j}P(j)E_{j}=k_{B}T^{2}{d \over dT}\ln Z$

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

$F=E-TS=-k_{B}T\ln Z$

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

$S=k_{B}\sum _{j}P(j)\ln P(j)=(E-F)/T=k_{B}T^{2}{d \over dT}{\ln Z \over T}$

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

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):

$Z=tre^{-\beta {\hat {H}}}$

If the Hamiltonian contains a dependence on a parameter $\lambda$, as in ${\hat {H}}={\hat {H}}_{0}+\lambda {\hat {A}}$ then the statistical average over ${\hat {A}}$ may be found from the dependence of the partition function on the parameter, by differentiation:

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

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.