Computer simulation expansion

Figure 9.47 Part of the observed 0[j band (top), an expansion of a small portion (middle) and a computer simulation (bottom) of (a) aniline and (b) aniline Ar. (Reproduced, with permission, from Sinclair, W. E. and Pratt, D. W., J. Chem. Phys., 105, 7942, 1996)...

For computer simulations, (5.35) leads to accurate estimates of free energies. It is also the basis for higher-order cumulant expansions [20] and applications of Bennett s optimal estimator [21-23], We note that Jarzynski s identity (5.8) follows from (5.35) simply by integration over w because the probability densities are normalized to 1 ... 

An analogy may be drawn between the phase behavior of weakly attractive monodisperse dispersions and that of conventional molecular systems provided coalescence and Ostwald ripening do not occur. The similarity arises from the common form of the pair potential, whose dominant feature in both cases is the presence of a shallow minimum. The equilibrium statistical mechanics of such systems have been extensively explored. As previously explained, the primary difficulty in predicting equilibrium phase behavior lies in the many-body interactions intrinsic to any condensed phase. Fortunately, the synthesis of several methods (integral equation approaches, perturbation theories, virial expansions, and computer simulations) now provides accurate predictions of thermodynamic properties and phase behavior of dense molecular fluids or colloidal fluids [1]. 

The approximations of the superposition-type like equation (2.3.54), are used in those problems of theoreticals physics when other-kind expansions (e.g., in powers of a small parameter) cannot be employed. First of all, we mean physics of phase transitions and critical phenomena [4, 13-15] where there are no small parameters at all. Neglect of the higher correlation forms a(ml in (2.3.54) introduces into solution errors which cannot be, in fact, estimated within the framework of the method used. That is, accuracy of the superposition-like approximations could be obtained by a comparison with either simplest explicitly solvable models (like the Ising model in the theory of phase transitions) or with results of direct computer simulations. Note, first of all, several distinctive features of the superposition approximations. 

Figure 5. a Excitation spectrum for supersonically expanded Na corresponding to the pumping of several vibrational levels in the A , state, b Computer simulation for an Na, vibrational temperature of 50 K and a rotational temperature of 30 K. c Computer simulation 99.75% of Na molecules at Tya — 50 K, Troi = 30 K 0.25% at T 500—1000 K (temperature range determined on basis of surface temperature measurement throughout oven expansion chamber). 

This also favours the occurrence of temperature-dependent disorder. Computer simulation of structural variations and comparison of both theoretical and experimental behaviour of the integral intensities of structure-sensitive diffraction reflections allow one to prove the presence of the temperature-dependent disorder of the Fe(C12Gm)3(B i-C4H9)2 molecules in crystals. The thermal expansion anomalies and the relative shift of the molecules preced the phase transition and, presumably, promot its realization in this crystal [282]. 

Oxide nanopartides, unlike nanopartides of metals, display an expansion in their lattice parameters in comparison with the bulk. Tsunekawa et al. have examined sub-10 nm Ce02 and sub-100 nm BaTi03 nanopartides using a combination of electron diffraction, X-ray photoelectron spectroscopy and ab initio computer simulations. They find that in the Ce02 system, the lattice expansion arises from a decrease in Ce valence, whilst in the BaTi03 system, the decreasing Ti-0 covalency with decreasing particle size results in the expanded lattice. 

Computer simulations, Monte Carlo or molecular dynamics, in fact appear to be the actual most effective way of introducing statistical averages (if one decides not to pass to continuous distributions), in spite of their computational cost. Some concepts, such as the quasi-structure model introduced by Yomosa (1978), have not evolved into algorithms of practical use. The numerous versions of methods based on virial expansion, on integral equation description of correlation functions, on the application of perturbation theory to simple reference systems (the basic aspects of these... 

