Displacement correlation function

With the availability of lasers, Brillouin scattering can now be used more confidently to study electron-phonon interactions and to probe the energy, damping and relative weight of the various hydro-dynamic collective modes in anharmonic insulating crystals.The connection between the intensity and spectral distribution of scattered light and the nuclear displacement-displacement correlation function has been extensively discussed by Griffin 236). 

One of the first applications of CMD to a realistic and important system was to study the quantum dynamical effects in water. It was found that, even at 300 K, the quantum effects are remarkably large. This finding, in turn, led us to have to reparameterize the flexible water model (called the SPC/F2 model) in order to obtain good agreement with a variety of experimental properties for the neat liquid. An example of the large quantum effects in water can be seen in Fig. 3 in which the mean-spared displacement correlation function, ( x(t) - x(0) 2) is plotted. (These are new results which are better converged than those in Ref 34.) Shown are the quantum CMD and the classical MD results for the SPC/F2 model. The mean-squared displacement for the quantized version of the model is 4.0 X 10- m s-f while the classical value is 4.0 x 10 m s-f The error in these numbers is about 15%. These results suggest that quantum effects increase the diffusivity of liquid water by a factor of two. 

Figure 3 Mean-squared displacement correlation function for liquid water at 300 K. Shown are the qaumtum CMD (solid line) and classical MD (dashed line) results.

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. 

Let us insert this equation into the displacement correlation function D(t, 0) = (0 q(t)q(f) ()) and apply once again the pair correlation approximation... 

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. 

We compute below the velocity and displacement correlation functions, first, of a classical, then, of a quantal, Brownian particle. In contrast to its velocity, which thermalizes, the displacement x(t) — x(l0) of the particle with respect to its position at a given time never attains equilibrium (whatever the temperature, and even at T = 0). The model allows for a discussion of the corresponding modifications of the fluctuation-dissipation theorem. 

Let us introduce the two-time displacement correlation function Cxx t, f t0) as defined by... 

The displacement correlation function of the overdamped free Brownian particle is proportional to the waiting time [40,41],... 

The symmetrized displacement correlation function C t, t t0) is defined by the formula ... 

The calculation of this quantity, and of the displacement-displacement correlation functions, involves a single summation over all wave vectors q in the Brillouin zone. 

Although (pc) clearly goes to zero in the thermodynamic limit, the weakness of this bound led Mermin to suggest that some sort of ordering might still be present. For a 2D harmonic lattice, Mermin showed that, although the displacement-displacement correlation function diverges. 

The dynamics of the solvent in the region near a protein can be characterized by a number of properties (e.g., solvent velocity correlation functions, mean-square displacement correlation functions, dipole orientation correlation functions, etc.). These properties provide information on a range of phenomena from local solute-solvent interactions (velocity correlation functions) to solvent mobility (mean-square displacement correlation functions) and dielectric behavior (dipole correlation functions). Here we focus on the diffusion constant, which provides a convenient measure of mobility for water molecules near protein atoms. The diffusion constant for solvent molecules may be computed directly from the slope of the mean-square displacement correlation function,... 

The atom-multiphonon component of the inelastic scattering can be obtained from the theory of Manson [44] by subtracting off the zero- and first-order terms from the power series expansion of the exponential displacement correlation function. With the assumption that the major contribution to the multiphonon scattering is due to low-energy, long-wavelength phonons, he is able to arrive at an expression with a very simple form for the transition rate. 

In the harmonic approximation, the displacement-displacement correlation function may be defined as... 

In the step from eqn [17] to eqn [18], we have explicitly used the time translation invariance property of the time-displaced correlation function. 

The Fy whose evolution is highly complicated for anharmonic forces, are thus systematically obtained from the simpler displacement-displacement correlation functions (DCFs for brevity), such as x"x . Each term of the cumulant expansion is of order of (k((a )) )", which implies fast convergence whenever k((a )) < 1. It is significant that this condition involves the unperturbed displacements, instead of their instantaneous values during the collision, which could be considerably larger. Many scattering experiments are carried out at initial temperatures so that only the... 

Shalaginov and Romanov [18] have recently developed a continuum model and obtained an explicit formula for the displacement-displacement correlation function. Furthermore, both the uniaxial properties of the SmA phase and multiple reflection from the interfaces have been included in this theoretical model. It offers excellent agreement with the X-ray scattering data [9] acquired at a temperature well into the SmA phase of FPP (or H7F6EPP) [5-n-heptyl-2-(4-/i-perfluorohexylethanophenyl)pyrimi-dine]. Both the specular reflectivity and off-specular diffuse diffraction from 4-, 20-, and... 

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... 

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