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By K. P. N. Murthy
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N. Murthy define a probability pi = exp(βJhi ) . exp(βJhi ) + exp(−βJhi ) (64) The heat-bath algorithm [51] consists of updating the spin Si (k) to Si (k + 1) where Si (k + 1) = ±1 with probabilities pi and 1 − pi respectively. This is implemented by drawing a random number ξ (uniformly distributed in the interval 0 to 1) and setting Si (k + 1) = SIGN(pi − ξ), where the function SIGN(η) equals +1 if η > 0 and −1 if η < 0. Thus the spin at site i behaves as if it is in its private heat bath constructed by its nearest neigbbours!
I said that λk and Ck can be obtained, in principle, from the transition matrix W . But in practice, we do not need the transition matrix; also it is not desirable nor feasible to work with the transition matrix since its size is rather large; as we have already seen, even for a small system of Ising spins on a 10 × 10 square lattice, the associated transition matrix would contain some 1060 elements! The practical implementation of the n-fold way proceeds as follows. All the spins in the system can be put into n classes where n is a small manageable number; Let ni be the number of spins in the class i when the system is in a microstate Ck at time k.
N. 5 1 0 0 Figure 5: Magnetization per spin ( |M |/L2 ) vs. J/kB T for various system sizes L. 5 1 0 0 Figure 6: Magnetic susceptibility per spin (χ/L2 ) v s. 5 1 0 0 Figure 7: Specific heat per spin (CV /L2 ) v s. J/kB T for various system sizes L So far so good; we know now how to assemble, employing random numbers, a Markov chain of microstates each produced from its predecessor through an attempted single flip. 19 The system has a natural tendency to remain stay put in a microstate step after step after step.