Correlation Function: Definition, Examples

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Many different types of correlation function exist. Their exact definitions depend on what field you’re working in. For example:

  • In statistics, a correlation function can find the correlation of two random variables or systems. “Correlation” is a measure of how one value or system responds to another.
    correlation function
    A correlation function can show how systems are correlated.

    The strength of correlation depends on the spatial or temporal distance between the random variables.
  • Also in statistics, the Auto Correlation Function (also called a correlogram) shows serial correlation in data (where error terms transfer from one period to another) that change over time.
  • In statistical mechanics, a mathematical correlation function measures order in a system and describes spatial correlation—how microscopic variables, like density or spin, are related at different positions. For example, the function is often used to describe how spins of ferro- and antiferromagnetic materials align with respect to their nearest neighbors. One application in physics assesses the probability of finding a particle’s center relative to another particle’s center [1]. This can be extended to galaxies, as a measure of the excess probability of finding a galaxy at a certain distance from another galaxy, compared to what you would expect with a random distribution of galaxies. See [2] for a slightly different definition and more details on cosmology applications.

Time Correlation Function

Time correlation functions, or time-dependent correlation functions, are used in the theory of noise and stochastic processes including statistical physics and spectroscopy. They are a measure of the correlation of two dynamical properties over time. Mathematically, the correlation function is defined as [3]:
Cαβ(t) = <α(0)β(t)>
The brackets <> indicate an average for the equilibrium ensemble.
If the two properties αβ are the same, the correlation function is called an autocorrelation function. If they are different, it’s a cross-correlation function.

Time-correlation functions are also used in quantum mechanics, where they represent the dynamics of a system. They give a statistical description of an ensemble variable’s time-evolution of at thermal equilibrium. These functions are often used to model both random and stochastic irreversible processes in condensed phases [4].


[1] Croker, J. & Weeks, E. What is the Pair Correlation Function? Retrieved April 4, 2021 from:
[2] B. The correlation function: galaxies. Retrieved April 4, 2021 from:
[3] Berne, B. & Harp, G. (1970). On the Calculation of Time Correlation Functions. Advance in Chemical Physics, Volume XVII.
[4] MIT Open Courseware. (2009). 5.74 Introductory Quantum Mechanics II. Retrieved April 4, 2021 from: CC Sharealike 4.0.

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