Time correlation function, equation

Now let s see how (u(O)u(t)), a property of the fluctuations in equilibrium, is related to a kinetic property, in this case the diffusion coefficient D. Integrate the time correlation function Equation (18.68) over all the possible time lags t. 

To complete the description and get the connection with the solute emission and absorption spectra, there is need of the correlation functions of the dipole operator pj= (a(t)+af(t))j and, consequently, the differential equation for the one solute mode has to be solved. The reader is referred to [133] for detailed analysis of this point as well as the equations controlling the relaxation to equilibrium population. The energy absorption and emission properties of the above model are determined by the two-time correlation functions ... 

From the above equation it appears convenient to characterize solvation dynamics by means of the solvation time correlation function C(t), defined asa)... 

The equations for dynamic LS require a more detailed outline. Here a time correlation function (TCE) of the scattering intensity is measured that is given as [60]... 

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 electron-spin time-correlation functions of Eq. (56) were evaluated numerically by constructing an ensemble of trajectories containing the time dependence of the spin operators and spatial functions, in a manner independent of the validity of the Redfield limit for the rotational modulation of the static ZFS. Before inserting thus obtained electron-spin time-correlation functions into an equation closely related to Eq. (38), Abernathy and Sharp also discussed the effect of distortional/vibrational processes on the electron spin relaxation. They suggested that the electron spin relaxation could be described in terms of simple exponential decay rate constant Ts, expressed as a sum of a rotational and a distortional contribution ... 

Collective Modes and Time-Correlation Functions. Our linear equations (6.15) and (6.16) describe two characteristic kinds of collective modes in gels the longitudinal part of u obeys Tanaka s equation (4.16) and g = V u is governed by the diffusion equation (4.18), while the transverse parts of u and are coupled to form a slow transverse sound at small wave numbers. By assuming the space-time dependence as exp(i[Pg.99]

Statement 1. Provided K(t) — K — const, i.e., neglecting change in time of the correlation functions, equations (8.2.12) and (8.2.13) of the concentration dynamics describe undamped concentration oscillations with the frequencies u < = y/aj3, dependent on the initial conditions. The de-... 

In this section some details of the static and dynamic structure factors and on the first cumulant of the time correlation function are given. Hie quoted equations are needed before the cascade theory can be applied. This section may be skipped on a first reading if the reader is concerned only with the application of the branching theory. 

There have been a number of attempts to calculate time-correlation functions on the basis of simple models. Notable among these is the non-Markovian kinetic equation, the memory function equation for time-correlation functions first derived by Zwanzig33 and studied in great detail by Berne et al.34 This approach is reviewed in this article. Its relation to other methods is pointed out and its applicability is extended to other areas. The results of this theory are compared with the results of molecular dynamics. 

Linear response theory is reviewed in Section II in order to establish contact between experiment and time-correlation functions. In Section III the memory function equation is derived and applied in Section IV to the calculation of time-correlation functions. Section V shows how time-correlation functions can be used to guess time-dependent distribution functions and similar methods are then applied in Section VI to the determination of time-correlation functions. In Section VII a succinct review is given of other exact and experimental calculations of time-correlation functions. 

It is possible to derive an equation which describes the time evolution of the time-correlation function Cn(t) where C stands for different autocorrelation functions depending on the definition of the scalar product (i), (ii), or (iii) of Eq. (73) adopted. 

Ku(t) is called the memory function/ and the equation for the time-correlation function that we derived is called the memory function equation.33,34,42 Note that the propagator in this equation contains the projection operator Pt. Further note that the memory function is an even function of the time,... 

Doob s theorem states that a Gaussian process is Markovian if and only if its time correlation function is exponential. It thus follows that V is a Gaussian-Markov Process. From this it follows that the probability distribution, P(V, t), in velocity space satisfies the Fokker-Planck equation,... 

The memory function equation for the time-correlation function of a dynamical operator Ut can be cast into the form of a continued fraction as was first pointed out by Mori.43 We prove this in a different way than Mori. In order to proceed it is necessary to define the set of memory functions K0 t),. .., Kn t). .., such that... 

Time-correlation functions Cu(t) obey the memory function equation... 

