Local moves

Sufficient conditions are that any local move away from the optimal point ti gives rise to an increase in the objective function. Expand F in a Taylor series locally around the candidate point ti up to second-order terms ... 

For this algorithm, one can prove that detailed balance is guaranteed and the exact average of any configuration-dependent property over the accessible space is obtained. Two key issues determine the detailed balance. The first is the fact that the trial probability to pick the displacement vector Dfc to go from the fcth to the Zth e-sphere equals the trial probability to pick the displacement vector D fc for the reverse step. The second issue is that the trial probability for a local MC step that moves the walker from a point inside an e-sphere to a point outside that sphere is the same as for the reverse move i.e., (1 - / ) times what it would be in a walk restricted to local moves. 

Mezei, M. Efficient Monte Carlo sampling for long molecular chains using local moves, tested on a solvated lipid bilayer. J. Chem. Phys. 2003, 118, 3874 9. 

Monte Carlo Algorithms In an MD simulation, the local movement of the atoms is performed due to the forces present at each step. In contrast, in a Monte Carlo (MC) simulation, the local moves of the atoms are performed randomly. 

A set of local moves for a one-dimensional spin-1/2 model is shown in Fig. 12 [49,50]. The two required moves are the insertion and removal of a pair of exchange processes (Fig. 12(a)) and the shift in time of an exchange process (Fig. 12(b)). Slightly more complicated local moves are needed for higher-dimensional models, for example to allow world lines to wind around elementary squares in a square lattice [51]. Since these local updates cannot... 

Fig. 16. Moves used to equilibrate coil configurations for the self-avoiding walk model of polymer chains on the simple cubic lattice (upper party end rotations, kinkjump motions and crankshaft rotations f 107]. From time to time these local moves alternate with a move (lower pan) where one attempts to replace an A-chain by a B-chain in an identical coil configuration, or vice versa. In the transition probability of this move, the chemical potential difference Ap as well as the energy change SjF enter. From Binder [2S8]...

The MC runs themselves were carried out with local moves in the familiar way. At each step a trial move is attempted for a randomly-chosen single particle within a small cubic box (the size chosen to give a moderate acceptance rate). A trial move from state m to state n is accepted as usual with probability amn = min(l, njnm) (using an obvious notation). Thus the run proceeds just as in conventional MC computations, except with the nonphysical sampling distribution ( ). The only extra computation time, compared to conventional MC runs at a single state, comes from having... 

The simulation of polyethylene melts proceeds by random jumps (of length 0.25 nm) by individual beads to unoccupied nearest-neighbor sites on the high coordination lattice, with retention of all connections to bonded beads. These single-bead moves correspond to a variety of local moves in the underlying fidly atomistic model that change the coordinates of 2 or 3 carbon atoms.4 The acceptance or rejection of proposed moves is via the customary Metropolis criteria." ... 

Flsu 9 Possible local motions for a linear chain. The original conformation of the chain is indicated by solid lines and filled circles. The dashed lines and open circles show the new conformations allowed by the indicated local moves. (From Ref 27.)... 

Panagiotopoulos and coworkers [51] use the same parameters as Larson for the study of phase behavior, but with two different simulation methodologies. The first technique is the Gibbs ensemble method, in which each bulk phase is simulated in a separate cell and molecules are interchanged and volumes adjusted between the two for equilibration of the system [52]. The second is a standard canonical ensemble simulation, like Larson s, but employs the configurational bias Monte Carlo method. The configurational bias Monte Carlo method is much more efficient than the ones based on reptation and other local moves but is not useful if any dynamic information is sought from the simulations. 

Fig. 3 In the BFM model each monomer occupies a unit cube in a 3D simple cubic lattice. Bonds between monomers are taken in a range between 2 and /lO lattice units which ensure cut-avoiding during local moves. In a Monte Carlo step a monomer is chosen randomly and tries to move into one of the six possible nearest neighbor positions. The move is rejected if the new lattice places are occupied by other monomers (excluded volume). If the energy difference between the new and the old monomer position is positive, the move is only accepted with the corresponding Metropohs rate. The interaction energy between different monomer species (A and B) is calculated in three shells around the given monomer as indicated in the figure. We implement only repulsive interaction between different monomer species. AH other interactions are athermal...

The Web site www.globalization.com points to the need to both internationalize and localize one s products. Internationalization is the development of products and software free of language and cultural bias. This enables adaptation of products to varied markets. Hopefully this book fulfills this intent. Localization moves in the other direction, taking a generic, location-neutral product and customizing it to a local market. Examples include end-user instmctions, labels, and adaptation to regional standards. 

FIGURE 5.6. A trapped conformation for the algorithm with only local moves for a chain on the simple square lattice (A). A longer distance move that guarantees the ergodicity of the algorithm, where a U shaped fragment at one part of the chain is cut-off and attached somewhere else (B). 

Figure 14 AFM (b-d) and fluorescence (a, e, f) images of PSSA structures formed in different points on the contact line, as shown on (g) in each point the nanostructures are oriented perpendicular to the local moving contact line.

---