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A function describing the influence of particles or groups of particles on one another and the effects due to the interaction of subsystems of the system under consideration.
 
A function describing the influence of particles or groups of particles on one another and the effects due to the interaction of subsystems of the system under consideration.
  
In classical statistical mechanics, the correlation functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c0265501.png" /> are defined by the relations
+
In classical statistical mechanics, the correlation functions $  G _ {2} ( 1, 2), G _ {3} ( 1, 2, 3) \dots $
 +
are defined by the relations
 +
 
 +
$$
 +
F _ {2} ( 1, 2)  = \
 +
F _ {1} ( 1) F _ {1} ( 2) +
 +
G _ {2} ( 1, 2),
 +
$$
  
<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/c/c026/c026550/c0265502.png" /></td> </tr></table>
+
$$
 +
F _ {3} ( 1, 2, 3)  = F _ {1} ( 1) F _ {1} ( 2)
 +
F _ {1} ( 3) + F _ {1} ( 1) G _ {2} ( 2, 3) +
 +
$$
  
<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/c/c026/c026550/c0265503.png" /></td> </tr></table>
+
$$
 +
+
 +
F _ {1} ( 2) G _ {2} ( 1, 3) + F _ {1} ( 3)
 +
G _ {2} ( 1, 2) + G _ {3} ( 1, 2, 3) \dots
 +
$$
  
<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/c/c026/c026550/c0265504.png" /></td> </tr></table>
+
where the symbols  $  1, 2 \dots $
 +
in the arguments of the functions denote the sets of coordinates  $  \mathbf r $
 +
and momenta  $  \mathbf p $
 +
of the 1st, 2nd  $  \dots $
 +
particles, respectively, and  $  F _ {s} ( 1 \dots s) $
 +
are the reduced distribution functions
  
where the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c0265505.png" /> in the arguments of the functions denote the sets of coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c0265506.png" /> and momenta <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c0265507.png" /> of the 1st, 2nd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c0265508.png" /> particles, respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c0265509.png" /> are the reduced distribution functions
+
$$
 +
F _ {s} ( 1 \dots s)  = \
 +
V \left (
 +
1 - {
 +
\frac{1}{N}
 +
}
 +
\right ) \dots
 +
$$
  
<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/c/c026/c026550/c02655010.png" /></td> </tr></table>
+
$$
 +
\dots \left ( 1 -  
 +
\frac{s - 1 }{N}
 +
\right ) \int\limits D _ {t}  d ( s + 1) \dots dN,
 +
$$
  
<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/c/c026/c026550/c02655011.png" /></td> </tr></table>
+
where  $  V $
 +
is the volume of the system,  $  N $
 +
is the number of particles and the  $  D _ {t} = D _ {t} ( 1 \dots N) $
 +
are the distribution functions in the phase space at time  $  t $,
 +
normalized so that
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655012.png" /> is the volume of the system, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655013.png" /> is the number of particles and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655014.png" /> are the distribution functions in the phase space at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655015.png" />, normalized so that
+
$$
 +
\int\limits D _ {t} ( 1 \dots N)
 +
d1 \dots dN  = 1.
 +
$$
  
<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/c/c026/c026550/c02655016.png" /></td> </tr></table>
+
The variation of  $  D _ {t} $
 +
in time is characterized by the Liouville equation  $  \partial  D _ {t} / \partial  t = \Lambda D _ {t} $,
 +
where  $  \Lambda $
 +
represents the Liouville operator, which is not explicitly dependent on time. One usually considers the case in which  $  \Lambda $
 +
is the sum of an additive part and a binary part characterizing the interactions of the particles:
  
The variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655017.png" /> in time is characterized by the Liouville equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655019.png" /> represents the Liouville operator, which is not explicitly dependent on time. One usually considers the case in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655020.png" /> is the sum of an additive part and a binary part characterizing the interactions of the particles:
+
$$
 +
\Lambda  = \
 +
\sum _ {1 \leq  j \leq  N }
 +
\Lambda ( j) +
 +
\sum _ {1 \leq  j _ {1} < j _ {2} \leq  N }
 +
\Lambda ( j _ {1} , j _ {2} ).
 +
$$
  
<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/c/c026/c026550/c02655021.png" /></td> </tr></table>
+
According to the principle of correlation damping, the correlation functions satisfy the boundary conditions  $  G _ {s} ( 1 \dots s) \rightarrow 0 $
 +
as  $  \max \{ | \mathbf r _ {1} - \mathbf r _ {2} | \dots | \mathbf r _ {1} - \mathbf r _ {s} | \dots | \mathbf r _ {s - 1 }  - \mathbf r _ {s} | \} \rightarrow \infty $.
  