Recently, the approximations underlying the perturbative DFTs have been called into question [130-132]. The solid density p (r) predicted by this theory and measured in computer simulations is very sharply peaked about lattice sites, as can be inferred from the relative smallness of the Lindemann parameter near melting. Thus, the expansion parameter Pj(r) - Pi is by no means small, so that the truncation of the functional... 

After these caveats, fig. 17 shows qualitatively the dimensionality dependence of the order parameter exponent /5, the response function exponent y, and correlation length exponent v. Although only integer dimensionalities d = 1,2, 3 are of physical interest (lattices with dimensionalities d = 4,5, 6 etc. can be studied by computer simulation, see e.g. Binder, 1981a, 1985), in the renormalization group framework it has turned out useful to continue d from integer values to the real axis, in order to derive expansions for critical exponents in terms of variables = du — d or e1 = d — dg, respectively (Fisher, 1974 Domb and Green, 1976 Amit, 1984). As an example, we quote the results for r) and v (Wilson and Fisher, 1972)... 

Of course, the above considerations may not be relevant to the problem at hand, since in solving the OZ equation, the important functions are y(l, 2), its Fourier transform and B(l,2). ° It is to be expected that y(l,2) will vary less quickly between different orientations and will be continuous even for hard core potentials. Thus, its expansion in spherical harmonics should be better behaved than that of gf(l,2). Computer simulation cannot be used to obtain y(l,2) but Lado has presented some evidence based upon his solution of the RHNC approximation for a hard diatomic fluid using a spherically averaged bridge function that the convergence is good. Nevertheless, the results he presents are, in our view, for a rather short diatomic bond length and may not be conclusive. 

An additional issue in the development of the density functional theory is the parameterization of the trial function for the one-body density. Early applications followed the Kirkwood-Monroe [17,18] idea of using a Fourier expansion [115-117,133]. More recent work has used a Gaussian distribution centered about each lattice site [122]. It is believed that the latter approach removes questions about the influence of truncating the Fourier expansion upon the DFT results, although departures from Gaussian shape in the one-body density can also be important as has been demonstrated in computer simulations [134,135]. 

DFT studies of binary hard-sphere mixtures predate the simulation studies by several years. The earliest work was that of Haymet and his coworkers [221,222] using the DFT based on the second-order functional Taylor expansion of the Agx[p]- Although this work has to some extent been superceded, it was a significant stimulus to much of the work that followed both with theory and computer simulations. For example, it was Smithline and Haymet [221] who first analyzed the Hume-Rothery rule in the context of hard sphere mixture behavior and who first investigated the stability of substitutionally ordered solid solutions. The most accurate DFT results for hard-sphere mixtures have come from the WDA-based theories. In particular the results of Denton and Ashcroft [223] and those of Zeng and Oxtoby [224] give qualitatively correct behavior for hard spheres forming substitutionally disordered solid solutions. 

An evaluation of the effect of thermal loading of optical elements is achieved by using finite element analysis computer simulation. Such a computer program calculates what happens over a given number of time steps when a certain heat load is applied to the surface of an optical element such as a mirror. It takes into account the heat conductivity in the mirror bulk and the emissivity of the reflecting and other mirror surfaces. Once the system has reached the calculated equilibrium the thermal expansion is calculated. 

Computer simulation results for S2 are somewhat sparse and involve the usual uncertainties involved in extrapolating results for a truncated T(r) used in a periodic box to untruncated T(r) in an infinite system." Nevertheless for polarizable hard-sphere and Lennard-Jones particles, it is probably safe to say that the estimates currently available from the combined use of analytic and simulation input are enough to provide a reliable guide to the p and dependence of Sj over the full fluid range of those variables. The most comprehensive studies of have been made by Stell and Rushbrooke" and by Graben, Rushbrooke, and Stell," for the hard-sphere and Lennard-Jones cases, respectively. Both these works utilize the simulation results of Alder, Weis, and Strauss," as well as exact density-expansion results, and numerical results of the Kirkwood superposition approximation... 

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