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If configuration A is equal to configuration B in a Metropolis Monte Carlo method, do you still do the attempted update?

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Was initially posted as a comment. (Comment removed now)

The micro-states are changing nonetheless, a different point in the phase space of your system, so the system is evolving, even though the two states are part of the same macro-state. Finally remember that the Metropolis probability criterion for accepting moves is (in one of the simplest schemes of it): $$\frac{N(n)}{N(o)}\propto exp[−β[U(n)−U(o)]]$$ Where $N(o)$ and $N(n)$ are the probability densities of the old and new states respectively. Finally in the case of equal energies we have $\frac{N(n)}{N(o)}\propto 1$ and the attempted move is accepted.

Bear in mind that, another way to go about this, is by considering the fact that even when your system has been equilibrated, you still need it to evolve and explore further micro-states, as there may still be fluctuating elements in your system, like an interface between two coexisting phases.

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