Time-correlation functions

The derivation is given in Appendix 3.II. Physically, the term dDIdx is needed to compensate for the flux caused by the random force which is dependent on particle position (see Fig. 3.2). 

The Langevin equation corresponding to the Smoluchowski equation in multidimensional phase space (eqn (3.21)) is given by  

The distribution of the random force is Gaussian, characterized by the moment 

The Langevin equation (3.39) represents the same motion as the Smoluchowski equation (3.21). However, each of the equations has advantages and disadvantages in solving our problems we shall therefore use both equations interchangeably. 

An important quantity characterizing the Brownian motion is the time correlation function, which is operationally defined in the following way. Suppose we measure a physic quantity A of a system of Brownian particles for many samples in the equilibrium state. Let A t) be the measured values of A at time t. Usually A(f) looks like a noise pattern as shown in Fig. 3.3a. The time correlation function Cyi (l) is defined as the average of the product A(r)A(0) over many measurements  

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Cao J and Voth G A 1995 A theory for time correlation functions in liquids J. Chem. Phys. 103 4211... 

Voth G A, Chandler D and Miller W H 1989 Time correlation function and path integral analysis of quantum rate constants J. Phys. Chem. 93 7009... 

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... 

The use of different time origins improroes the accuracy with which time correlation functions can be ted. 

IV. Time Correlation Function Expressions for Transition Rates... 

The first-order El "golden-rule" expression for the rates of photon-induced transitions can be recast into a form in which certain specific physical models are easily introduced and insights are easily gained. Moreover, by using so-called equilibrium averaged time correlation functions, it is possible to obtain rate expressions appropriate to a... 

In this form, one says that the time dependenee has been reduee to that of an equilibrium averaged (n.b., the Si pi I)i ) time correlation function involving the eomponent of the dipole operator along the external eleetrie field att = 0(Eo p) and this eomponent at a different time t (Eo p (t)). 

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)... 

All of these time correlation functions contain time dependences that arise from rotational motion of a dipole-related vector (i.e., the vibrationally averaged dipole P-avejv (t), the vibrational transition dipole itrans (t) or the electronic transition dipole ii f(Re,t)) and the latter two also contain oscillatory time dependences (i.e., exp(icofv,ivt) or exp(icOfvjvt + iAEi ft/h)) that arise from vibrational or electronic-vibrational energy level differences. In the treatments of the following sections, consideration is given to the rotational contributions under circumstances that characterize, for example, dilute gaseous samples where the collision frequency is low and liquid-phase samples where rotational motion is better described in terms of diffusional motion. 

If the rotational motion of the molecules is assumed to be entirely unhindered (e.g., by any environment or by collisions with other molecules), it is appropriate to express the time dependence of each of the dipole time correlation functions listed above in terms of a "free rotation" model. For example, when dealing with diatomic molecules, the electronic-vibrational-rotational C(t) appropriate to a specific electronic-vibrational transition becomes ... 

If the Bath relaxation constant, t, is greater than O.I ps, you should be able to calculate dynamic properties, like time correlation functions and diffusion constants, from data in the SNP and/or CSV files (see Collecting Averages from Simulations on page 85). 

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... 

From the theory of neutron scattering [62], S JQ, CO) may be written as the Fourier transfonn of a time correlation function, the intermediate scattering function, hJQ, t) ... 

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... 

In mean field approximation we obtain for the imaginary-time correlation functions [296]... 

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). 

Zatsepin V. M. Time correlation functions of one-dimensional rotational Brownian motion in n-fold periodical potential. Theor. and Math. Phys. 

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). 

The approach to the evaluation of vibrational spectra described above is based on classical simulations for which quantum corrections are possible. The incorporation of quantum effects directly in simulations of large molecular systems is one of the most challenging areas in theoretical chemistry today. The development of quantum simulation methods is particularly important in the area of molecular spectroscopy for which quantum effects can be important and where the goal is to use simulations to help understand the structural and dynamical origins of changes in spectral lineshapes with environmental variables such as the temperature. The direct evaluation of quantum time- correlation functions for anharmonic systems is extremely difficult. Our initial approach to the evaluation of finite temperature anharmonic effects on vibrational lineshapes is derived from the fact that the moments of the vibrational lineshape spectrum can be expressed as functions of expectation values of positional and momentum operators. These expectation values can be evaluated using extremely efficient quantum Monte-Carlo techniques. The main points are summarized below. 

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. 

Amides, alkaline hydrolysis, 215 Anharmonic systems, direct evaluation of quantum time-correlation functions, 93 Apollo DSP—160, CHARMM performance, 129/ simulations, solvent effects, 83... 

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). ... 

FIG. 4 Time-resolved fluorescence Stokes shift of coumarin 343 in Aerosol OT reverse micelles, (a) normalized time-correlation functions, C i) = v(t) — v(oo)/v(0) — v(oo), and (b) unnormalized time-correlation functions, S i) = v i) — v(oo), showing the magnitude of the overall Stokes shift in addition to the dynamic response, wq = 1.1 ( ), 5 ( ), 7.5 ( ), 15 ( ), and 40 (O) and for bulk aqueous Na solution (A)- Points are data and lines that are multiexponential fits to the data. (Reprinted from Ref 38 with permission from the American Chemical Society.)... 

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. 

Moving downward to the molecular level, a number of lines of research flowed from Onsager s seminal work on the reciprocal relations. The symmetry rule was extended to cases of mixed parity by Casimir [24], and to nonlinear transport by Grabert et al. [25] Onsager, in his second paper [10], expressed the linear transport coefficient as an equilibrium average of the product of the present and future macrostates. Nowadays, this is called a time correlation function, and the expression is called Green-Kubo theory [26-30]. 

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). 

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