Correlation function in statistical mechanics

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[edit]

[1] |  N.N. Bogolyubov, "Problems of a dynamical theory in statistical physics" , North-Holland (1962) (Translated from Russian)

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[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)