Time correlation function decay

A common assumption in the relaxation theory is that the time-correlation function decays exponentially, with the above-mentioned correlation time as the time constant (this assumption can be rigorously derived for certain limiting situations (18)). The spectral density function is then Lorentzian and the nuclear spin relaxation rate of Eq. (7) becomes ... 

For Li, the time correlation function decays slowly compared to Na", due to the strong Li -water interaction. Thus, at low temperatures we could not detect differences in the dynamics given by the 12-6 or 10-3 functions. However, at high temperature, the 10-3 potential yields shorter times than the 12-6. For Mg", we observe that the 6-4 function yields a larger coordination... 

Molecular dynamics is a useful technique for probing transport and relaxation processes in materials, provided the relevant time correlation functions decay appreciably over times shorter than 100 ns. Unfortunately, many dynamical processes in real-life materials, especially polymers, are governed by time scales appreciably longer than this. Techniques alternative to brute force MD must be developed to predict the kinetics of such processes from molecular constitution. 

Mandelshtam V A and Taylor H S 1997 Spectral analysis of time correlation function for a dissipative dynamical system using filter diagonalization application to calculation of unimolecular decay rates Phys. Rev. Lett. 78 3274... 

TOWARDS THE HYDRODYNAMIC LIMIT STRUCTURE FACTORS AND SOUND DISPERSION. The collective motions of water molecules give rise to many hydrodynamical phenomena observable in the laboratories. They are most conveniently studied in terms of the spatial Fourier ( ) components of the density, particle currents, stress, and energy fluxes. The time correlation function of those Fourier components detail the decay of density, current, and fluctuation on the length scale of the Ijk. 

From this expression we see that the friction cannot be determined from the infinite-time integral of the unprojected force correlation function but only from its plateau value if there is time scale separation between the force and momentum correlation functions decay times. The friction may also be estimated from the extrapolation of the long-time decay of the force autocorrelation function to t = 0, or from the decay rates of the momentum or force autocorrelation functions using the above formulas. 

Lindenberg and West conclude, after analysis of Eq.(59) at low temperatures where kTccIly, that the correlation function decays on a time scale li/ kT rather than 1/y. Thus, the bath can dissipate excitations whose energies lie in the range (0/fi.y), while the spontaneous fluctuations occur only in the range (0,kT) if kTcorrelation time of the fluctuations is therefore the longer of fi/ kT and 1/y. The idea advanced by these authors is that fluctuations and dissipation can have quite distinct time scales [133], This is important if the two quantum states of the system of interest correspond to chemical interconverting states [139, 144, 145],... 

Time Correlation Function for a Dissipative Dynamical Systems Using Filter Diagonalization Application to Calculation of Unimolecular Decay Rates. 

Different equilibrium, hydrodynamic, and dynamic properties are subsequently obtained. Thus, the time-correlation function of the stress tensor (corresponding to any crossed-coordinates component of the stress tensor) is obtained as a sum over all the exponential decays of the Rouse modes. Similarly, M[rj] is shown to be proportional to the sum of all the Rouse relaxation times. In the ZK formulation [83], the connectivity matrix A is built to describe a uniform star chain. An (f-l)-fold degeneration is found in this case for the f-inde-pendent odd modes. Viscosity results from the ZK method have been described already in the present text. 

The Lyapunov exponents and the Kolmogorov-Sinai entropy per unit time concern the short time scale of the kinetics of collisions taking place in the fluid. The longer time scales of the hydrodynamics are instead characterized by the decay of the statistical averages or the time correlation functions of the... 

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-... 

Sharp and Lohr proposed recently a somewhat different point of view on the relation between the electron spin relaxation and the PRE (126). They pointed out that the electron spin relaxation phenomena taking a nonequilibrium ensemble of electron spins (or a perturbed electron spin density operator) back to equilibrium, described in Eqs. (53) and (59) in terms of relaxation superoperators of the Redfield theory, are not really relevant for the PRE. In an NMR experiment, the electron spin density operator remains at, or very close to, thermal equilibrium. The pertinent electron spin relaxation involves instead the thermal decay of time correlation functions such as those given in Eq. (56). The authors show that the decay of the Gr(T) (r denotes the electron spin vector components) is composed of a sum of contributions... 

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 ... 

