Equal time density correlation function

The interfacial pair correlation functions are difficult to compute using statistical mechanical theories, and what is usually done is to assume that they are equal to the bulk correlation function times the singlet densities (the Kirkwood superposition approximation). This can be then used to determine the singlet densities (the density and the orientational profile). Molecular dynamics computer simulations can in... 

This equality implies that p". >neq (y, t2 — t ) is not positive definite, a price that we have to pay to ensure the equivalence between the density and trajectory picture in the non-Poisson case. Thus, the two-time correlation function is evaluated using only density prescriptions, and the result turns out to be identical to Eq. (148), which is known to correspond to the prescription of renewal theory [see Eq. (147)]. In the Poisson case the equilibrium distribution is flat. Thus, the contributionp s " cq(y, t2 — t ) vanishes. 

First of all, let us discuss the case of equal concentrations nA(t) = nB(t) = n t) when two kinds of similar correlation functions coincide Xjj r,t) = X r,t), u = A,B. In Fig. 5.2 the concentration development in the one-dimensional case is presented [26]. The curve (a) gives averaged (over 10 simulations) computer-calculated density. Stripped lines demonstrate dispersion of results they correspond to the curves n t)) s t), where ( (0) is standard deviation. Curve (b) shows the numerical solution of a set (4.1.19), (4.1.28), and (5.1.14) to (5.1.16) derived in the framework of the superposition approximation. Curve (c) gives results of the linear approximation (4.1.41) and (4.1.42). At last, the additional curve (d) is drawn Just to illustrate concentration behaviour at short times. In the linear approximation we neglect similar reactant correlation, X r,t) — 1, whereas in curve (d) dissimilar (AB) reactant correlations (4.1.40) are also... 

The many-body ground and excited states of a many-electron system are unknown hence, the exact linear and quadratic density-response functions are difficult to calculate. In the framework of time-dependent density functional theory (TDDFT) [46], the exact density-response functions are obtained from the knowledge of their noninteracting counterparts and the exchange-correlation (xc) kernel /xcCf, which equals the second functional derivative of the unknown xc energy functional ExcL i]- In the so-called time-dependent Hartree approximation or RPA, the xc kernel is simply taken to be zero. 

Dynamic Density Functional Theory (DDFT), Fig. 1 Illustration of the local equilibrium approximation involved in the development of the DDFT. The left-hand side illustrates the nonequilibrium evolution of the density p(r t) thin lines) up to time t thick line). For the time evolution, the equal-time correlation function g(r, r t) is... 

The range of relaxation times allowed in the fitting was usually between 0.5 ps and 1 s with a density of 12 points per decade. Relaxation rates are obtained from the moments of the peaks in the relaxation time distribution or, if the peaks overlap, from the peak maximum position. With a broad distribution of relaxation times, these inversion methods yield multiple peaks in the "unsmoothed" analysis. The "smoothing" parameter (P) was selected as 0.5 in all cases, after it was established that the number of peaks did not increase with further increase in smoothing. As a further check, an analysis was made on a simulated correlation function consisting of a broad continuous distribution of relaxation times with noise added equal to the residuals from the analysis of the experimental correlation curve. REPES recovers the original distribution except when a very low smoothing parameter (P 0) is used. 

This chapter is organized as follows. In section 1.1, we introduce our notation and present the details of the molecular and mesoscale simulations the expanded ensemble-density of states Monte Carlo method,and the evolution equation for the tensor order parameter [5]. The results of both approaches are presented and compared in section 1.2 for the cases of one or two nanoscopic colloids immersed in a confined liquid crystal. Here the emphasis is on the calculation of the effective interaction (i.e. potential of mean force) for the nanoparticles, and also in assessing the agreement between the defect structures found by the two approaches. In section 1.3 we apply the mesoscopic theory to a model LC-based sensor and analyze the domain coarsening process by monitoring the equal-time correlation function for the tensor order parameter, as a function of the concentration of adsorbed nanocolloids. We present our conclusions in Section 1.4. 

Later studies showed the same phenomena in deuterium and deuterium-rare gas mixtures [335, 338, 305], and also in nitrogen and nitrogen-helium mixtures [336] in nitrogen-argon mixtures the feature is, however, not well developed. The intercollisional dip (as the feature is now commonly called) in the rototranslational spectra was identified many years later see Fig. 3.5 and related discussions. The phenomenon was explained by van Kranendonk [404] as a many-body process, in terms of the correlations of induced dipoles in consecutive collisions. In other words, at low densities, the dipole autocorrelation function has a significant negative tail of a characteristic decay time equal to the mean time between collisions see the theoretical developments in Chapter 5 for details. 

The accuracy obtained in all cases depends on the details of the method used. The most accurate calculations are those obtained by HF methods with full correlation energy corrections. (The correlation energy is defined as the difference between the HF energy and the exact energy.) But these are only practical for very small values of N, since computer times now scale as N. The best DFT methods available at present are equal to the best practical HF-based methods available that is, there is some correlation energy included. At the same time the computer time required is 10 to 100 times less for DFT calculations, depending on N. It is hard to avoid the conclusion that density functional theory will almost completely replace wave function theory in the area of ab-initio calculations on molecules. 

The power available within a molecular system to induce transitions by virtue of its molecular tumbling is referred to as the spectral density J(co) (Section 2.5) and this provides a measure of how the relaxation rates Wq, W] and W2 vary as a function of tumbling rates. This is illustrated schematically in Fig. 8.6 for three different correlation times. An alternative description of the spectral density is that it represents the probability of finding a fluctuating magnetic component at any given frequency as a result of the motion and as such the area under each of the curves of Fig. 8.6 must then be equal. Thus, for a molecule with a short tc (rapid tumbling) there exists an almost... 

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