Pivot algorithm

N. Madras, A. D. Sokal. The pivot algorithm A highly efficient Monte Carlo method for the self-avoiding walk. J Stat Phys 50 109, 1988. 

According to the results, it is determined that the asphericities can be described in terms of polynomials in Forni et al. [140] also used an off-lattice model and an MC Pivot algorithm to determine the star asphericity for ideal, theta, and EV 12-arm star chains. They also found that the EV stars chains are more spherical than the ideal and theta star chains. In these simulations the theta chains exhibit a remarkable variation of shape with arm length, so that short chains (where core effects are dominant for all chains with intramolecular interactions) have asphericities closer to those to those found with EV, while longer chains asymptotically approach the ideal chain value(see Fig. 10). 

In order to determine a system thermodynamically, one has to specify some independent parameters (e.g. N, T, P or V) besides the composition of the system. The most common choice in MC simulation is to specify N, V and T resulting in the canonical ensemble, where the Helmholtz free energy A is the natural thermodynamical potential. However, MC calculations can be performed in any ensemble, where the suitable choice depends on the application. It is straightforward to apply the Metropolis MC algorithm to a simple electric double layer in the iVFT ensemble. It is however, not so efficient for polymers composed of more than a few tens of monomers. For long polymers other algorithms should be considered and the Pivot algorithm [21] offers an efficient alternative. MC simulations provide thermodynamic and structural information, but time-dependent properties are not accessible. If kinetic or time-dependent properties are of interest one has to use molecular dynamic or brownian dynamic simulations. 

On the subject of stars and linear chains, the same authors have employed MC calculations based on Bishop and Clarke s pivot algorithm to study the validity of scaling and group renormalization theories of these interesting molecules. Dimensions and intrinsic viscosities were also calculated. - ... 

Using the best numerical values of v = 0.588 and y = 1.1619 (hence, 6 = 0.275 and f = 2.427), and with the help of a technique of the pivot algorithm, Valleau [41] could show that the empirical equation remarkably well fits the Monte Carlo simulations on a 3d-lattice, in support of the conjecture for a simple closed form of the excluded volume chain in the asymptotic limit, N— °°. 

Written in this form, (74)-(76) are an example of a linear complementarity problem, which also has applications in game theory. In this context, (76) is referred to as the complementarity condition, and all Wi, Zi pairs are said to be complementary variables. A method for finding a solution to this system is the complementary pivoting algorithm credited to Lemke (1968). Under certain assumptions on the matrix M, the algorithm determines a solution or finds a direction indicating unboundedness in a finite number of iterations. 

Nevertheless, the pivot algorithm has one distinctive advantage. It exhibits very good ergodicity properties [5], while slithering snake and... 

Bernardini, R Mittleman, J. Rushmeier, H. Silva, C. Taubin, G. The Ball-Pivoting Algorithm for Surface Reconstruction, Visualization and Computer Graphics, 5(4), 1999,349-359. DOI 10.1109/2945.817351... 

Fig. 1.4 Various examples of dynamic Monte Carlo algorithms for SAWs sites taken by beads are shown by dots, and bonds connecting the bead are shown by lines. Bonds that are moved are shown as a wavy line (before the move) or broken line (after the move), while bonds that are not moved are shown as full lines, (a) Generalized Verdier-Stockmayer algorithm on the simple cubic lattice showing three type of motions end-bond motion, kink-jump motion, 90° crankshaft rotation (b) slithering snake algorithm (c) pivot algorithm. (From Kremer and Binder )...

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