Correlation length

The correlation length is a term from statistical mechanics . It denotes the distance between two particles within which their dynamics are correlated with each other , i.e. commonalities occur in their random fluctuations. These similarities are expressed in correlations of measurable quantities such as density or magnetization . These correlations come about through interactions between the particles, so the correlation length is also a measure of the range of these interactions. In the case of continuous or quantum mechanical systems, in which no discrete particles are considered, the correlation length only refers to the quantities measured at different locations. ${\displaystyle \xi }$

Mathematical description

Mathematically, the correlation length is defined using a correlation function as a function of the distance . ${\displaystyle G(r)}$ ${\displaystyle r}$

Let's consider a size , e.g. B. the alignment of a spin , then: ${\displaystyle a}$

${\displaystyle G(r)=\langle \mathbf {a} (R)\cdot \mathbf {a} (R+r)\rangle \ -\langle \mathbf {a} (R)\rangle \cdot \langle \mathbf {a} (R+r)\rangle \,,}$

where the angle brackets stand for the averaging over all locations . ${\displaystyle R}$

Is above the correlation length

${\displaystyle r>\xi \Rightarrow G(r)\approx 0\,,}$

there is practically no correlation left.

In many cases the correlation function drops exponentially with distance:

${\displaystyle G(r)\propto \exp {\left(-{\frac {|r|}{\xi }}\right)}}$

Another model for the correlation function is the Gaussian correlation, in which the correlation falls in the form of a Gaussian curve as a function of the distance :

${\displaystyle G(r)=\exp {\left(-\left({\frac {r}{\xi }}\right)^{2}\right)}}$

The simplest form of correlation is triangular correlation. The correlation decreases linearly from to up to the correlation length : ${\displaystyle 1}$ ${\displaystyle 0}$

${\displaystyle G(r)=\left\{{\begin{array}{ll}1-{\frac {|r|}{\xi }}&|r|\leq \xi \\0&{\text{sonst}}\end{array}}\right.}$

With liquids

In liquids , the correlation length is usually very small (around 0.1 to 1 nm). In the vicinity of phase transitions (e.g. between the liquid and the gaseous phase ) it increases strongly, so that the correlations here reach macroscopic orders of magnitude.

At critical points

The behavior of the correlation length near a critical point , e.g. B. a critical temperature , can be expressed by a power law : ${\displaystyle T_{\mathrm {c} }}$

${\displaystyle \xi \propto |T-T_{\mathrm {c} }|^{-\nu }\,,}$

where is a critical exponent that is the same for many systems and is therefore a universal property. ${\displaystyle \nu }$

If more and more approaches (regardless of whether from above or from below), it goes towards infinity and towards 1: ${\displaystyle T}$ ${\displaystyle T_{\mathrm {c} }}$ ${\displaystyle \xi }$ ${\displaystyle G(r)}$

${\displaystyle T\to T_{\mathrm {c} }\quad \Rightarrow \xi \to \infty \quad \Rightarrow G(r)\to 1\,.}$

The correlation can therefore be measured across the entire system very close to the critical point. This leads to critical phenomena such as critical opalescence , a milky cloudiness of a substance at the transition between liquid and gaseous state.

literature

• Klaus Stierstadt, Wilhelm T. Hering, Thomas Dorfmüller: Mechanics, Relativity, Warmth (= Bergmann-Schaefer: Textbook of Experimental Physics . Volume 1). Walter de Gruyter, Berlin / New York 1998.
• Klaus Stierstadt: Thermodynamics - From Microphysics to Macrophysics . Springer, Berlin, 2010.
• ME Fisher : Renormalization Group in Theory of Critical Behavior . In: Reviews of Modern Physics . 46, No. 4, 1974, pp. 597-616. bibcode : 1974RvMP ... 46..597F . doi : 10.1103 / RevModPhys.46.597 .
• AR Its, VE Korepin, AG Izergin, NA Slavnov: Temperature Correlation of Quantum Spins. In: arXiv - Quantum Physics (quant-ph). 2009, arxiv : 0909.4751 .
• JM Yeomans: Statistical Mechanics of Phase Transitions . Oxford Science Publications, 1992, ISBN 0-19-851730-0 .

Web links

• The Correlation Length in Spin Systems Statistical Physics, Summer Semester 2004, Prof. Dr. Igor M. Sokolov, Humboldt University Berlin
• Fluctuations in macroscopic quantities Online script Thermodynamics and Statistical Physics from the 1996 summer semester, Prof. H. Brand, University of Bayreuth.

Individual evidence

1. ↑ ^{a b} Jason Papaioannou. Stochastic finite element methods. Engineering Risk Analysis Group, Technical University of Munich, 2017.