Appendix C Parallel tempering

In parallel tempering, N independent systems are simulated simultaneously. The total partition function of this system (Q) equals 

Displacement. A system i is selected with a fixed probability Pi. For this system, a new point in phase-space (xi (u)) is generated from Xi (o), we assume here that this generation is symmetric and that the other systems j are not affected (xj (n) = Xj (o), j i). The ratio of acceptance probabilities equals 

Note that this expression does not depend on f (xj, pj). For the canonical ensemble, this equation reduces to 

To sample all N systems equally well, pi should be proportional to the time constant of a characteristic autocorrelation function. 

Such trial moves will be accepted when there is enough overlap between f (xi, pi) and 

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