More on Detailed Balance

The Metropolis choice is to maximize the $A$ values, that is

$$\begin{equation} A_{j \rightarrow i} = \min \left ( 1, \frac{p_iT_{i\rightarrow j}}{ p_jT_{j\rightarrow i}}\right ). \tag{9} \end{equation}$$

Other choices are possible, but they all correspond to multilplying $A_{i\rightarrow j}$ and $A_{j\rightarrow i}$ by the same constant smaller than unity. The penalty function method uses just such a factor to compensate for $p_i$ that are evaluated stochastically and are therefore noisy.

Having chosen the acceptance probabilities, we have guaranteed that if the ${\cal w}_i^{(n)}$ has equilibrated, that is if it is equal to $p_i$, it will remain equilibrated. Next we need to find the circumstances for convergence to equilibrium.

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