Whenever the absorbing species undergoes one or more processes that depletes its numbers, we say that it has a finite lifetime. For example, a species that undergoes unimolecular dissociation has a finite lifetime, as does an excited state of a molecule that decays by spontaneous emission of a photon. Any process that depletes the absorbing species contributes another source of time dependence for the dipole time correlation functions C(t) discussed above. This time dependence is usually modeled by appending, in a multiplicative manner, a factor exp(-ltl/x). This, in turn modifies the line shape function I(co) in a manner much like that discussed when treating the rotational diffusion case ... 

Nonequilibrium Aging State (NEAS). The system is initially prepared in a nonequilibrium state and put in contact with the sources. The system is then allowed to evolve alone but fails to reach thermal equilibrium in observable or laboratory time scales. In this case the system is in a nonstationary slowly relaxing nonequilibrium state called aging state and is characterized by a very small entropy production of the sources. In the aging state two-times correlations decay slower as the system becomes older. Two-time correlation functions depend on both times and not just on their difference. 

It is the spread of oscillation frequencies a>j that causes the environment response to decohere after a (typically short) correlation time t (Figure 4.3b). Hence, the Markovian assumption that the correlation function decays to 0 instantaneously, d>(t) 8 t), is widely used it is, in particular, the basis for the... 

From the above linear theory the time-correlation function for the thermal fluctuation < ,(1) decays exponentially with the decay rate rth(q) given by... 

We decouple the above four-body correlation function in the integrand and set 3(x, t) = S(x, 0) because Stime-correlation function of the Fourier component ,(t) is given by Eqs. (6.24) and (6.25) in equilibrium. Then, Eq. (6.58) is transformed into an integration over the wave vector,... 

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. 

We have done a study by time-resolved hole-burning spectroscopy for dye molecules in polar solvents and found that the time correlation function of the hole width decays much slower than that of the peak shift of the hole, which occurs very rapidly, as you observed in the case of the fluorescence Stokes shift [K. Nishiyama, Y. Asano, N. Hashimoto, and T. Okada, J. Mol. Liquids 65/66, 41 (1995)]. 

In the treatment of a rigid dumbbell, where the whole time-correlation functions (TCF) can be solved exactly, Stockmayer and Burchard21 disclosed the origin for the discrepancy between theory and experiments. They recognized that all measurements of the TCF can be carried out down only to a limiting minimum delay time. With common instruments, this lower limit lies at about 100 ns but the lowest time is often much higher under conditions such that the TCF should have decayed to e"2 at channel 8Q220). These experimental condition imply that only an apparent first cumulant is determined defined by... 

Measurements of static light or neutron scattering and of the turbidity of liquid mixtures provide information on the osmotic compressibility x and the correlation length of the critical fluctuations and, thus, on the exponents y and v. Owing to the exponent equality y = v(2 — ti) a 2v, data about y and v are essentially equivalent. In the classical case, y = 2v holds exactly. Dynamic light scattering yields the time correlation function of the concentration fluctuations which decays as exp(—Dk t), where k is the wave vector and D is the diffusion coefficient. Kawasaki s theory [103] then allows us to extract the correlation length, and hence the exponent v. 

MCT can be best viewed as a synthesis of two formidable theoretical approaches, namely the renormalized kinetic theory [5-9] and the extended hydrodynamic theory [10]. While the former provides the method to treat both the very short and the very long time responses, it often becomes intractable in the intermediate times. This is best seen in the calculation of the velocity time correlation function of a tagged atom or a molecule. The extended hydrodynamic theory provides the simplicity in terms of the wavenumber-dependent hydrodynamic modes. The decay of these modes are expressed in terms of the wavenumber- and frequency-dependent transport coefficients. This hydrodynamic description is often valid from intermediate to long times, although it breaks down both at very short and at very long times, for different reasons. None of these two approaches provides a self-consistent description. The self-consistency enters in the determination of the time correlation functions of the hydrodynamic modes in terms of the... 

The movements capable of relaxing the nuclear spin that are of interest here are related to the presence of unpaired electrons, as has been discussed in Section 3.1. They are electron spin relaxation, molecular rotation, and chemical exchange. These correlation times are indicated as rs (electronic relaxation correlation time), xr (rotational correlation time), and xm (exchange correlation time). All of them can modulate the dipolar coupling energy and therefore can cause nuclear relaxation. Each of them contributes to the decay of the correlation function. If these movements are independent of one another, then the correlation function decays according to the product... 

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