Elementary moves

Monte Carlo simulations generate a large number of confonnations of tire microscopic model under study that confonn to tire probability distribution dictated by macroscopic constrains imposed on tire systems. For example, a Monte Carlo simulation of a melt at a given temperature T produces an ensemble of confonnations in which confonnation with energy E. occurs witli a probability proportional to exp (- Ej / kT). An advantage of tire Monte Carlo metliod is tliat, by judicious choice of tire elementary moves, one can circumvent tire limitations of molecular dynamics techniques and effect rapid equilibration of multiple chain systems [65]. Flowever, Monte Carlo... 

With either method, for an energy calculation associated with an elementary move in a large system, a vast majority of the interatomic distances involving the moving parts are never computed. In contrast, Verlet lists [52] are only useful in MC calculations of dense fluids because of the local nature of elementary moves in this particular case. 

Motion of the chain, represented by a random walk on a lattice, is defined by the set of allowed elementary moves of monomers between neighbouring sites that preserve chain connectivity and do not allow the chain to... 

In numerical simulations of systems of N polarizable dipolar particles the induced dipoles must be computed by solving a set of N linear equations depending on the positions and orientations of the N dipoles of the system. The solution of this set of equations requires of the order of operations and deteriorates considerably the efficiency of MC simulations since, in principle, for each MC elementary move involving, for instance, the displacement of one particle, the N linear equations have to be solved. The vaUdity of numerical procedures allowing us to overcome this problem is discussed in [94]. The procedures are based on the choice of an adequate cut-off of the interparticle distances such that the computation of the induced dipole of a displaced particle depends only on the positions and orientations of dipoles located at a distance of the trial particle position smaller than the cut-off. 

We have used the CONROT method in the melt simulations discussed here. Rather than going into the geometric and sampling subtleties of the CONROT move, we will confine ourselves to a Mef description of the elementary moves, referring the reader to [27] for more details on the method. 

Before moving on to the PVT and chain characteristics, some points should be inade concerning the reptation elementary move. We found that if a reptation... 

Second, do the following for each attempted elementary move. 

If, and only if, the elementary move is accepted and the skin was broken, then rebuild neighbor lists. 

The efficiency of a MC algorithm depends on the elementary moves it employs to go from one configuration to the next in the sequence. An attempted move typically involves changing a small number of degrees of freedom it is accepted or rejected according to selection criteria designed so that the sequence ultimately conforms to the probability distribution of interest. In addition to usual moves of molecule translation and rotation practiced for small-molecule fluids, special moves have been invented for polymers. The reptation (slithering... 

A) We must prove ergodicity (irreducibility), i.e., we must prove that we can get from any state to G S to any other state w e S by some finite sequence of allowed elementary moves. 

In this chapter we always measure time in units of attempted elementary moves. Much of the literature on dynamic SAW models uses a time scale of attempted elementary moves per bead autocorrelation times expressed in this way should be multipUed by N (actually A+ 1) before comparing them to the present paper. 

The elementary moves in a SAW Monte Carlo algorithm can be classified according to whether they are... 

Scheme References Elementary moves Autocorrelation time t (see Figs 2.1 and 2.2) (in elementary moves) ...

This nonergodicity is in fact quite easy to see consider the double cul-de-sac configuration shown in Fig. 2.10(a). This SAW is completely frozen under elementary moves A, B, D and F it cannot transform itself into any other state, nor can it be reached from any other state. It follows that the original Verdier-Stockmayer algorithm and most of its generaliza-tions are nonergodic (in d=l) already for 11. 

Let us consider, as a typical quantity, the evolution of the squared radius of gyration of the chain. At each elementary move, a few beads of the chain move a distance of order 1 it follows from eq. (2.5c) that the change in is of order. In order to traverse its equilibrium distribution, must change by something of order its standard deviation, which is N. Assuming that R executes a random walk, it takes about j 2+2v elementary moves for this to occur. So we predict r This basic estimate ought to be applicable to the dynamics of... 

The slithering-snake (reptation) algorithm was invented independently by Kron and Wall and Mandel. " The elementary moves of this algorithm are shthering motions one step is appended at one end of the walk, and one step is simultaneously deleted from the other end (Fig. 2.4). Such a move is W-conserving and Aflocal (but not local). 

The algorithm s elementary moves are as follows either one attempts to append a new step to the walk, with equal probability in each of the 2d possible directions or else one deletes the last step from the walk. In the former case, one must check that the proposed new walk is self-avoiding if it is not, then the attempted move is rejected and the old configuration is counted again in the sample ( null transition ). If an attempt is made to delete a step from an already-empty walk, then a null transition is also made. The relative probabilities of AN = H-1 and AN = -1 attempts are chosen to be... 

The dynamic critical behavior of the pivot + join-and-cut algorithm was studied in Ref. 165 by both analytical and numerical methods. For the relevant observable 2T= log[A i(JV(o( — A i)], the autocorrelation time in units of elementary moves is found to grow as Ti , x in d = 2. On... 

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