According to the principle of correlation damping, the correlation functions satisfy the boundary conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655022.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655023.png" />.
+
The correlation functions  $  G _ {1} ( 1) = F _ {1} ( 1), G _ {2} ( 1, 2) \dots G _ {s} ( 1 \dots s) $
 +
are the functional derivatives,
  
The correlation functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655024.png" /> are the functional derivatives,
+
$$
 +
G _ {s} ( 1 \dots s)  = \
 +
\left (
  
<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/c/c026/c026550/c02655025.png" /></td> </tr></table>
+
\frac{\delta  ^ {s} A _ {t} ( u) }{\delta u ( 1) \delta u ( 2) \dots \delta u ( s) }
  
of a functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655026.png" /> which is related to the so-called generating functional
+
\right ) _ {u = 0 }  ,
 +
$$
  
<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/c/c026/c026550/c02655027.png" /></td> </tr></table>
+
of a functional  $  A _ {t} ( u) $
 +
which is related to the so-called generating functional
 +
 
 +
$$
 +
L _ {t} ( u)  = \
 +
\int\limits \left \{
 +
\prod _ {1 \leq  j \leq  N }
 +
\left ( 1 +
 +
{
 +
\frac{V}{N}
 +
} u ( j)
 +
\right )  \right \}
 +
D _ {t}  d1 \dots dN
 +
$$
  
 
by the relation
 
by the relation
  
<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/c/c026/c026550/c02655028.png" /></td> </tr></table>
+
$$
 +
L _ {t} ( u)  = \
 +
e ^ {A _ {t} ( u) } .
 +
$$
  
The functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655029.png" /> satisfies the equation
+
The functional $  A _ {t} ( u) $
 +
satisfies the equation
  
<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/c/c026/c026550/c02655030.png" /></td> </tr></table>
+
$$
  
<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/c/c026/c026550/c02655031.png" /></td> </tr></table>
+
\frac{\partial  A _ {t} ( u) }{\partial  t }
 +
  = \
 +
\int\limits u ( 1) \Lambda ( 1)
  
<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/c/c026/c026550/c02655032.png" /></td> </tr></table>
+
\frac{\delta A _ {t} ( u) }{\delta u ( 1) }
 +
\
 +
d1 +
 +
$$
 +
 
 +
$$
 +
+
 +
{
 +
\frac{1}{2}
 +
} \int\limits \left \{ u ( 1) u ( 2) +
 +
{
 +
\frac{N}{V}
 +
} u ( 1) + {
 +
\frac{N}{V}
 +
} u ( 2) \right \} \Lambda ( 1, 2)
 +
$$
 +
 
 +
$$
 +
\left \{
 +
\frac{\delta A _ {t} ( u) }{\delta u
 +
( 1) }
 +
 +
\frac{\delta A _ {t} ( u) }{\delta u ( 2) }
 +
+
 +
\frac{\delta
 +
^ {2} A _ {t} ( u) }{\delta u ( 1) \delta u ( 2) }
 +
\right \}  d1  d2.
 +
$$
  
 
In quantum statistical mechanics, the correlation functions are operator quantities, defined as follows:
 
In quantum statistical mechanics, the correlation functions are operator quantities, defined as follows:
  
<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/c/c026/c026550/c02655033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
F _ {2} ( 1, 2)  = \
 +
S ( 1, 2) \{
 +
F _ {1} ( 1) F _ {1} ( 2) \} +
 +
G _ {2} ( 1, 2),
 +
$$
 +
 
 +
$$
 +
F _ {3} ( 1, 2, 3)  = S ( 1, 2, 3)
 +
\{ F _ {1} ( 1) F _ {1} ( 2) F _ {1} ( 3) \} +
 +
$$
  
<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/c/c026/c026550/c02655034.png" /></td> </tr></table>
+
$$
 +
+
  
<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/c/c026/c026550/c02655035.png" /></td> </tr></table>
+
\frac{1}{2}
 +
S ( 1, 2, 3) \{ F _ {1} ( 1) G _ {2} ( 2, 3) + F _ {1} ( 2) G _ {2} ( 1, 3) +
 +
$$
  
