Correlation function (physics)

In the physics of many particles , the correlation function is defined in order to quantify the correlations between particles as a function of distance. Their behavior in the vicinity of phase transitions is often of particular interest.

definition

The density-density correlation function is defined as

${\displaystyle {\begin{aligned}G({\vec {r}},{\vec {r}}')&=\langle [n({\vec {r}})-\langle n({\vec {r}})\rangle ][n({\vec {r}}')-\langle n({\vec {r}}')\rangle ]\rangle \\&=\langle n({\vec {r}})\,n({\vec {r}}')\rangle -\langle n({\vec {r}})\rangle \langle n({\vec {r}}')\rangle \end{aligned}}}$

with the density . ${\displaystyle n({\vec {r}})}$

The angle brackets indicate a mean value calculation :

• In quantum mechanics through trace formation:

${\displaystyle \langle {\hat {A}}\rangle =\operatorname {Tr} ({\hat {\rho }}{\hat {A}})}$

with the density operator ; ${\displaystyle {\hat {\rho }}}$

• in classical statistical mechanics through the weighted sum over all possible system configurations:

${\displaystyle \langle A\rangle ={\frac {1}{\mathcal {Z}}}\sum \limits _{\text{Konfig.}}A\mathrm {e} ^{-\beta {\mathcal {H}}}}$

With

• ${\displaystyle \beta ={\frac {1}{k_{\mathrm {B} }\,T}}}$
  • the Boltzmann constant k _B
  • the absolute temperature ${\displaystyle T}$
• ${\displaystyle A}$and the total Hamilton function depend on the respective system configuration. ${\displaystyle {\mathcal {H}}}$
• the state sum ${\displaystyle {\mathcal {Z}}=\sum \limits _{\text{Konfig.}}\mathrm {e} ^{-\beta {\mathcal {H}}}}$
• “System configurations” here means every possible state of the system, so that in a many-body system consisting of spins there are straight combinations or “system configurations”, e.g. In the Ising model , for example, there are two spin setting options, up (+1) or down (−1). ${\displaystyle N}$ ${\displaystyle 2^{N}}$

Examples

For translation-invariant systems, the correlation function does not depend on the specific locations, but only on the difference vector:

${\displaystyle G({\vec {r}},{\vec {r}}')=G({\vec {r}}-{\vec {r}}')}$

Such systems should always be considered in the following.

For large distances, the particles are typically uncorrelated so that the correlation function approaches zero.

A typical (but not necessarily present) behavior is then an exponential decay towards zero, so that one can specify a typical length scale to which the system reacts, the correlation length : ${\displaystyle \xi }$

${\displaystyle G({\vec {r}}-{\vec {r}}')\propto \mathrm {e} ^{-|{\vec {r}}-{\vec {r}}'|/\xi }}$

A diverging correlation length when approaching the temperature of the phase transition is typical for phase transitions. The relative temperature is defined , which approaches zero at the phase transition. ${\displaystyle \xi \to \infty }$ ${\displaystyle T_{C}}$ ${\displaystyle t={\frac {T}{T_{C}}}-1}$

Often one is interested in the critical exponents , i.e. how quickly the correlation length or other quantities diverge at the phase transition, e.g. B. . ${\displaystyle \xi \propto t^{-\nu }}$

• The XY model does not show a phase transition in space ( ), but a transition in the asymptotic behavior of the correlation function: At low temperatures it decays algebraically ( with ), at high temperatures it decays exponentially, with the correlation length increasing exponentially ( ). A Kosterlitz-Thouless transition is observed in this model . ${\displaystyle d=2}$ ${\displaystyle k_{\mathrm {B} }T_{C}=\pi d(d-2)=0}$ ${\displaystyle \propto \beta ^{-|{\vec {r}}-{\vec {r}}'|}}$ ${\displaystyle \beta =1/(k_{\mathrm {B} }T)}$ ${\displaystyle \xi \propto \mathrm {e} ^{\mathrm {const.} /{\sqrt {t}}}}$
• The Ornstein-Zernike theory provides a correlation function in the form of a Yukawa potential : . The correlation length at the phase transition again approaches infinity and long-range correlations are obtained. In this theory one also considers the structure factor , the Fourier transform of the correlation function. It is proportional to the scattering intensity of the incident radiation. Experimentally, it is found that this becomes arbitrarily large, which ultimately reflects the diverging correlation length. The reference to the scattering intensity also makes it possible to understand the phenomenon of critical opalescence , in which a special material appears increasingly "milky" when it cools. This is because such a material comes below the critical temperature when it cools, so that the correlation length becomes so great that it comes in the order of magnitude of visible light. This is then scattered so effectively that you can no longer see a “clear picture”. ${\displaystyle G(r)\propto {\frac {\mathrm {e} ^{-r/\xi }}{r}}}$
  ${\displaystyle S(q)}$ ${\displaystyle q\to 0,T\to T_{C}}$
  
• The mean field approximation for the Ising model provides an exponentially decaying correlation function with a critical exponent . In spatial dimension 2, this corresponds to the exact solution of the Ising model according to Onsager . A renormalization group method for the Ising model provides in spatial dimensions . ${\displaystyle \nu =1}$ ${\displaystyle d}$ ${\displaystyle \nu ={\frac {1}{d-1}}}$

Individual evidence

1. ↑ ^{a b c} Stanley, H. Eugene. "Introduction to phase transitions and critical phenomena." Introduction to Phase Transitions and Critical Phenomena, by H Eugene Stanley, pp. 336. Foreword by H Eugene Stanley. Oxford University Press, July 1987. ISBN 0195053168 .
2. ↑ ^{a b c} Chaikin, Paul M., and Tom C. Lubensky. Principles of condensed matter physics . Vol. 1. Cambridge: Cambridge university press, 2000.