Traditionally, stochastic processes are modelled one of two ways: a continuum Fokker-Planck approach, where a PDE is solved to determine the time evolution of the probability density, or a Langevin approach, where the SDE describing the system is sampled, and multiple simulations are used to collect statistics. There is also a third way: the functional or path integral. Originally developed by Wiener in the 1920s to model Brownian motion, path integrals were famously applied to quantum mechanics by Feynman in the 1950s. However, they also have much to offer classical stochastic processes (and statistical physics).
In this talk I will introduce the formalism at a physicist’s level of rigour, and focus on determining the dominant contribution to the path integral when the noise is weak. There exists a remarkable correspondence between the most-probable stochastic paths and Hamiltonian dynamics in an effective potential [1,2]. I will then discuss some applications, mainly in the general area of rare events. The first concerns reaction pathways conditioned on finite time [2], and demonstrates that the most probable pathway at a finite time may be very different from the usual minimum energy path used to calculate the average reaction rate. Secondly, I will review the path integral hyperdynamics introduced by Chen and Horing in 2007 [3], and discuss its application to non-equilibrium processes [4]. Finally, if time permits, I will share some more speculative ideas on exploiting the path integral to accelerate the sampling of rare events.
[1] Ge, Hao, and Hong Qian. Int. J. Mod. Phys. B 26.24 1230012 (2012)
[2] Fitzgerald, Steve et al, arXiv:2210.11280 (2022)
[3] Chen, L. Y., and Horing, N. J. M. J. Chem. Phys. 126.22 224103 (2007)
[4] Huang, Shenglin, et al. J. Mech. Phys. Solids 161 104779 (2022)
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