<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/c/c026/c026550/c02655036.png" /></td> </tr></table>
+
$$
 +
+
 +
{} F _ {1} ( 3) G _ {2} ( 1, 2) \} + G _ {3} ( 1, 2, 3) \dots
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655038.png" /> are the symmetrization operator for Bose systems and the anti-symmetrization operator for Fermi systems. The correlation functions (*), forming the density matrix, satisfy the quantum-mechanical Liouville equation (see [[#References|[2]]]).
+
where $  S ( 1, 2) $,
 +
$  S ( 1, 2, 3) $
 +
are the symmetrization operator for Bose systems and the anti-symmetrization operator for Fermi systems. The correlation functions (*), forming the density matrix, satisfy the quantum-mechanical Liouville equation (see [[#References|[2]]]).
  
 
In quantum statistical mechanics, besides the correlation function (*) one considers correlation functions based on conventional thermodynamical averages (see [[#References|[3]]]), and correlation functions based on quasi-averages (see [[#References|[3]]]).
 
In quantum statistical mechanics, besides the correlation function (*) one considers correlation functions based on conventional thermodynamical averages (see [[#References|[3]]]), and correlation functions based on quasi-averages (see [[#References|[3]]]).

Latest revision as of 17:31, 5 June 2020


A function describing the influence of particles or groups of particles on one another and the effects due to the interaction of subsystems of the system under consideration.

In classical statistical mechanics, the correlation functions $ G _ {2} ( 1, 2), G _ {3} ( 1, 2, 3) \dots $ are defined by the relations

$$ F _ {2} ( 1, 2) = \ F _ {1} ( 1) F _ {1} ( 2) + G _ {2} ( 1, 2), $$

$$ F _ {3} ( 1, 2, 3) = F _ {1} ( 1) F _ {1} ( 2) F _ {1} ( 3) + F _ {1} ( 1) G _ {2} ( 2, 3) + $$

$$ + F _ {1} ( 2) G _ {2} ( 1, 3) + F _ {1} ( 3) G _ {2} ( 1, 2) + G _ {3} ( 1, 2, 3) \dots $$

where the symbols $ 1, 2 \dots $ in the arguments of the functions denote the sets of coordinates $ \mathbf r $ and momenta $ \mathbf p $ of the 1st, 2nd $ \dots $ particles, respectively, and $ F _ {s} ( 1 \dots s) $ are the reduced distribution functions

$$ F _ {s} ( 1 \dots s) = \ V \left ( 1 - { \frac{1}{N} } \right ) \dots $$

$$ \dots \left ( 1 - \frac{s - 1 }{N} \right ) \int\limits D _ {t} d ( s + 1) \dots dN, $$

where $ V $ is the volume of the system, $ N $ is the number of particles and the $ D _ {t} = D _ {t} ( 1 \dots N) $ are the distribution functions in the phase space at time $ t $, normalized so that

$$ \int\limits D _ {t} ( 1 \dots N) d1 \dots dN = 1. $$

The variation of $ D _ {t} $ in time is characterized by the Liouville equation $ \partial D _ {t} / \partial t = \Lambda D _ {t} $, where $ \Lambda $ represents the Liouville operator, which is not explicitly dependent on time. One usually considers the case in which $ \Lambda $ is the sum of an additive part and a binary part characterizing the interactions of the particles:

$$ \Lambda = \ \sum _ {1 \leq j \leq N } \Lambda ( j) + \sum _ {1 \leq j _ {1} < j _ {2} \leq N } \Lambda ( j _ {1} , j _ {2} ). $$

According to the principle of correlation damping, the correlation functions satisfy the boundary conditions $ G _ {s} ( 1 \dots s) \rightarrow 0 $ as $ \max \{ | \mathbf r _ {1} - \mathbf r _ {2} | \dots | \mathbf r _ {1} - \mathbf r _ {s} | \dots | \mathbf r _ {s - 1 } - \mathbf r _ {s} | \} \rightarrow \infty $.

