Two coupled spins have unnormalised joint weight with

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Monte Carlo and lattice simulation of physical systems · Monte Carlo and lattice simulation of physical systems

Error, Chance and the Machine, Ch Markov-chain Monte Carlo

Problem

Two coupled spins s1,s2∈{+1,−1}s_1, s_2 \in \{+1,-1\} have unnormalised joint weight w(s1,s2)=eKs1s2w(s_1,s_2) = e^{K s_1 s_2} with K=0.4K = 0.4. A Gibbs sampler updates s1s_1 from its conditional distribution given s2s_2, then s2s_2 from its conditional distribution given the new s1s_1. ((a)) Show that P(s1=+1∣s2)=11+e−2Ks2P(s_1 = +1\mid s_2) = \dfrac{1}{1+e^{-2Ks_2}}, and evaluate it for s2=+1s_2=+1 and s2=−1s_2=-1. ((b)) Starting from (s1,s2)=(−1,−1)(s_1,s_2) = (-1,-1), perform one Gibbs sweep. To update a spin, draw a uniform random number uu on [0,1)[0,1) and set the spin to +1+1 if u<P(+1∣other spin)u < P(+1\mid\text{other spin}), otherwise to −1-1. Use u1=0.85u_1 = 0.85 for the update of s1s_1 and, after that update, u2=0.30u_2 = 0.30 for the update of s2s_2. Give the state after the sweep.

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