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Astron. Astrophys. 326, 433-441 (1997)

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3. Correlation functions

The mathematical tools that allow important filtered information to be extracted from the distribution of particles are the correlation functions. These will enable us to ascertain whether the distribution is Poissonian, or how large the difference is with respect to a Poissonian distribution.

Hereafter, I define certain functions that contain the symbol [FORMULA] as meaning an average of the quantity [FORMULA]. There are two different ways of calculating these averages: (i) by volume, and (ii) statistically-mechanically. I now describe the two ways:

(i) In this case, the average is calculated according to the expression

[EQUATION]

where [FORMULA] represents the quantity whose average we wish to obtain. It is clear that we are calculating a volume average. This is the method that describes the distribution macroscopically. The observational correlations are extracted with this algorithm.

(ii) In this case, the average calculation is conducted by means of the expression

[EQUATION]

[EQUATION]

where [FORMULA] is the same as in (i). This method appeals to the microscopic properties of the physical system.

The definitions that follow include an expansion with the second averaging method.

3.1. Particle-particle two-point correlation function

The density of the particles is [FORMULA].

The two-point correlation function for particles without autocorrelation ( [FORMULA] in the following expression) is, from ( 6 ) and ( 9 ),

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

In order to simplify the notation, the mass dependence of U will not be indicated in what follows. A straightforward calculation shows that

[EQUATION]

[EQUATION]

If we assume isotropy, the function only depends on [FORMULA] ; I denote this as [FORMULA].

3.2. Mass-mass two-point correlation function

The mass density is [FORMULA]. The two-point correlation function for mass without autocorrelation is

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

With isotropy, the function depends on [FORMULA].

3.3. Mass-particle two-point correlation function

With the same philosophy in mind, we define other functions as the two-point correlation function mass-particle:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

With isotropy, the function depends on [FORMULA].

3.4. Mass-mass-particle three-point correlation function

This function is defined as follows:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

With isotropy, this function depends on three variables: [FORMULA], [FORMULA] and [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: October 15, 1997
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