Four-point correlation function

Having determined the effect of the diffusive interfaces on the structure parameters, we now turn to the calculation of H and K in microemulsions. In the case of oil-water symmetry three-point correlation functions vanish and = 0. In order to calculate K from (77) and (83) we need the exphcit expressions for the four-point correlation functions. In the Gaussian approximation... 

The above results show that the structure of the system with the molecules self-assembled into the internal films is determined by their correlation functions. In contrast to simple fluids, the four-point correlation functions are as important as the two-point correlation functions for the description of the structure in this case. The oil or water domain size is related to the period of oscillations A of the two-point functions. The connectivity of the oil and water domains, related to the sign of K, is determined by the way four moleeules at distanees eomparable to their sizes are eorrelated. For > 0 surfactant molecules are correlated in such a way that preferred orientations... 

A. Ciach. Four-point correlation functions and average Gaussian curvature in microemuisions. Phys Rev E 55 1954-1964, 1997. 

Appendix C Four-Point Correlation Function Expression for Fluorescence Spectra Appendix D Phase-Space Doorway-Window Wavepackets for Fluorescence Appendix E Doorway-Window Phase-Space Wavepackets for Pump-Probe Signals References... 

This formula resembles Eq. (2.6) for the autocorrelation signal. We can further expand 5(l)( ,r) to second order in the pump field and express the result in terms of the four-point correlation function (2.8) (see Appendix E). 

APPENDIX C FOUR-POINT CORRELATION FUNCTION EXPRESSION FOR FLUORESCENCE SPECTRA... 

We now derive the expression for the fluorescence signal in terms of the doorway and window wavepackets instead of the four-point correlation function. We start with Eq. (3.1) and write the four-point correlation function F(4) explicitly as the trace with respect to the equilibrium density matrix. We then use the cyclic invariance of the trace and obtain... 

We shall calculate here the pump-probe signal (4.1) using the doorway window wavepackets representation. The polarization 3 2(/> to third order in the external field is given in Ref. 17 and is shown to be expressed in terms of the four-point correlation function (2.8) ... 

It should be noted that although in Eq. (90) only the connected motion of the solute and the solvent is retained, in the argument presented on the time scale it is the disconnected parts which have been considered. This is because in the latter part, for the derivation of the expression of Ci. the solute and the solvent motions are assumed to be disconnected. This assumption is the same as those made in the density functional theory and also in mode coupling theories where a four-point correlation function is approximated as the product of two two-point correlation functions. This approximation when incorporated in Ci. means that after the binary collision takes place, the disturbances in the medium will propagate independently. A more exact calculation would be to consider the whole four-point correlation function, thus considering the dynamics of the solute and the solvent to be correlated even after the binary collision is over. Such a calculation is quite cumbersome and has not been performed yet. 

P. A. Fleury. Light Scattering Determinations of Four Point Correlation Functions. In J. Woods Halley (ed.). Proceedings of the NATO Advanced Study Institute on Correlation Functions and Quasiparticle Interactions, Plenum, New York, 1978, pp. 325-361. 

In terms of this four-point correlation function, we have6,19... 

In previous simulation studies of collision induce effects in rare gases, y=0 and only the first term in (56b) contributes to AA2 [92]. The first and second rank collision induced correlation functions, (57) and (60), are notoriously difficult to calculate they involve four point correlation functions, (i induced by j correlated with k induced by Z and the simulation runs in [90, 91, 92] are for tens of thousands of timesteps). 

The molecular bond direction is defined by an imaginary line between one pair of the nearest neighbour molecules [23]. This means that the measurement of bond-orientation correlation requires a four-point correlation function. To date no appropriate experimental tool allows us to conduct this kind of measurement. Thus the degree of bond-orientation correlation can only be inferred indirectly. Moreover, no available physical field exists which can couple directly to the hexatic order in the hexatic B phase. Consequently, the heat-capacity measurement is one of the most powerful experimental probes used to investigate the nature of the SmA-SmBhex transition and to complement structural identification by X-ray or electron diffraction. 

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