ABSTRACT
The complexity of a self-avoiding walk (SAW) pointed out in Chapter 9 is reflected also by the difficulty to handle it with computer simulation. Chapter 14 deals with methods for generating SAWs step-by-step (from nothing), with the help of transition probabilities (TPs); thus, unlike the case of Monte Carlo, the construction probability, P = Π k TP k , is known and the entropy, S is known as well. With the earliest technique, “simple sampling” (Wall et al.), at each step k on a square lattice, a direction, ν, is chosen “democratically” out of the four directions neighbor to step k – 1; while the procedure is exact, it is very inefficient (N < 90 in d = 2) due to the high chance to hit an already occupied site. With the Rosenbluth & Rosenbluth (RR) method (N < 160, d = 3), ν is chosen at random, but out of the unoccupied sites around step k – 1, a procedure which introduces a bias should be removed. More efficient are “the enrichment method” (Wall & Erpenbeck) (N ~ 1000) and “the dimerization method” (Alexandrowicz) (N ~ 104). With the “scanning method” (SM), the one step ahead (f = 1) scanning of RR is replaced by a longer scanning based on f > 1 steps ahead (f < 20 for d = 2), where the bias decreases with increasing f. SM (N ~ 1500, d = 2) provides also a lower and upper bounds for S, and a procedure for estimating accuracy; it is, in particular, efficient for chains under geometrical constraints (e.g., SAWs adsorbed to a surface) and for Ising models. SM is the basis for methods for extracting S from MC/MD samples (see Chapter 19).
