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''partition function''
 
''partition function''
  
 
A function used in equilibrium statistical physics (cf. [[Statistical physics, mathematical problems in|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|Gibbs statistical aggregate]]; [[Statistical ensemble|Statistical ensemble]]).
 
A function used in equilibrium statistical physics (cf. [[Statistical physics, mathematical problems in|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|Gibbs statistical aggregate]]; [[Statistical ensemble|Statistical ensemble]]).
  
1) In a classical system, the density of the Gibbs distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s0874701.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s0874702.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s0874703.png" /> is the phase space of the system), relative to the natural measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s0874704.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s0874705.png" />, is defined by the formula
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s0874706.png" /></td> </tr></table>
+
$$
 +
p( \omega )  = ( \Xi )  ^ {-} 1  \mathop{\rm exp} \{ - \beta [ H _ {0} ( \omega ) + \mu _ {1} H _ {1} ( \omega ) + \dots + \mu _ {k} H _ {k} ( \omega )] \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s0874707.png" /> is the Hamilton function (energy) of the system and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s0874708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s0874709.png" />, is a set of quantities which are conserved when the system defined by the Hamiltonian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747010.png" /> evolves in time; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747012.png" /> are real parameters. The normalization factor
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747013.png" /></td> </tr></table>
+
$$
 +
\Xi ( \beta , \mu _ {1} \dots \mu _ {k} )  = \int\limits _  \Omega  \mathop{\rm exp} \left \{ - \beta \left [ H _ {0} ( \omega ) + \sum _ { i= } 1 ^ { k }  \mu _ {i} H _ {i} ( \omega )
 +
\right ] \right \}  d \omega
 +
$$
  
 
is also called a statistical sum (or a partition function).
 
is also called a statistical sum (or a partition function).
Line 15: Line 43:
 
2) In a quantum system the canonical Gibbs state is defined by the density matrix
 
2) In a quantum system the canonical Gibbs state is defined by the density matrix
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747014.png" /></td> </tr></table>
+
$$
 +
\rho  = ( \Xi )  ^ {-} 1  \mathop{\rm exp} \{ - \beta ( \widehat{H}  _ {0} + \mu _ {1} \widehat{H}  _ {1} + \dots +
 +
\mu _ {k} \widehat{H}  _ {k} ) \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747015.png" /> is the Hamiltonian (energy operator) of the system, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747017.png" />, are commuting operators, corresponding to quantities conserved in the course of time; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747019.png" /> are real parameters. The normalization factor (called the statistical sum or partition function) is equal to
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747020.png" /></td> </tr></table>
+
$$
 +
\Xi ( \beta , \mu _ {1} \dots \mu _ {k} )  = \
 +
\mathop{\rm Sp}  \mathop{\rm exp} \left \{
 +
- \beta _ {0} \left ( \widehat{H}  _ {0} + \sum _ { i= } 1 ^ { k }  \mu _ {i} H
 +
hat _ {i} \right ) \right \} .
 +
$$
  
 
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.).
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747021.png" /> and the energy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747022.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747023.png" />), as well as the other quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747025.png" /> (respectively, the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747027.png" />) that appear in the definition of a Gibbs ensemble are invariant relative to shifts in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747028.png" /> and are almost additive, i.e. (in a classical system)
+
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)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747029.png" /></td> </tr></table>
+
$$
 +
H _ {i} ( \omega _ {1} , \omega _ {2} )  \approx  H _ {i} ( \omega _ {1} ) + H _ {i} ( \omega _ {2} ),\ \
 +
i = 0 \dots k,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747031.png" /> are two configurations of particles sufficiently far apart (for an exact formulation of this condition and its quantum analogue, see [[#References|[2]]]), in a passage to the thermodynamic limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747032.png" /> the statistical sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747033.png" /> has the following asymptotics:
+
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 [[#References|[2]]]), in a passage to the thermodynamic limit $  \Lambda \uparrow \mathbf R  ^ {2} $
 +
the statistical sum $  \Xi $
 +
has the following asymptotics:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747034.png" /></td> </tr></table>
+
$$
 +
\Xi ( \beta , \mu _ {1} \dots \mu _ {k} )  = \
 +
\mathop{\rm exp} \{ | \Lambda | \chi ( \beta , \mu _ {1} \dots \mu _ {k} ) + o( | \Lambda | ) \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747035.png" /> is the volume of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747036.png" />; and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087470/s08747037.png" /> — 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.).
+
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.D. Landau,  E.M. Lifshitz,  "Statistical physics" , Pergamon  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Ruelle,  "Statistical mechanics: rigorous results" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Balescu,  "Equilibrium and non-equilibrium statistical mechanics" , '''1–2''' , Wiley  (1975)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.D. Landau,  E.M. Lifshitz,  "Statistical physics" , Pergamon  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Ruelle,  "Statistical mechanics: rigorous results" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Balescu,  "Equilibrium and non-equilibrium statistical mechanics" , '''1–2''' , Wiley  (1975)</TD></TR></table>

Revision as of 08:23, 6 June 2020


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)
How to Cite This Entry:
Statistical sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Statistical_sum&oldid=48820
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article