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Consider a parallel-tempering Markov-Chain Monte Carlo (MCMC) simulation of a system, with inverse temperatures $\beta_1 < \cdots < \beta_N$ and energies $E_1, \cdots, E_N$.

If there are two temperatures, $N=2$, the temperature-swap procedure is clear: First, let each replica evolve with the MCMC dynamics, and once in a while exchange $\beta_1 \leftrightarrow \beta_2$ with probability $p_{1,2} =\min \{ 1, \exp[-(\beta_1-\beta_2)(E_1-E_2)]\}$.

However, by looking in the literature, I could not find any detailed information on how to swap temperatures in the more general case where there are $N>2$ temperatures. One possible choice would be to do, once in a while in the simulation,

for $n=1,...N-1$

$\,\;$Exchange $\beta_n \leftrightarrow \beta_{n+1}\,$ with probability $p_{n,n+1}$

end

Is this choice correct? Does it satisfy ergodicity and detailed balance?

I am worried by the fact that this choice does not treat the temperatures $\beta_1$ and $\beta_N$ at the boundaries in the same way as those in the bulk. In fact, it attempts to swap a temperature in the bulk twice as often as those at the boundary.

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