Pair correlation function time-dependent

Figure 31. The domain growth during the phase separation process reflected by the shift of the first zero in the pair correlation function (a) and by the surface area reduction (b). Although the surface area and first zero of the pair-correlation functions are equivalent lengthscales, the time dependence of the surface area is less affected by the finite lattice size affects.

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

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. 

Figure 6.15. Time dependent pair correlation function Gj(r,t) in composition 35iO.5Li2O-O.5K2O] 65Si02 for (a) Lf ion migration to Lf vacancy, (b) ion migration to Li vacancy, (c) Li ion migration to vacancy, and (d) ion...

Next we define the time-dependent pair correlation function or van Hove correlation function5 G(r,t) as the inverse Fourier transform of F(q,t) in space, that is,... 

Another important point in the entropy calculation is the time of the MD simulation. It has been discussed earlier in this chapter that the entropy of the system depends on the number of states visited by the system. The accuracy of the pair correlation function and thus the accuracy of the entropy increase with increase in the simulation time. One can plot the calculated entropy for different simulation times and find that the value of the entropy asymptotically saturates at longer times. Thus it is important to report the total simulation time along with other details while reporting the value of the entropy. 

Step fluctuations have been observed for both Ag and Cu surfaces in both vacuum and electrolytes [8]. As shown in Fig. 11, the steps on an immersed Ag(lll) actually appear to be friz2y due to kink motion, which is rapid compared to the tip raster speed [8,91,92]. In x — t images, the fluctuations can be quantitatively analyzed by means of a step correlation function, G(t) = [x t) — x(0)] >, where x defines the step position at a particular time, t. If image drift is a problem, the step pair correlation function may be used [8, 93]. The evolution of the correlation function and its dependence on step spacing is a reflection of the mass transport mechanism, which is dependent on both the potential and electrolyte composition. Furthermore, an assessment of the temperature dependence of the fluctuations allows the activation energy of the rate-limiting process to be evaluated. As shown in Fig. 11,... 

In the list of the various correlation functions we have introduced at item (b) the so-called site-site pair correlation functions gap(ro ) since they depend only on the radial variables r p (between sites), they are naturally simpler than g(r,2 co,(02) but, at the same time, they contain less information. The theories for g p fall into two categories based on site-site or particle-particle OZ equations, respectively. Various closures can be used with either category. 

Elementary excitations also include single particle diffusive excitations beside quantized vibrations (i.e., molecular vibrations and vibrations of the crystal as a whole associated with phonons/magnons). Consider the incoherent dynamic structure factor 5snc(Q,(o), which is the Fourier transform pair of the time-dependent self-correlation function, compare... 

The pair correlation function g(r) specifies how often the intermolecular distance r = Tj — Tk occurs in the scattering volume K in time average. Thus, the pair correlation function g(r) strongly depends on the degree of periodicity within the material. In Fig. 4.11 two examples of characteristic pair correlation functions g(r) and the resulting scattering intensities /(q) are shown. Due to the periodic strucmre, the pair correlation function g(r) of ideal crystalline materials (Fig. 4.11a) exhibits 5-functions in multiple distances of a set value d, which corresponds to the periodicity distance within the material. In consequence, the Fourier transform of this... 

G(r, t) - space- and time-dependent pair correlation function = Debye-Waller temperature factor Ge(r) = equilibrium spatial pair cor relation function for atoms Go(r) == instantaneous spatial pair correlation function... 

The function, /, r), is the time-dependent intermediate scattering function. It contains time-dependent spatial structural information, and can be attributed to scattering that arises from the non-periodic pair correlation function within the electron density function and it is further defined by... 

The relation between collective and self-motion in simple monoatomic liquids was theoretically deduced by de Gennes [233] applying the second sum rule to a simple diffusive process. Phenomenological approaches like those proposed by Vineyard [ 194] and Skbld [234] also relate pair and single particle motions and may be applied to non-exponential functions. The first clearly fails to describe the PIB results since it considers the same time dependence for both correlators. Taking into account the stretched exponential forms for Spair(Q.t) (Eq. 4.21) and Sseif(Q>0 (Eq 4.9), the Skold approximation ... 

The indices k in the Ihs above denote a pair of basis operators, coupled by the element Rk. - The indices n and /i denote individual interactions (dipole-dipole, anisotropic shielding etc) the double sum over /x and /x indicates the possible occurrence of interference terms between different interactions [9]. The spectral density functions are in turn related to the time-correlation functions (TCFs), the fundamental quantities in non-equilibrium statistical mechanics. The time-correlation functions depend on the strength of the interactions involved and on their modulation by stochastic processes. The TCFs provide the fundamental link between the spin relaxation and molecular dynamics in condensed matter. In many common cases, the TCFs and the spectral density functions can, to a good approximation, be... 

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