Ergodic walk

Figure 8.16 from Frantz, D D, D L Freeman and J D DoU 1990. Reducing quasi-ergodic behavior in Monte Carlo simulations by J-walking applications to atomic clusters. The Journal of Chemical Physics 93 2769-2784. 

In J-walking [20] the periodic MC trial probability for a simulation at temperature T is taken to be a Boltzmann distribution at a high temperature, Tj ( 3j = IkTj), The jumping temperature, 7), is sufficiently high that the Metropolois walk can be assumed to be ergodic. This results in the acceptance probability. 

Frantz, D.D. Freeman, D.L. Doll, J.D., Reducing quasi-ergodic behavior in Monte Carlo simulation by J-walking Applications to atomic clusters, J. Chem. Phys. 1990, 93, 2769... 

How general are our results From a stochastic point of view ergodicity breaking, Levy statistics, anomalous diffusion, aging, and fractional calculus, are all related. In particular ergodicity breaking is found in other models with power-law distributions, related to the underlying stochastic model (the Levy walk). For example, the well known continuous time random walk model also... 

G. Bel and E. Barkai, Weak ergodicity breaking in the continous-time random walk, Phys. Rev. Left. 94 240602 (2005). 

Tackling the Problem of Quasi-ergodicity J-walking and Multicanonical Monte Carlo... 

Quasi-ergodic Behavior in Monte Carlo Simulations by J-walking Applications to Atomic Clusters. 

It is easily verified that this transition matrix satisfies detailed balance for the distribution irp. It is also easy to see that the algorithm is ergodic to get from a walk u to another walk (p, it suffices to use AN = —1 moves to transform u into the empty walk, and then use AN = H-1 moves to build up the walk w. ... 

Since the algorithm used in step (a) is ergodic in the ensemble of fixed-length walks, it is easy to see that the full algorithm is ergodic. (If and uP- are perpendicular rods, then the join-and-cut move will always succeed.) It is also easy to see that the algorithm satisfies the detailed-balance condition with respect to the equal-weight measure... 

For the BFACF algorithm applied to unrooted polygons (ring polymers), the ergodic classes are precisely the knot classes. This is probably true also for the BFACF algorithm applied to walks with x oo = 1, but it has apparently not yet been proven. 

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