Random hopping

Fig. 5.13. Relaxation time r3 plotted vs. temperature for the coarse-grained model of PE with N = 20, using the random hopping algorithm (upper set of data) or the slithering snake algorithm (lower set of data), respectively. The time r3 is of the same order as the Rouse relaxation time of the chains, and is defined in terms of a crossing criterion for the mean-square displacements [41], g3(t = r3) = g2(t = r3) [See Eqs. (5.2) and 5.3)]. From [32]...

When one implements an MC stochastic dynamics algorithm in this model (consisting of random-hopping moves of the monomers by one lattice constant in a randomly chosen lattice direction), the chosen set of bond vectors induces the preservation of chain connectivity as a consequence of excluded volume alone, which thus allows for efficient simulations. This class of moves... 

An adsorbed phase can exhibit monolayer adsorption as well as multilayer adsorption. Surface flow in the presence of multilayer adsorption can be accounted for in the models described in the previous section. For example Okazaki and Tamon (1981) describe multilayer diffusion in their random hopping model. 

Li" ion enters the adjacent, face-sharing site. It is clear that, in these materials, Li ions do not move by means of isolated, random hops in the LISICONS there is clear evidence for clustering of lithium ions and indeed, migration may involve a process of continual reorganisation of the clusters (Bruce and Abrahams, 1991). 

Fig. 3 Dynamic PDF calculated for a diatomic molecule a oscillating harmonically at a frequency a>o, and b two atoms randomly hopping between two states with different bond lengths. (Dmowski W, Vakhrushev SB, Jeong I-K, Hehlen M, Trouw F, Egami T (2006) Abstracts American conference on neutron scattering, St. Charles, IL, 18-22 June 2006)...

Albrecht U, Bassler H (1995) Yield of geminate pair dissociation in an energetically random hopping system. Chem Phys Lett 235 389... 

At small radiation doses (the number of radiation-produced defects), the mean distance l between components of such geminate pairs (the vacancy and an interstitial atom) is much less than the mean distance between different pairs Iq = n-1/3, where n is defect concentration. The initial defect distribution is described by the distribution function f(r). Below a certain temperature (typically < 30 K for interstitial atoms and 200 K for vacancies in alkali halides), defects are immobile. With a temperature increase, the defects perform thermally activated random hops between the nearest lattice sites. This is usually considered to be continuous diffusion. 

Linear polymers move a distance of order of their own size during their relaxation time, leading to a diffusion coefficient D R /r [Eq. (9.12)]. However, the diffusion of entangled stars is different because at the time scale of successful arm retraction, the branch point can only randomly hop between neighbouring entanglement cells by a distance of order one tube diameter a. For this reason, diffusion of an entangled star is much slower than diffusion of a linear polymer with the same number of monomers ... 

Fig. 15. Approximate mapping of a chemically realistic polymer (polyethylene in this example) to the bond fluctuation model on the (simple cubic) lattice. In this coarse-graining one integrates n successive chemical monomers (e.g. n = 3) into one effective monomer which blocks 8 adjacent sites on the simple cubic lattice (or 4 on the square lattice in d = 2 dimensions) from occupation by other monomers. The chemical bonds 1, 2, 3 then correspond to effective bond I, bonds 4, 5, 6 to effective bond II. Some information on the chemical structure can be kept indirectly by using suitable distributions P (9) for the angle between subsequent effective bonds, but so far this has been done for homopolymer melts only [94-99]. In the simplest version of the bond fluctuation model [84-88] studied for blends in d = 3 dimensions [88, 91, 92, 99], bond lengths t are allowed to fluctuate freely from i = 2 to t = v/l0, with t = being excluded to maintain that chains do not cut through each other in the course of the random hops of the effective monomers. From Binder [95]...

Fig. 25.2 gives a representation of the curves a((o) vs. log((w) and e" vs. s for the main models. First, non-interacting or free charges we observe a constant value for a and a semicircle for s" vs. s. constant conductivity is related to the fact that a randomly hopping defect has no memory hence... 

Each compound synthesized is structurally similar to the previous one, since lead optimization projects do not randomly hop around the entirety of chemical space. Therefore, each step between two molecules in chemical space is small. Once synthesized and tested, no compound is considered twice during the advance of a lead optimization project. Thus the trajectory in chemical space is self-avoiding. 

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