Quasi-ergodicity An Insidious Problem

While the methods discussed thus far are sufficiently powerful to allow the simulation of many complex phenomena, there are circumstances where their direct implementation must be modified for efficient sampling. In many important cases the direct application of the preceding strategies with a (necessarily) finite set of points can give misleading or incorrect results. 

To illustrate the kinds of problems to which we refer, we make use of a classical double-well potential that has proved to be usefuF for the study of quasi-ergodicity  

This potential has a minimum of zero energy at x = 1, a second minimiun of variable energy atx = —a and a barrier of vmit height separating the minima at X = 0. For 0 a 1 it is useful to define the relative well depth by 

Monte Carlo points. As the temperature is increased beyond the heat capacity maximum in the exact result, the MMC simulated points begin to rise but they are in poor agreement with the exact data. Additionally, the calculated error bars increase, but are artificially large in this calculation and do not accurately reflect the true asymptotic fluctuations of the heat capacity. Finally, at the highest calculated temperatures, both the Metropolis Monte Carlo and exact data are in agreement. 

Several methods have been developed to remove such difficulties from MMC simulations. One obvious but not very useful method is to include more Metropolis Monte Carlo points. In the limit of an infinite simulation 

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