At nonzero temperature, thermal fluctuations play a role in the behaviour of any physical, chemical, or biological system. The use of stochastic processes to model such systems dates back over 100 years, to the work of Einstein, Langevin, and Smoluchowski. Modern computer simulation techniques, in particular kinetic Monte Carlo, are built on these foundations, and a whole field of mathematical research has grown from these seeds.
Conventionally, two approaches are used in their study: 1) stochastic differential equations, where a random component is explicitly included in the forces acting on the system, and 2) deterministic partial differential equations for the system’s probability density function. In this work, we will investigate a third, less widely known, approach: path or functional integrals [1] [2]. This technique expresses the probability density as a sum over system trajectories, with a statistical weight attached to each one.
Inspired by semiclassical quantum-mechanical path integrals, which allows certain quantities to be expressed in closed form as ℏ->0 we develop a weak noise approximate theory for classical stochastic processes. This reveals a remarkable correspondence between the most probable stochastic paths and Hamiltonian mechanics in an effective potential. We investigate several previously overlooked subtleties, such as the role played by the functional Jacobian, and the necessity of turning paths, to correctly treat the long-time limit.
Armed with these tools, we derive, for the first time, closed-form expressions for the first passage density in a one-dimensional stochastic system subject to a general, nontrivial potential, and investigate simple potentials in detail. We revisit the ubiquitous problem of fluctuation-driven escape over an energy barrier, and derive the full first passage density, where only the mean was previously available. The extension to higher dimensions is then briefly explored, with the simplest free diffusion results returned to demonstrate the technique's validity beyond one dimension.
