A 'path-by-path' monotone extrapolation sequence for Feynman path integral calculations of quantum mechanical free energies

Feynman path integral methods based on the Trotter approximation represent paths by a set of P discrete points. We prove that the M-point partition function is an upper bound of the P-point one if M is a divisor of P. Also for this case, we show that, during calculations using P-point paths, it is possible - at negligible additional cost - to obtain M-point estimators of the partition function that, for each individual path, converge monotonically. This permits accurate extrapolation to infinite P, which greatly improves the accuracy of calculations of thermodynamic quantities.