General time correlation function

Time correlation functions of more complicated form can be calculated in a similar way. For example, the time correlation function 

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Thus /4(tto) contains both the Hamiltonian and its own intrinsic evolution built in, and this may be recognized as defining the Heisenberg representation of any arbitrary operator. A general time correlation function associated with two general TD operators A t) and B t representing two different physical entities at different times t and t, is defined then by the formula... 

General Time Correlation Functions Centroid Molecular Dynamics Method... 

The latter connection is important for the mathematical justification [5] of CMD (cf. Section III.B.l) and for one particular CMD-based approach [4,5] for computing general time correlation functions (cf. Section III.B.3). 

Stochastic equation for reptation dynamics Although the above probabilistic description is quite useful in understanding the essence of reptation dynamics, it becomes progressively more difficult to proceed with the calculation for other types of time correlation function. For example, it is not easy to calculate the mean square displacement of a primitive chain segment (R(s, t)-R(s, 0)) ) by this method. In this section we shall describe a convenient method" for calculating general time correlation functions. 

Notice that in this general case, correlation functions cannot be solved for directly instead, there is an entire hierarchy of lower-order correlations expressed as functions of higher-order correlations. For example if we take an average of equation 7.79 over all space-time histories, and assume that we have a steady-state so... 

Fig. 18. Stress time-correlation function of EV linear chains and stars with different functionalities. Comparison of Brownian dynamics (crosses) and generalized Zimm calculations from MC averages (solid lines). Reprinted with permission from [89]. Copyright (1996) American Institute of Physics...

Much less attention has been paid to the dynamic properties of water at the solution/metal interface (or other interfaces). Typical dynamic properties that are of interest include the diffusion constant of water molecules and several types of time correlation functions. In general, the time correlation function for a dynamic variable of interest A(t) is defined as... 

Consider a general system described by the Hamiltonian of Eq. (5), where = Huif) describes the interaction between the spin system (7) and its environment (the lattice, L). The interaction is characterized by a strength parameter co/i- When deriving the WBR (or the Redfield relaxation theory), the time-dependence of the density operator is expressed as a kind of power expansion in Huif) or (17-20). The first (linear) term in the expansion vanishes if the ensemble average of HiL(t) is zero. If oo/z,Tc <5c 1, where the correlation time, t, describes the decay rate of the time correlation functions of Huif), the expansion is convergent and it is sufficient to retain the first non-zero term corresponding to oo/l. This leads to the Redfield equation of motion as stated in Eq. (18) or (19). In the other limit, 1> the expan-... 

The important step of identifying the explicit dynamical motivation for employing centroid variables has thus been accomplished. It has proven possible to formally define their time evolution ( trajectories ) and to establish that the time correlations ofthese trajectories are exactly related to the Kubo-transformed time correlation function in the case that the operator 6 is a linear function of position and momentum. (Note that A may be a general operator.) The generalization of this concept to the case of nonlinear operators B has also recently been accomplished, but this topic is more complicated so the reader is left to study that work if so desired. Furthermore, by a generalization of linear response theory it is also possible to extract certain observables such as rate constants even if the operator 6 is linear. 

As a new subject we have considered the effect of the frequency-dependence of the elastic moduli on dynamic light scattering. The resultant nonexponential decay of the time-correlation function seems to be observable ubiquitously if gels are sufficiently compliant. Furthermore, even if the frequency-dependent parts of the moduli are very small, the effect can be important near the spinodal point. The origin of the complex decay is ascribed to the dynamic coupling between the diffusion and the network stress relaxation [76], Further scattering experiments based on the general formula (6.34) should be very informative. 

Spectroscopic techniques have been applied most successfully to the study of individual atoms and molecules in the traditional spectroscopies. The same techniques can also be applied to investigate intermolecular interactions. Obviously, if the individual molecules of the gas are infrared inactive, induced spectra may be studied most readily, without interference from allowed spectra. While conventional spectroscopy generally emphasizes the measurement of frequency and energy levels, collision-induced spectroscopy aims mainly for the measurement of intensity and line shape to provide information on intermolecular interactions (multipole moments, range of exchange forces), intermolecular dynamics (time correlation functions), and optical bulk properties. 

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... 

S. Mukamel In general, multipulse experiments depend on a multitime correlation function of the dipole operator [1], The term x(n) depends on a combination of n + 1 time correlation functions. Their behavior for large n will depend on the model. In some cases (e.g., the accumulated photon echo used by Wiersma) the multiple-pulse sequence is simply used to accumulate a large signal and the higher... 

At present the complete time dependence of only a few time-correlation functions have been determined experimentally. Furthermore, the theory of time-dependent processes is such that we know in principle which experiments can be used to determine specific correlation functions, and in addition certain general properties of these correlation functions. However, one of the major difficulties encountered in developing a theory of time-correlation functions arises from the fact that there seems to be, at least at present, no simple way of bypassing the complex many-body dynamics in a realistic fashion. Consequently both theoretically and experimentally there are difficult obstacles impeding progress towards a satisfactory understanding of the dynamic behavior of liquids, solids, and gases. 

Spectral lineshapes were first expressed in terms of autocorrelation functions by Foley39 and Anderson.40 Van Kranendonk gave an extensive review of this and attempted to compute the dipolar correlation function for vibration-rotation spectra in the semi-classical approximation.2 The general formalism in its present form is due to Kubo.11 Van Hove related the cross section for thermal neutron scattering to a density autocorrelation function.18 Singwi et al.41 have applied this kind of formalism to the shape of Mossbauer lines, and recently Gordon15 has rederived the formula for the infrared bandshapes and has constructed a physical model for rotational diffusion. There also exists an extensive literature in magnetic resonance where time-correlation functions have been used for more than two decades.8... 

In the previous section it was shown how classical many-body systems can be studied by computer experiments. Actual laboratory experiments probe real systems which are, strictly speaking, entirely quantum-mechanical in nature. What, then, is the relationship between the classical and quantum-mechanical time-correlation function of the dynamical variable 0, To expedite this discussion consider the one sided function < /+,(0) /j(t)>. This correlation function is in general complex with real part , (0 and imaginary part [Pg.78]

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

There exists another prescription to extend the hydrodynamical modes to intermediate wavenumbers which provides similar results for dense fluids. This was done by Kirkpatrick [10], who replaced the transport coefficients appearing in the generalized hydrodynamics by their wavenumber and frequency-dependent analogs. He used the standard projection operator technique to derive generalized hydrodynamic equations for the equilibrium time correlation functions in a hard-sphere fluid. In the short-time approximation the frequency dependence of the memory kernel vanishes. The final result is a... 

A centroid trajectory for a given set of initial centroid conditions must contain some degree of dynamical information due to the nonstationarity of the ensemble created by the centroid constraints. It is therefore important to explore the correlations in time of these trajectories. In the centroid dynamics perspective, a general quantum time correlation function can be expressed as... 

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