The Time Correlation Function

In dynamic or quasi-elastic light scattering, a time dependent correlation function (i (0) i (t)) = G2 (t) is measured, where i (0) is the scattering intensity at the beginning of the experiment, and i (t) that at a certain time later. Under the conditions of dilute solution (independent fluctuation of different small volume elements), the intensity correlation function can be expressed in terms of the electric field correlation function gi (t) 

The denominator in Eq. (B.25) is recognized as the static scattering function, and the numerator is called the dynamic structure factor S (q, t) which for a homopolymer is given as 

The angle brackets now denote the average over the space-time distribution, i.e. the average has to be taken over all possible positions of the j-th element at time zero and over all possible positions of the k-th element at a delayed time t. 

For a polydisperse system of homopolymers, one finds in the same manner as outlined for the static scattering66,70 72) 

We have already given in Eq. (B.24) the average of the exponential function for the static scattering assuming Gaussian statistics. Before the corresponding average for the 

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In effect, i is replaced by the vibrationally averaged electronic dipole moment iave,iv for each initial vibrational state that can be involved, and the time correlation function thus becomes ... 

Here, I(co) is the Fourier transform of the above C(t) and AEq f is the adiabatic electronic energy difference (i.e., the energy difference between the v = 0 level in the final electronic state and the v = 0 level in the initial electronic state) for the electronic transition of interest. The above C(t) clearly contains Franck-Condon factors as well as time dependence exp(icOfvjvt + iAEi ft/h) that produces 5-function spikes at each electronic-vibrational transition frequency and rotational time dependence contained in the time correlation function quantity <5ir Eg ii,f(Re) Eg ii,f(Re,t)... 

A powerful analytical tool is the time correlation function. For any dynamic variable A (it), such as bond lengths or dihedral angles, the time autocorrelation function Cy) is defined... 

We discuss the rotational dynamics of water molecules in terms of the time correlation functions, Ciit) = (P [cos 0 (it)]) (/ = 1, 2), where Pi is the /th Legendre polynomial, cos 0 (it) = U (0) U (it), u [, Is a unit vector along the water dipole (HOH bisector), and U2 is a unit vector along an OH bond. Infrared spectroscopy probes Ci(it), and deuterium NMR probes According to the Debye model (Brownian rotational motion), both... 

M uj) is the default model, by which additional knowledge about system properties can be incorporated. Minimum additional knowledge is equivalent to M uS) = const. Without data, 5" is maximized by A uj) = M uj). measures the deviation of the time correlation function Q computed from a proposed A via Eq. (32) from the PIMC value G at the point in imaginary time,... 

Theor. Phys. 33, 423-55 (1965) A continued-fraction representation of the time-correlation functions, Ibid. 34, 399-416 (1965). 

Kometani K., Shimizu H. Study of the dipolar relaxation by a continued fraction representation of the time correlation function, J. Phys. Soc. Japan 30, 1036-48 (1971). 

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. 

For liquids, few simple and widely accepted theories have been developed. The shear viscosity can be related to the way in which spontaneous fluctuations relax in an equilibrium system, leading to the time correlation function expression " " ... 

Equilibrium molecular dynamics simulations have been performed to obtain the solution of the time correlation function (Table 14). ... 

That the time correlation function is the same using the terminal velocity or the coarse velocity in the intermediate regime is consistent with Eqs (53) and (54). 

It was necessary periodically to generate an adiabatic trajectory in order to obtain the odd work and the time correlation functions. In calculating E (t) on a trajectory, it is essential to integrate E)(t) over the trajectory rather than use the expression for E (T(f)) given earlier. This is because is insensitive to the periodic boundary conditions, whereas j depends on whether the coordinates of the atom are confined to the central cell, or whether the itinerant coordinate is used, and problems arise in both cases when the atom leaves the central cell on a trajectory. 

Consequently, the time correlation function given by Onsager-Machlup theory... 

Arguably a more practical approach to higher-order nonequilibrium states lies in statistical mechanics rather than in thermodynamics. The time correlation function gives the linear response to a time-varying field, and this appears in computational terms the most useful methodology, even if it may lack the... 

Starting with a crude model of a polymer melt, consisting of anharmonic springs connecting repulsive beads, with a bending potential to represent the polymer stiffness, the authors show how the time correlation function, C(f), should be expected to behave as a function of the polymer stiffness. 

Linear response theory10 provides a link between the phenomenological description of the kinetics in term of reaction rate constants and the microscopic dynamics of the system [33]. All information needed to calculate the reaction rate constants is contained in the time correlation function... 

The slope of (7(f) in the time regime rmoi < f forward reaction rate constant. Thus, for the calculation of reaction rate constants it is sufficient to determine the time correlation function (7(f). In the following paragraphs we will show how to do that in the transition path sampling formalism. 

To make this idea more precise we rewrite the time correlation function C t) in terms of integrals over pathways... 

This probability distribution is similar to the probability distribution from (7.28), except that here we average only over configurations that have evolved from configurations in. (/ a time t earlier. By integrating this probability distribution over all order parameter values corresponding to region 23 we obtain the total probability that a system initially in 2 is in 23 at time t later. This is nothing other than the time correlation function C(t) and we can write... 

Finally, in Sect. 7.6, we have discussed how various free energy calculation methods can be applied to determine free energies of ensembles of pathways rather than ensembles of trajectories. In the transition path sampling framework such path free energies are related to the time correlation function from which rate constants can be extracted. Thus, free energy methods can be used to study the kinetics of rare transitions between stable states such as chemical reactions, phase transitions of condensed materials or biomolecular isomerizations. 

The content of (3.11) can be clarified by considering the time correlation function of the solvent coordinate itself, when the solute coordinate is fixed at its Transition State value x=0. It is then a straighforward exercise to show from (3.10) that... 

We should point out here the great analogy between and the friction coefficient studied in the Brownian motion problem of Section IV (see Eq. (242)) instead of having the time autocorrelation function of the force F , we now have the time correlation function between F and Fe. 

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. 

Sikorsky and Romiszowski [172,173] have recently presented a dynamic MC study of a three-arm star chain on a simple cubic lattice. The quadratic displacement of single beads was analyzed in this investigation. It essentially agrees with the predictions of the Rouse theory [21], with an initial t scale, followed by a broad crossover and a subsequent t dependence. The center of masses displacement yields the self-diffusion coefficient, compatible with the Rouse behavior, Eqs. (27) and (36). The time-correlation function of the end-to-end vector follows the expected dependence with chain length in the EV regime without HI consistent with the simulation model, i.e., the relaxation time is proportional to l i+2v The same scaling law is obtained for the correlation of the angle formed by two arms. Therefore, the model seems to reproduce adequately the main features for the dynamics of star chains, as expected from the Rouse theory. A sim-... 

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... 

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... 

The theory of statistical mechanics provides the formalism to obtain observables as ensemble averages from the microscopic configurations generated by such a simulation. From both the MC and MD trajectories, ensemble averages can be formed as simple averages of the properties over the set of configurations. From the time-ordered properties of the MD trajectory, additional dynamic information can be calculated via the time correlation function formalism. An autocorrelation function Caa( = (a(r) a(t + r)) is the ensemble average of the product of some function a at time r and at a later time t + r. 

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 ... 

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-... 

A similar approach, also based on the Kubo-Tomita theory (103), has been proposed in a series of papers by Sharp and co-workers (109-114), summarized nicely in a recent review (14). Briefly, Sharp also expressed the PRE in terms of a power density function (or spectral density) of the dipolar interaction taken at the nuclear Larmor frequency. The power density was related to the Fourier-Laplace transform of the time correlation functions (14) ... 

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