Time-space correlation function

It was recently shown that a formal density expansion of space-time correlation functions of quantum mechanical many-body systems is possible in very general terms [297]. The formalism may be applied to collision-induced absorption to obtain the virial expansions of the dipole... 

Both Pecora (16) and Komarov and Fisher (17) adapted van Hove s space-time correlation function approach for neutron scattering (18) to the light-scattering problem to calculate the spectral distribution of the light scattered from a solution. Using a molecular analysis, Pecora assumed the scattering particles to be undergoing Brownian motion, and predicted a Lorentzian line shape for the spectral distribution of the... 

Gs(r, t) and Gd(r, t) are called the Van Hove self and distinct space-time correlation functions.18 It clearly follows that probability distribution describing the event that a molecule is at the origin at t = 0 and at the point r at the time t. Gs(r, t) is consequently the probability distribution characterizing the net displacement or diffusion of a particle in the time t. Gd(r, t) on the other hand is a probability distribution describing the event that a molecule is at the origin at t = 0 and a different molecule is at the point r at the time t. Gd(r, t) describes the correlated motion of two molecules. It should be noted that the initial value of Gs(r, t) is... 

Hove self space-time correlation function. [8] In other words,... 

The basic formulation of this problem was given by Van Hove [25] in the form of his space-time correlation functions, G ir, t) and G(r, t). He showed that the scattering functions, as defined above, for a diffusing system are given by the Fourier transformation of these correlation functions in time and space. Incoherent scattering is linked to the self-correlation function, Gs(r, t) which provides a full definition of tracer diffusion while coherent scattering is the double Fourier transform of the full correlation function which is similarly related to chemical or Fick s law diffusion. Formally the equations can be written ... 

GS(R, t) appears so frequently that it has been given the name Van-Hove self space-time correlation function after Leon Van-Hove, who first demonstrated its relationship to neutron scattering (Van Hove, 1954). It was asserted above that G (R, t) is related to Fs(q, t). What is this relationship ... 

The first cumuleint F is generally calculated by assuming the hydro namic interaction eis described by Oseen (3) where no knowledge of the space-time correlation function is needed (4-6>. The purpose of the present contribution is an experimental test of theoretical relationships which are based on the Flory-Stockmayer (FS) breinching theory (7) of the solution properties from reui-domly crossllnked monodisperse primary chains. Most of the theoretical work emd peirt of the e q>erlmental work has been piA>llshed previously (8-13). We, therefore, bring here only a short outline of the theory and confine ourselves mainly to the discussion of the namlc properties. 

Dynamic structure information can be obtained from the Space-Time Correlation Function, G(r, t).G( r, t ) expresses the probability that an atom is at position r at time t, given that an atom was at the origin r = 0 at initial time t — Q. G(r, t)can be separated into two parts. Terms having i = j yields the self space-time correlation function, for which the atom at... 

Figure 22. Distinct space-time correlation function for 0,2Na,2O, cOa,O.(0.8 —x)Si02 glass.

In Sections 2 and 3, we set up a formalism for dealing with the dynamics of dense fluids at the molecular level. We begin in Section 2 by focusing attention on the phase space density correlation function from which the space-time correlation functions of interest in scattering experiments and computer simulations can be obtained. The phase space correlation function obeys a kinetic equation that is characterized by a memory function, or generalized collision kernel, that describes all the effects of particle interactions. The memory function plays the role of an effective one-body potential and one can regard its presence as a renormalization of the motions of the particles. 

We will construct our theory in terms of C(12). Once we find C(12) all the space-time correlation functions of interest can be calculated. For example, the density-density correlation function, " also known as the dynamic structure factor, is given by... 

This is a set of coupled integral equations that can be readily converted into a set of coupled algebraic equations for the space-time correlation functions Fy ... 

The space-time correlation function like the radial distribution function has a simple physical meaning which is most helpful in suggesting simple models for the construction of this function. As we wiD show below the construction of this function is the first step in caloilating the frequency distribution of the scattered light. 

The Van Hove space-time correlation function, G(Ro, x), is the conditional probability that given the fact that a scattering particle is at the point r(0) at t = 0, the same or another particle will be at the point... 

Thus if one wishes to calculate the spectrum of light scattered from moving particles one must do the calculation in three steps (1) Construct Gs(l o, r) the Van Hove space-time correlation function. For many types of motion such a construction is quite straightforward from the physical significance of G (Ro,z). (2) Evaluate the integral, Eq.(39) using Gs(Ro> previously constructed to obtain Co(t) the auto-correlation function. Since this quantity can in fact be measured one may wish to stop here. (3) Using the Wiener-Kinchine theorem, Eq. (36), obtain the spectrum I(co). 

The space-time correlation function of the response is obtained by average of the product of v(zi, t) and v(z2, t + x), which are given by the impulse response function... 

The reflection coefficient 5(A T, co) can finally be written in a convenient form, namely as the Fourier transform in r and time t over a space-time correlation function G(r, t)... 

In applications of LLS, we often need to understand theories for both scattering techniques and properties of complex fluids, partly because scattering is an observation not in real space but in Fourier space. Therefore, the results do not yield direa images of stmctures and dynamics. Furthermore, soft matter often does not have definite stmctures and/or simple dynamics, but exhibits stochastic behavior described theoretically in terms of the space-time correlation function. In this sense, scattering methods are suitable and irreplaceable tools for investigating soft condensed matter. 

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