Nonlocal moves

Figure 3 The collapse of the peptide Ace-Nle30-Nme under deeply quenched poor solvent conditions monitored by both radius of gyration (Panel A) and energy relaxation (Panel B). MC simulations were performed in dihedral space 81% of moves attempted to change angles, 9% sampled the w angles, and 10% the side chains. For the randomized case (solid line), all angles were uniformly sampled from the interval —180° to 180° each time. For the stepwise case (dashed line), dihedral angles were perturbed uniformly by a maximum of 10° for 4>/ / moves, 2° for w moves, and 30° for side-chain moves. In the mixed case (dash-dotted line), the stepwise protocol was modified to include nonlocal moves with fractions of 20% for 4>/ J/ moves, 10% for to moves, and 30% for side-chain moves. For each of the three cases, data from 20 independent runs were combined to yield the traces shown. CPU times are approximate, since stochastic variations in runtime were observed for the independent runs. Each run comprised of 3 x 107 steps. Error estimates are not shown in the interest of clarity, but indicated the results to be robust.

The spin stiffness ps can be obtained from fluctuations of the winding numbers of the world lines [45], a measurement which obviously requires nonlocal moves that can change these winding numbers. 

An algorithm that incorporates large nonlocal moves of bonds and works for dense polymer systems (even without any vacancies, = 1) is the collective motion algorithm where one transports beads from kinks or chain ends along the chain contour to another position along the chain, for several chains simultaneously, so that in this way this rearrange-... 

The best can hope for is an autocorrelation time r (N) for even if the non-local moves were to cause instant equilibration at fixed N, the local moves would still have to carry out a random walk in N. Such a behavior, if achieved, would be a significant improvement over the pure BFACF algorithm. This estimate refers, however, to physical time units since the nonlocal moves require a mean CPU time per move that grows as a fractional power of (N), it is a subtle matter to choose p i so as to minimize the autocorrelation time as measured in computer (CPU) time units. 

Batoulis and Kremer were able to make very accurate estimates of the exponent 7 as well as p r) and (R ) for/< 6 in a good solvent. Dynamic MC also works well in this limit, particularly if one invokes nonlocal moves, such the pivot algorithm.However as /increases, the interior becomes very dense and many of these methods fail or become inefficient. In this case, one can use either MD methods or a local stochastic MC method such as the bond fluctuation method on a lattice or a simple off-lattice MC in which one attempts to move one monomer at a time. It is also possible to use nonlocal moves in the dilute, outer regions of the star and local moves near the interior, though this has not been done to the best of our knowledge. 

This formulation results very insightful according to Equation 8.30, particles move under the action of an effective force — We , i.e., the nonlocal action of the quantum potential here is seen as the effect of a (nonlocal) quantum force. From a computational viewpoint, these formulation results are very interesting in connection to quantum hydrodynamics [21,27]. Thus, Equations 8.27 can be reexpressed in terms of a continuity equation and a generalized Euler equation. As happens with classical fluids, here also two important concepts that come into play the quantum pressure and the quantum vortices [28], which occur at nodal regions where the velocity field is rotational. 

At the other extreme, we have a molecule that upon adsoiption loses only one of the translational degrees of freedom that it had in solution [Fig. 6.95(b)]. This translational degree of freedom is transformed into a vibration perpendicular to the surface. What about the other degrees of freedom In this case the molecule retains the rest of them, and thus it can move freely on the surface of the electrode and rotate as it pleases. The molecule is adsoibed in a nonlocalized fashion. 

This approach is very general. For example, it is not restricted to monodis-perse systems, and Krauth and co-workers have applied it successfully to binary [17] and polydisperse [18] mixtures. Indeed, conventional simulations of size-asymmetric mixtures typically suffer from jamming problems, in which a very large fraction of all trial moves is rejected because of particle overlaps. In the geometric cluster algorithm particles are moved in a nonlocal fashion, yet overlaps are avoided. 

Nonlocal bending resistance A resistance to bending resulting from the differential expansion and compression of the two adjacent leaflets of a lipid bilayer. It is termed nonlocal because the leaflets can move laterally relative to one another to relieve local strains, such that the net resistance to bending depends on the integral of the change in curvature of the entire membrane capsule. 

N = 60 and A N = 1000. Both local and nonlocal (ie, reptation) moves have been used in our Monte-Carlo simulations. Filled-in symbols are reproduced from Reference 35 ( , N = 100) and Reference 36 (B, Af = 64). The curves are drawn to guide the eye. Reprinted with permission from Ref 40. Copyright (1997) American Chemical Society. 

Thus as we proceed through the orbitals of an atom from the K-shell outwards, the screening seen by each shell increases even though all the electrons are moving in the same nonlocal potential. To illustrate this consider the advantages of the choice for the single particle field for helium. It can be written as... 

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