The correlation functions $ G _ {1} ( 1) = F _ {1} ( 1), G _ {2} ( 1, 2) \dots G _ {s} ( 1 \dots s) $ are the functional derivatives,

$$ G _ {s} ( 1 \dots s) = \ \left ( \frac{\delta ^ {s} A _ {t} ( u) }{\delta u ( 1) \delta u ( 2) \dots \delta u ( s) } \right ) _ {u = 0 } , $$

of a functional $ A _ {t} ( u) $ which is related to the so-called generating functional

$$ L _ {t} ( u) = \ \int\limits \left \{ \prod _ {1 \leq j \leq N } \left ( 1 + { \frac{V}{N} } u ( j) \right ) \right \} D _ {t} d1 \dots dN $$

by the relation

$$ L _ {t} ( u) = \ e ^ {A _ {t} ( u) } . $$

The functional $ A _ {t} ( u) $ satisfies the equation

$$ \frac{\partial A _ {t} ( u) }{\partial t } = \ \int\limits u ( 1) \Lambda ( 1) \frac{\delta A _ {t} ( u) }{\delta u ( 1) } \ d1 + $$

$$ + { \frac{1}{2} } \int\limits \left \{ u ( 1) u ( 2) + { \frac{N}{V} } u ( 1) + { \frac{N}{V} } u ( 2) \right \} \Lambda ( 1, 2) $$

$$ \left \{ \frac{\delta A _ {t} ( u) }{\delta u ( 1) } \frac{\delta A _ {t} ( u) }{\delta u ( 2) } + \frac{\delta ^ {2} A _ {t} ( u) }{\delta u ( 1) \delta u ( 2) } \right \} d1 d2. $$

In quantum statistical mechanics, the correlation functions are operator quantities, defined as follows:

$$ \tag{* } F _ {2} ( 1, 2) = \ S ( 1, 2) \{ F _ {1} ( 1) F _ {1} ( 2) \} + G _ {2} ( 1, 2), $$

$$ F _ {3} ( 1, 2, 3) = S ( 1, 2, 3) \{ F _ {1} ( 1) F _ {1} ( 2) F _ {1} ( 3) \} + $$

$$ + \frac{1}{2} S ( 1, 2, 3) \{ F _ {1} ( 1) G _ {2} ( 2, 3) + F _ {1} ( 2) G _ {2} ( 1, 3) + $$

$$ + {} F _ {1} ( 3) G _ {2} ( 1, 2) \} + G _ {3} ( 1, 2, 3) \dots $$

where $ S ( 1, 2) $, $ S ( 1, 2, 3) $ are the symmetrization operator for Bose systems and the anti-symmetrization operator for Fermi systems. The correlation functions (*), forming the density matrix, satisfy the quantum-mechanical Liouville equation (see [2]).

In quantum statistical mechanics, besides the correlation function (*) one considers correlation functions based on conventional thermodynamical averages (see [3]), and correlation functions based on quasi-averages (see [3]).

Bilinear combinations of correlation functions (both quantum-mechanical and classical) yield the Green functions (see [5]). Correlation functions possess spectral representations; they satisfy the Bogolyubov inequality and a variation of the mean-value theorem (see [4]).

Correlation functions corresponding to the Kirkwood decomposition are sometimes used (see [6]); another version is a space-time correlation function (see [8]).

Correlation functions may be interpreted as characteristic functions of probability measures (see [9]).

References

[1] N.N. Bogolyubov, "Problems of a dynamical theory in statistical physics" , North-Holland (1962) (Translated from Russian)
[2] N.N. Bogolyubov, K.P. Gurov, Zh. Eksp. i Teoret. Fiziki , 17 : 7 (1947) pp. 614–628
[3] N.N. Bogolyubov, "Selected works" , 3 , Kiev (1971) (In Russian)
[4] N.N. Bogolyubov jr., B.I. Sadovnikov, "Some questions in statistical mechanics" , Moscow (1975) (In Russian)
[5] N.N. Bogolyubov, S.B. Tyablikov, Dokl. Akad. Nauk SSSR , 159 : 1 (1959) pp. 53–56
[6] R. Libov, "Introduction to the theory of kinetic equations" , Wiley (1969)
[7] A. Isihara, "Statistical physics" , Acad. Press (1971)
[8] D. Ruelle, "Statistical mechanics: rigorous results" , Benjamin (1974)
[9] C.J. Preston, "Gibbs states on countable sets" , Cambridge Univ. Press (1974)
How to Cite This Entry:
Correlation function in statistical mechanics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Correlation_function_in_statistical_mechanics&oldid=17541
This article was adapted from an original article by A.N. ErmilovA.M. Kurbatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article