Dynamical quantities autocorrelation function

The autocorrelation function quantifies, on a unit scale, the degree to which a quantity is correlated with values of the same quantity at later times. The function can be meaningfully calculated for any dynamical simulation, in the sense defined earlier, and therefore including MC. We must consider a set of time-ordered values of the observable of interest, so that / — f(t — /At), with j — 1,2,..., N and At the time step between frames. (For MC simulations, one can... 

The friction on a tagged particle is expressed in terms of the time-dependent force-force autocorrelation function. Although the bare, short-time part of the friction that arises from binary collisions can be calculated from kinetic theory, the long-time part needs the knowledge of the solvent and the solute dynamics and the coupling between them. The solvent dynamic quantities... 

The other dynamic variables required to calculate Rpp(t) and Rrr(t) are the dynamic structure factor of the solvent, F(q, t), the inertial part of the dynamic structure factor, Fo(q, t), the transverse current autocorrelation function of the solvent, C (q,t), the inertial part of the same, Ctf0(q, t), the self-dynamic structure factor of the solute, Fs(q, t), and the inertial part of the self-dynamic structure factor of the solute, Fs0(q,t). The expressions for all the above-mentioned dynamic quantities are similar to that given in Section IX but in two dimensions. 

Dynamical variables that correspond to observables are real, however it is sometimes convenient to work with complex quantities such as the variables a and a in Eq. (6.38). For a complex dynamical variable the autocorrelation function is conventionally defined as... 

The intensity I of the light scattered from a dilute macromolecular or supra-molecular solution is a fluctuating quantity due to the Brownian motion of the scattering particles. These fluctuations can be analysed in terms of the normalised autocorrelation function y1 ( t ) of the scattered electrical field Es, which contains information about the structure and the dynamics of the scattering particles [80]. 

Whether the inactive region is a true continuum (e.g., photofragmentation) or a quasi-continuum comprised of an enormous density of rigorously bound eigenstates (polyatomic molecule dynamics, Section 9.4.14) is often of no detectable consequence. The dynamical quantities discussed in Section 9.1.4 (probability density, density matrix, autocorrelation function, survival probability, transfer probability, expectation values of coordinates and conjugate momenta) describe the active space dynamics without any reference to the detailed nature of the inactive space. 

To understand the reason for this fundamental, qualitative difference between Nos6-Hoover-Langevin and Langevin dynamics, and to compare it with Velocity Rescaling, one may study a quantity that relates the rate of convergence to equilibrium to the rate of growth of the error in the autocorrelation function. In [225], this precise quantity is introduced and termed the efficiency of the thermostat ... 

The dynamics of different modes of molecular librations (hindered rotations) and intramolecular vibrations in supercritical water can now be analyzed in terms of velocity autocorrelation functions for the corresponding projections (Eqns. 22-27) (Kalinichev and Heinzinger 1992, 1995 Kalinichev 1993). The velocity autocorrelation functions calculated for the quantities Qi (Eqns. 25-27) are shown in Figure 19 for two extreme cases of high-density and low-density supercritical water. The Fourier transforms of these functions result in the spectral densities of the corresponding vibrational modes. They are shown in Figure 20 for the supercritical thermodynamic states listed in Table 5. 

The autocorrelation function is a very useful quantity, because it reflects the dynamics of the wave packets, that is how fast they depart from the region, where they were launched, how often they recur to the place of birth, for how long the wave packets remain compact and localized and on which timescale the bonds are broken, Eventually, when the entire wave packet has left the interaction region, i.e. when all molecules are dissociated, S t) becomes zero. Even though the initial wave packet is a real function, it becomes complex, because the time evolution operator is complex. Thus, S t) is a complex function and fulfills the symmetry relation... 

A semiclassical description is well established when both the Hamilton operator of the system and the quantity to be calculated have a well-defined classical analog. For example, there exist several semiclassical methods for calculating the vibrational autocorrelation function on a single excited electronic surface, the Fourier transform of which yields the Franck-Condon spectrum. ° In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator,which circumvent the cumbersome root-search problem in boundary-value based semiclassical methods, have been successfully applied to a variety of systems (see, for example, the reviews Refs. 85, 86 and references therein). These methods cannot directly be applied to nonadiabatic dynamics, though, because the Hamilton operator for the vibronic coupling problem [Eq. (1)] involves discrete degrees of freedom (discrete electronic states) which do not possess an obvious classical counterpart. 

Residence Times. The dynamic behavior of water is frequently characterized by the self diffusion coefficient (sdc) D, which can be calculated from the particle mean square displacements via the Einstein relation or from the velocity autocorrelation functions (acf) via the Kubo relation. Near an interface this quantity D is not the self diffusion coefficient, since there are no free boundary conditions for the surface layer. Sonnenschein and Heinzinger [52] calculate a property called residence autocorrelation function... 

In order to get information about dynamic properties of the system various quantities have been monitored with time at equilibrium states corresponding to various temperatures [38 0] the mean squared displacement of monomers, (r ), the mean squared displacement of the center of mass of chains, the bond autocorrelation function, the end-to-end... 

The dynamieal quantities used to study the dynamics of glass transitions are generally the MSD, the intermediate seattering function, or the density autocorrelation function, presented above. The MSD is given by... 

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