Phase space correlation function

While writing the above expression, the facts that have been used are GsD = CSC, GsBD = CqC0, and (G ) 1 = (CsylC l. Note that the definition of GsBD is different from the previous formulation [9] this follows from the difference in definition of CB, which has been discussed before. Here Cs and C are the phase space correlation functions defined as... 

The phase space correlation function can be expressed in terms of Maxwellian distribution and static pair correlation function and can be written as... 

The kinetic equation for the A species phase-space correlation function... 

The reactive pair phase-space correlation functions can be constructed from these fields as... 

An equation with the form of the macroscopic law in (2.16) can be obtained from the singlet field kinetic equation by projecting out the velocity dependence of the phase-space correlation functions. A comparison of the resulting equation with this macroscopic law can then yield a microscopic correlation function expression for the rate kernel. 

Before concluding this section on the implications of the pair kinetic theory for configuration space descriptions, we show that the kinetic equation may also be used to obtain the kinetic theory result for the rate kernel. This can be accomplished by projecting out the position and velocity dependence of the pair phase-space correlation function ab,ab( 2> 1 2 /) to obtain an equation for... 

From (B.14), we see that when + acts on the doublet field n 2), a contribution proportional to the triplet field n gs(123) is obtained. Thus according to the general formulation set out in Section VII.B, we consider a description based on the two fields 6 ab> abs)- Since the only nonzero damping matrix occurs in the triplet field equation, we may immediately write the following two coupled equations for the phase-space correlation functions. For the doublet field, we have... 

Now let us see how an approximate form for the memory function for F k,t), i.e., Ki, k,t) in Eq. (5.23), comes about on the basis of the results of the generalized kinetic theory. This can be done by relating the phase-space correlation function to more familiar ones, such as F k,t), Ch k,t) and Ct(A , t). For this purpose it is convenient to switch from a continuous to a discrete matrix representation of the phase-space correlation function by introducing a complete set of orthonormal momentum functions these are generally chosen to be the Hermite... 

The expansion (5.47) for the phase-space density implies a similar decomposition (mode expansion) for the phase-space correlation function... 

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. 

The phase space correlation function C(12 t) is not the appropriate quantity for the discussion of single-particle motions in a fluid. For the study of particle diffusion one should consider the van Hove self-correlation function Ss(fc, ft>), which is measured by incoherent neutron scattering experi-ments. We can calculate 5 using phase space correlation functions by introducing the single-particle density... 

In the formalism of fully renormalized kinetic theory (FRKT) one directs one s attention not to the phase space correlation function 0(12) but to the associated memory function < (12), which is defined by " ... 

Let s analyze the relation between a so-called phase space correlation function and the conjugate response (susceptibility), which is a measure of the system s reaction to an externally imposed perturbation. Suppose one takes a system at equilibrium with Hamiltonian Hq and applies a time dependent external force starting at t=0, so that (McQuarrie,... 

There are two distinct contributions to the flux. The initial 3-correlated contribution, which gives rise to the transition state rate, and a retarded backflow j t) associated with third-body collisions. The temporal characteristics of the flux can be determined from the phase space distribution function R, t R 0), R(0)) which, for the inverted parabolic potential, is ... 

The low-order phase-space density correlation functions are the quantities of primary interest to us. For example, the singlet correlation function C .(X, x, z) = C -(l, 1 z) yields information about the configuration space correlation functions, discussed in connection with the phenomenological rate laws in Section II, when integrations over field point velocities are carried out [cf. (2.15)],... 

We have studied a kinetic theory approach to the calculation of dynamical correlations in a dense fluid. By considering density correlations in phase space one can discuss the dynamics in terms of renormalized molecular interactions that take into account the existence of structural correlations in the fluid. The advantage of this approach is that through the properties of a single function, the phase space memory function, one can treat in a unified manner a variety of correlation functions without restriction to either wavelength-frequency domain or fluid density. 

The description of chemical reactions as trajectories in phase space requires that the concentrations of all chemical species be measured as a function of time, something that is rarely done in reaction kinetics studies. In addition, the underlying set of reaction intennediates is often unknown and the number of these may be very large. Usually, experimental data on the time variation of the concentration of a single chemical species or a small number of species are collected. (Some experiments focus on the simultaneous measurement of the concentrations of many chemical species and correlations in such data can be used to deduce the chemical mechanism [7].)... 

For strongly structured microemulsions, g is negative, and the structure functions show a peak at nonzero wavevector q. As long as g < 2 /ca, inverse Fourier transform of S q) still reveals that the water-water correlation functions oscillate rather than decay monotonically. The lines in phase space where this oscillating behavior sets in are usually referred to as disorder lines, and those where the maximum of S q) moves away from zero as Lifshitz lines. ... 

The present theory can be placed in some sort of perspective by dividing the nonequilibrium field into thermodynamics and statistical mechanics. As will become clearer later, the division between the two is fuzzy, but for the present purposes nonequilibrium thermodynamics will be considered that phenomenological theory that takes the existence of the transport coefficients and laws as axiomatic. Nonequilibrium statistical mechanics will be taken to be that field that deals with molecular-level (i.e., phase space) quantities such as probabilities and time correlation functions. The probability, fluctuations, and evolution of macrostates belong to the overlap of the two fields. 

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 equation gives the dynamics of the quantum-classical system in terms of phase space variables (R, P) for the bath and the Wigner transform variables (r,p) for the quantum subsystem. This equation cannot be simulated easily but can be used when a representation in a discrete basis is not appropriate. It is easy to recover a classical description of the entire system by expanding the potential energy terms in a Taylor series to linear order in r. Such classical approximations, in conjunction with quantum equilibrium sampling, are often used to estimate quantum correlation functions and expectation values. Classical evolution in this full Wigner representation is exact for harmonic systems since the Taylor expansion truncates. 

Finally, we return to the case of antisymmetric surfaces, i.e. a situation where one surface of the thin film favors the A-rich phase and the other the B-rich phase (Fig. Id). Simulations were recently carried out [266] in order to test the predictions Eq. (127) on the anomalous interfacial broadening (Sect. 2.5). Figure 24 demonstrates that this phenomenon can indeed be readily observed. Using the interfacial tension a that has been independently measured [215], o= 0.015, and the correlation length 3.6 lattice spacings from a direct study of the bulk correlation function, one can evaluate Eq. (127) quantitatively, using... 

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