Space and time correlation functions

Elastic and inelastic scattering experiments measure spatial and temporal correlations in the scattering medium and are therefore strongly related to the respective correlation functions describing the sample. This section will introduce some of the important correlation functions [90-92]. 

The easiestexample for establishing the relationship between the structure of a sample and its scattering behaviour is the case of monoatomic liquid [93]. In this section, the static neutron scattering function (static structure factor) will be derived. In a simple homogeneous monoatomic liquid the probability to find a certain atom in volume element d3r at the position r is given by P(r)dir. Since the liquid is homogeneous P(r = and 

N is the number of atoms. Based on this the probability to find a specific atom at r and a second one at r2 is P (n, r2). For non-interacting particles P (n, r2) can be factorised in 

If a distance-dependant interaction between the atoms of the liquid occurs this can be described in terms of the pair correlation function g(r 2) 

In the limit r —0, g( r) vanishes due to the excluded volume of the atoms. For large values of r the pair correlation function approaches 1. The differential neutron cross-section, 

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A complete insight into the statistical turbulence and turbulence process not only requires the study of the axial and transverse fluctuations but also to measure space and time correlation functions. 

Space and time-correlation functions of macrovariables around nonequilibrium steady states have recently been calculated from nonequilibrium statistical mechanics [2]. Despite this progress, fluctuation theory for nonequilibrium states remains based, in its essential aspects, on stochastic theory. Specifically, it is assumed that one can define an appropriate set of discrete variables... 

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