Statistical sum

partition function

A function used in equilibrium statistical physics (cf. Statistical physics, mathematical problems in), equal to a normalization constant in an expression for the density (or for the density matrix in a quantum system) in a canonical Gibbs ensemble (cf. Gibbs statistical aggregate; Statistical ensemble).

1) In a classical system, the density of the Gibbs distribution $ p( \omega ) $, $ \omega \in \Omega $( $ \Omega $ is the phase space of the system), relative to the natural measure $ d \omega $ on $ \Omega $, is defined by the formula

$$ p( \omega ) = ( \Xi ) ^ {-1} \mathop{\rm exp} \{ - \beta [ H _ {0} ( \omega ) + \mu _ {1} H _ {1} ( \omega ) + \dots + \mu _ {k} H _ {k} ( \omega )] \} , $$

where $ H _ {0} ( \omega ) $ is the Hamilton function (energy) of the system and $ H _ {i} ( \omega ) $, $ i = 1 \dots k $, is a set of quantities which are conserved when the system defined by the Hamiltonian $ H _ {0} ( \omega ) $ evolves in time; $ \beta > 0 $ and $ \mu _ {1} \dots \mu _ {k} $ are real parameters. The normalization factor

$$ \Xi ( \beta , \mu _ {1} \dots \mu _ {k} ) = \int\limits _ \Omega \mathop{\rm exp} \left \{ - \beta \left [ H _ {0} ( \omega ) + \sum_{i=1}^ { k } \mu _ {i} H _ {i} ( \omega ) \right ] \right \} d \omega $$

is also called a statistical sum (or a partition function).

2) In a quantum system the canonical Gibbs state is defined by the density matrix

$$ \rho = ( \Xi ) ^ {-1} \mathop{\rm exp} \{ - \beta ( \widehat{H} _ {0} + \mu _ {1} \widehat{H} _ {1} + \dots + \mu _ {k} \widehat{H} _ {k} ) \} , $$

where $ \widehat{H} _ {0} $ is the Hamiltonian (energy operator) of the system, and $ \widehat{H} _ {i} $, $ i = 1 \dots k $, are commuting operators, corresponding to quantities conserved in the course of time; $ \beta > 0 $ and $ \mu _ {1} \dots \mu _ {k} $ are real parameters. The normalization factor (called the statistical sum or partition function) is equal to

$$ \Xi ( \beta , \mu _ {1} \dots \mu _ {k} ) = \ \mathop{\rm Sp} \mathop{\rm exp} \left \{ - \beta _ {0} \left ( \widehat{H} _ {0} + \sum_{i=1}^ { k } \mu _ {i} H hat _ {i} \right ) \right \} . $$

Statistical sums are defined in the same way for other Gibbs ensembles (micro-canonical and small canonical), as well as for Gibbs ensembles defined for various simplified modifications of real physical systems (lattice systems, configuration systems, etc.).

In the typical case where the system is enclosed in a bounded domain $ \Lambda \subset \mathbf R ^ {3} $ and the energy $ H _ {0} ( \omega ) $( or $ \widehat{H} _ {0} $), as well as the other quantities $ H _ {i} ( \omega ) $, $ i = 1 \dots k $( respectively, the operators $ \widehat{H} _ {i} $, $ i = 1 \dots k $) that appear in the definition of a Gibbs ensemble are invariant relative to shifts in $ \mathbf R ^ {3} $ and are almost additive, i.e. (in a classical system)

$$ H _ {i} ( \omega _ {1} , \omega _ {2} ) \approx H _ {i} ( \omega _ {1} ) + H _ {i} ( \omega _ {2} ),\ \ i = 0 \dots k, $$

where $ \omega _ {1} $ and $ \omega _ {2} $ are two configurations of particles sufficiently far apart (for an exact formulation of this condition and its quantum analogue, see [2]), in a passage to the thermodynamic limit $ \Lambda \uparrow \mathbf R ^ {2} $ the statistical sum $ \Xi $ has the following asymptotics:

$$ \Xi ( \beta , \mu _ {1} \dots \mu _ {k} ) = \ \mathop{\rm exp} \{ | \Lambda | \chi ( \beta , \mu _ {1} \dots \mu _ {k} ) + o( | \Lambda | ) \} , $$

where $ | \Lambda | $ is the volume of the domain $ \Lambda $; and the function $ \chi ( \beta , \mu _ {1} \dots \mu _ {k} ) $— the so-called thermodynamic potential — is an important thermodynamic characteristic of the system: many other thermodynamic characteristics can be expressed in terms of it (specific energy, density, specific entropy, etc.).

References

[1]	L.D. Landau, E.M. Lifshitz, "Statistical physics" , Pergamon (1980) (Translated from Russian)
[2]	D. Ruelle, "Statistical mechanics: rigorous results" , Benjamin (1969)
[3]	R. Balescu, "Equilibrium and non-equilibrium statistical mechanics" , 1–2 , Wiley (1975)