Partition function: Difference between revisions

Revision as of 20:50, 14 February 2003

Revision as of 20:55, 14 February 2003

Line 30:

=== Partition function in statistical mechanics ===

In [[statistical mechanics]], the partition function <math>Z</math> 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>

Line 38 ⟶ 40:

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)=\exp({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).

Line 56 ⟶ 58:

:<math>S=(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):