Algebraic decay

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

Recently, Miller and co-workers have obtained a generalized form of the distribution of unimolecular decay rates for the case of coupled open channels contributing with unequal partial half-widths [139]. Further results have also recently been obtained in the statistical theory of reactions where the possibility of algebraic decay besides the RRKM exponential decay has been discussed [140]. ... 

In this context, it should be pointed out that an algebraic decay has also been numerically observed in classical Coulomb-type models of atomic autoionization processes by Blumel [141]. This might turn out to be relevant for Rydberg molecules, which also represent Coulomb-type systems. For the recent observation of algebraic decays in Rydberg atoms, see Ref. 142. 

Since the Bohr radius of the electron centres in ionic crystals is typically rather small, (ro 1 A), usually the ratio ro/R = 0.01-0.05 and equation (4.2.4) reveals the algebraic decay law of intensity. This decay kinetics has been observed more than once (see [18, 34] for more details where other kinds of spatial distributions are also discussed). 

The difference in the kinetics for two limiting cases Da = 0 and Da = Db becomes more obvious in terms of the current critical exponents defined earlier, equation (4.1.68). It yields the slope of decay curves shown in Fig. 6.39. The conclusion can be drawn from Fig. 6.40 that in the symmetric case we indeed observe well-known algebraic decay kinetics with a(oo) = 1 corresponding to time-independent reaction rate. However, in the asymmetric case the critical exponent increases in time thus indicating the peculiarity of the kinetics as we qualitatively estimated in the beginning of this Section 6.4. 

Note, that with this result we have neglected the algebraic decay for small x < XT. Therefore a better interpolation formula for the correlation length is ( t + Xt), which takes the slow decay for small x into account. In terms of the length-scale dependent t(l) this rewrites to... 

In the following section we prove that, despite the observed algebraic decay, the fractal is indeed a skinny fractal, i.e. it does not contain any regions of finite measure. 

Lai, Y.-C., Ding, M., Grebogi, C. and Bliimel, R. (1992a). Algebraic decay and fluctuations of the decay exponent in Hamiltonian systems, Phys. Rev. A46, 4661 669. 

The stiffness constant then entirely governs the algebraic decay of the antiferromagnetie eorrelations. The spin response at 2k° is given by... 

To assess properly the significance of these results, and to test simulation methods, a benchmark fluid system is required which can exhibit classical or non-classical criticality, depending on the parameters. To this end, we have examined the liquid/vapor criticality in a fluid of hard spheres with algebraically decaying attractive interactions we will refer to this system as attractive hard spheres (AHSs). The pair potential is,... 

An alternative picture was first introduced by Aharony and Pytte in the context of random magnets. In this picture the order parameter correlation function exhibits algebraic decay with distance instead. This situation, intermediate between SRO and LRO, has come to be known as quasi-long-range order (QLRO). The most well-known example of QLRO, due to Berezinsky and to Kosterlitz and Thouless occurs in the low temperature phase of the two-dimensional XY model. A number of recent theoretical and computational studies have supported this point of view in random spin systems in a higher dimensionality . 

Schiessel and Blumen (1995) describe a mechanical-ladder model based on the algebraic decay (decay1 ) profile of viscoelastic properties at gelation. The mechanical model does not relate to the underlying physics, but is based on a mechanical-ladder model that mimics fractional relaxation equations and is useful in determining viscoelastic decay properties of gelled systems. 

The power law singularities in S(k) reflect the power law (algebraic) decay of the Debye-Waller correlation function Cg(R) for large R [47,48],... 

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