The Schofield approximation is useful insofar as it gives an approximate quantum-mechanical time-correlation function which satisfies the condition of detailed balance as it must. Needless to say if (f) is equated to v / (/) the condition of detailed balance will not hold. It should be noted that the Schofield approximation does not satisfy the moment sum rules on <]>, (/). It was for this reason that Egelstaff proposed his y time approximation. Egelstaff showed that if y2 — t2 — ihfit, then taking... 

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... 

Thus the equation for the time correlation function can be written as... 

In the previous section we find that the time correlation function can be obtained from the relaxation equation. When A is not a single variable but represents a column matrix containing a set of coupled variables, then the... 

The quantities in (4.40) are single time quantities. According to Eq. (4.38) we need the special correlation function (8pay(PP t)8 Uya( p pt)) this function is closely connected to the correlation function (SpSp). According to the discussions presented in Refs. 12 and 28 for the determination of such correlation functions, one has to start from a differential equation for the corresponding two-time correlation functions and use the relevant single-time correlation function as initial condition. From the equation for the fluctuations 8p, which reads... 

However, delayed dissipation typically occurs after the s-region has settled into its thermal equilibrium, with RDOp peq, and the more familiar equations for the dissipative rate can be used. Then yVl( -u)(/, //) is obtained to second order from a p-s coupling Hps = A(p B(SJ>, with trs(r(s>Bs3>) = 0 chosen for convenience, and can be written in terms of the time-correlation functions Cp-7 t ))) = trs[B t)B t )feq] of the atomic... 

We see that the first two terms in the square parenthesis in Eq. (F.68) are identical to the first two terms in Eq. (F.69). The next two terms in the equations are also almost identical, except for the sign. Taking this into account we see that the coordinate representation of the time-correlation function in Eq. (F.58) may finally be written as... 

For the dynamical distribution it will in general be necessary to consider both the auto and cross time correlation functions of the 0-1 and the 1-2 frequencies (117). For example, if the fluctuations, <5A(t), in the anhar-monicity are statistically independent of the fluctuations in the fundamental frequency, the oscillating term (1 — elAt3) in Equation (18) would be damped. In a Bloch model the fluctuations in anharmonicity translate into different dephasing rates for the 0-1 and 1-2 transitions that were discussed previously for two pulse echoes of harmonic oscillators. Thus we see that even if A vanishes, the third-order response can be finite (94). 

Each molecular vibration factor in Equation (3) is a type of molecular time correlation function for the internal vibrational dynamics. In the harmonic approximation, i) and f) would reduce to the harmonic vibrational eigenstates and the qj would be the actual molecular normal modes. Then one has the simplification... 

The first system we consider is the solute iodine in liquid and supercritical xenon (1). In this case there is clearly no IVR, and presumably the predominant pathway involves transfer of energy from the excited iodine vibration to translations of both the solute and solvent. We introduce a breathing sphere model of the solute, and with this model calculate the required classical time-correlation function analytically (2). Information about solute-solvent structure is obtained from integral equation theories. In this case the issue of the quantum correction factor is not really important because the iodine vibrational frequency is comparable to thermal energies and so the system is nearly classical. 

There are also situations when one is not in the classical limit, and so Equation (13) would not seem applicable, and instead one would like to approximate one of the quantum mechanical expressions for Ti by relating the relevant quantum time-correlation function to its classical analog. For the sake of definiteness, let us consider the case where the oscillator is harmonic and the oscillator-bath coupling is linear in q, as discussed above. In this case k 0 can be written as... 

In the oxygen VER experiments (3) the n = 1 vibrational state of a given oxygen molecule is prepared with a laser, and the population of that state, probed at some later time, decays exponentially. Since in this case tiojo kT, we are in the limit where the state space can be truncated to two levels, and 1/Ti k, 0. Thus the rate constant ki o is measured directly in these experiments. Our starting point for the theoretical discussion is then Equation (14). For reasons discussed in some detail elsewhere (6), for this problem we use the Egelstaff scheme in Equation (19) to relate the Fourier transform of the quantum force-force time-correlation function to the classical time-correlation function, which we then calculate from a classical molecular dynamics computer simulation. The details of the simulation are reported elsewhere (4) here we simply list the site-site potential parameters used therein e/k = 38.003 K, and a = 3.210 A, and the distance between sites is re = 0.7063 A. 

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