Path probability distribution of stochastic motion of non dissipative systems: a classical analog of Feynman factor of path integral

摘要

We investigate, by numerical simulation, the path probability of nondissipative mechanical systems undergoing stochastic motion. The aim is tosearch for the relationship between this probability and the usual mechanicalaction. The model of simulation is a one-dimensional particle subject toconservative force and Gaussian random displacement. The probability that asample path between two fixed points is taken is computed from the number ofparticles moving along this path, an output of the simulation, devided by thetotal number of particles arriving at the final point. It is found that thepath probability decays exponentially with increasing action of the samplepaths. The decay rate increases with decreasing randomness. This resultsupports the existence of a classical analog of the Feynman factor in the pathintegral formulation of quantum mechanics for Hamiltonian systems.