A **correlation functions** can determine the correlation of two random variables or systems. “Correlation” is a measure of how one value or system responds to another. Many different types of correlation function exist. Their exact definitions depend on what field you’re working in.

For example, one correlation function used 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.

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 changes over time.

## 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]:

**C _{αβ}(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].

## References

[1] Croker, J. & Weeks, E. What is the Pair Correlation Function? Retrieved April 4, 2021 from: http://www.physics.emory.edu/faculty/weeks//idl/gofr.html

[2] B. The correlation function: galaxies. Retrieved April 4, 2021 from: https://ned.ipac.caltech.edu/level5/March04/Jones/Jones5_2.html

[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: https://ocw.mit.edu/courses/chemistry/5-74-introductory-quantum-mechanics-ii-spring-2009/lecture-notes/MIT5_74s09_lec05.pdf CC Sharealike 4.0.

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