Displacement correlation function time-dependent

A non-perturbative theory of the multiphonon relaxation of a localized vibrational mode, caused by a high-order anharmonic interaction with the nearest atoms of the crystal lattice, is proposed. It relates the rate of the process to the time-dependent non-stationary displacement correlation function of atoms. A non-linear integral equation for this function is derived and solved numerically for 3- and 4-phonon processes. We have found that the rate exhibits a critical behavior it sharply increases near a specific (critical) value(s) of the interaction. 

The relaxation of a local mode is characterized by the time-dependent anomalous correlations the rate of the relaxation is expressed through the non-stationary displacement correlation function. The non-linear integral equations for this function has been derived and solved numerically. In the physical meaning, the equation is the self-consistency condition of the time-dependent phonon subsystem. We found that the relaxation rate exhibits a critical behavior it is sharply increased near a specific (critical) value(s) of the interaction the corresponding dependence is characterized by the critical index k — 1, where k is the number of the created phonons. In the close vicinity of the critical point(s) the rate attains a very high value comparable to the frequency of phonons. 

A substantial recent theme has been comparison of the dependences of one-particle and two-particle displacement correlation functions Sxi(t)) ) and Sxi(t)8xj t)) (for i 7 j). Crocker, et al. used videomicroscopy to capture the positions of pairs of particles at an extended series of times(82). Cross-correlations... 

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-... 

This relation holds only if the rate of the process is sufficiently small as compared to k — 1 )T. The fact that in this case equation (3) holds also for k > 2, means that phonons remain almost harmonic. This allows one to use in equation (7) the pair correlation approximation Dk-1(t1, t2) k — 1) > 1 (Ti, f2), where D(t, t/) = (0 q(t)q(t ) Q) is the displacement pair correlation function. The same time pairings are neglected while they give contribution to k — 2, k — 4,. ..-phonon transitions and, therefore, result in small change of the anharmonic constants Vk. Note also that the validity of equation (3) with a non-zero value of vt(t) means the existence of anomalous correlations (bf(t)) = vitfiit, these correlations depend on time. 

MD simulation is advantageous for obtaining dynamic properties directly, since the MD technique provides not only particle positions but also particle velocities that enable us to utilize the response theory (e.g., the Kubo formula [175,176]) to calculate the transport coefficients from time-dependent correlation functions. For example, we will examine the self-diffusion process of a tagged PFPE molecular center of mass (Fig. 1.49) from the simulation to gain insight into the excitation of translational motion, specifically, spreading and replenishment. The squared displacement of the center mass of a molecule or a bead is used as a measure of translational movement. The self-diffusion coefficient D can be represented as a velocity autocorrelation function... 

Therefore, before describing the modification of the equilibrium FDT, we need to study in details the behavior of D(t). Note, however, that the integrated velocity correlation function [, Cvv(/) df takes on the meaning of a time-dependent diffusion coefficient only when the mean-square displacement increases without bounds (when the particle is localized, this quantity characterizes the relaxation of the mean square displacement Ax2 t) toward its finite limit Ax2(oo)). 

To obtain the diffusion constant, D, we consider two alternative equilibrium time correlation function approaches. First, D can be obtained from the long time limit of the slope of the time-dependent mean square displacement of the electron from its starting position. The quantum expression for this estimator is... 

Now we can show the explicit relation with experiment. What is usually measured in spectroscopic or scattering experiments is the spectral density function /(to), which is the Fourier transform of some correlation function. For example, the absorption intensity in infrared spectroscopy is given by the Fourier transform of the time-dependent dipole-dipole correlation function <[/x(r), ju,(0)]>. If one expands the observables, i.e., the dipole operator in the case of infrared spectroscopy, as a Taylor series in the molecular displacement coordinates, the absorption or scattering intensity corresponding to the phonon branch r at wave vector q can be written as (Kobashi, 1978)... 

FIGURE 16.7 Temperature dependence of protein/lipid and water dynamical properties from MD simulations of purple membrane [9,10]. (a) MSFs of protein/lipid nonexchangeable H atoms averaged over 5 ns blocks of the trajectories, (b) Temperature dependence of the inverse of the correlation times, of the protein-water (filled circles) and lipid-water (open circles) hydrogen bond correlation functions, (c) Value of the mean-squared displacement of water O atoms at f = 100 ps. 

