Locating stationary paths in functional integrals: An optimization method utilizing the stationary phase Monte Carlo sampling function

A method is presented for determining the stationary phase points for multidimensional path integrals employed in the calculation of finite-temperature quantum time correlation functions. The method can be used to locate stationary paths at any physical time; in the case that t≫βℏ, the stationary points are the classical paths linking two points in configuration space. Both steepest descent and simulated annealing procedures are utilized to search for extrema in the action functional. Only the first derivatives of the action functional are required. Examples are presented first of the harmonic oscillator for which the analytical solution is known, and then for anharmonic systems, where multiple stationary phase points exist. Suggestions for Monte Carlo sampling strategies utilizing the stationary points are made. The existence of many and closely spaced stationary paths as well as caustics presents no special problems. The method is applicable to a range of problems involving functional integration, where optimal paths linking two end points are desired.