Partition function (statistical mechanics)

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.