The analysis of the results concentrated on the time-averaged structure and on correlation functions that illustrate the time-dependence of internal motions. A key conclusion was that the internal motion was fluidlike at ordinary temperatures, i.e. that the dynamics of atomic displacements are dominated by collisions with neighboring atoms, at least on the picosecond time scale. Hence, it is argued, ...many of the dynamical properties (though not necessarily the correct average structure) can be obtained from any potential function which includes the forces that depend strongly upon distance (covalent and hydrogen bonds, nonbonded repulsions) and provides sufficient attractive interactions to preserve the compact structure of the native system. ... 

The field correlation function g directly reflects any changes in the microstructure of the suspension. In the case of purely diffusive processes, i.e. when the displacement of scattering objects follows Pick s second law, an exponential time dependency is found. 

Here at = [ciqf/kT], a and qt are, respectively, the concentration and charge of species i (anions and cations). Vi 0)Vi(t)) and AFf(t)) are, respectively, the velocity correlation function (VCF) and mean-squared displacement in time of species i. The steady current behavior at long times in the step-on experiment (see above) means that ABf t)) becomes linearly dependent on time, giving the Nernst-Einstein equation that connects the low-/" conductivity cr (O) (=diffusion coefficients A for the translational motions of the ions (9,10) ... 

The calculation of collisional cross sections for phenomena involving atoms and molecules is particularly difficult because many quantum states of the colliding partners are coupled by the interaction forces. Even in cases involving electronically adiabatic phenomena, where one can assume that the electronic states of the system remain the same while the nuclei move, one must yet deal with the coupling of translational, rotational and vibrational degrees of freedom of the nuclei. The interaction forces furthermore depend intricately on the molecular orientations and on the atomic displacements within molecules, and change extensively with the atomic composition of molecules. It is therefore usually impossible to invoke physical considerations to make a preliminary selection of the quantum states that are relevant to the collision. We describe here the computational aspects of an alternative approach, based on the time evolution of operators for scattering, and on their time-correlation functions, which eliminates the need for basis set expansions. 

Taking the Intermolecular potential in a liquid as decomposable into a sum of a repulsive potential depending only on nuclear coordinates and the Coulomb potential between the electronic and nuclear charges of different molecules, the latter taken as a perturbation to a classical liquid with a purely repulsive Intermolecular potential, one can perform a quantum perturbation expansion of the trace of the statistical operator exp(- H) and similarly expansions of the thermally averaged, imaginary time displaced, molecular charge correlation functions pO, t )>. The individual terms in the expansions consist of multipolar interaction tensors and functionals of molecular polarizabilities of various orders. Summation of all terms depending only on linear molecular polarizabilities lead to ... 

More time-dependent structural correlation functions can be constructed, depending on the chemical nature of the system and of the phenomenon under investigation. For example, correlation functions over the positions and velocities of centers of mass yield information on translational diffusion. Molecular diffusion properties are often described by the self-diffusion coefiticient, D, using a simple formula that involves the mean square displacement of the centers of mass [9,10] ... 

Equation (524) is an exact result for the displacement SA(r, t)) fromequilibriumarising from the external force F (f, t). As evident from Eq. (524), the response of the system to the external force F (r, t) is nonlocal in space and time. Moreover, the response is nonlinear with the nonlinear dependence parried by the nonequilibrimn time correlation function r t ) through pf). 

MD simulations provide the means to solve the equations of motion of the particles and output the desired physical quantities in the term of some microscopic information. In a MD simulation, one often wishes to explore the macroscopic properties of a system through the microscopic information. These conversions are performed on the basis of the statistical mechanics, which provide the rigorous mathematical expressions that relate macroscopic properties to the distribution and motion of the atoms and molecules of the N-body system. With MD simulations, one can study both thermodynamic properties and the time-dependent properties. Some quantities that are routinely calculated from a MD simulation include temperature, pressure, energy, the radial distribution function, the mean square displacement, the time correlation function, and so on (Allen and Tildesley 1989 Rapaport 2004). 

---