View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c0265501.png" /> are defined by the relations

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<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>

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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

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<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 <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

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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:

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655024.png" /> are the functional derivatives,

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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

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by the relation

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The functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026550/c02655029.png" /> satisfies the equation

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<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>

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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>

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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]]]).

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]]]).

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

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

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Bogolyubov,   "Problems of a dynamical theory in statistical physics" , North-Holland  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.N. Bogolyubov,   K.P. Gurov,   ''Zh. Eksp. i Teoret. Fiziki'' , '''17''' :  7  (1947)  pp. 614–628</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.N. Bogolyubov,   "Selected works" , '''3''' , Kiev  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.N. Bogolyubov jr.,   B.I. Sadovnikov,   "Some questions in statistical mechanics" , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.N. Bogolyubov,   S.B. Tyablikov,   ''Dokl. Akad. Nauk SSSR'' , '''159''' :  1  (1959)  pp. 53–56</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R. Libov,   "Introduction to the theory of kinetic equations" , Wiley  (1969)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A. Isihara,   "Statistical physics" , Acad. Press  (1971)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  D. Ruelle,   "Statistical mechanics: rigorous results" , Benjamin  (1974)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  C.J. Preston,   "Gibbs states on countable sets" , Cambridge Univ. Press  (1974)</TD></TR></table>