Phase space maximal entropy random walk: Langevin-like ensembles of physical trajectories

Abstract

As written by statistician George Box "All models are wrong, but some are useful", standard diffusion derivation or Feynman path ensembles use nonphysical infinite velocity/kinetic energy nowhere differentiable trajectories - what seems wrong, might be only our approximation to simplify mathematics. This article introduces some basic tools to investigate this issue. To consider ensembles of more physical finite velocity trajectories, we can work in $(x,v)$ phase space like in Langevin equation with velocity controlling spatial steps, here also controlled with spatial potential $V(x)$. There are derived and compared 4 approaches to predict stationary probability distributions: using Boltzmann ensemble of steps/points in space (GRW - generic random walk) or in phase space (psGRW), and analogously Boltzmann ensemble of paths in space (MERW - maximal entropy random walk) and in phase space (psMERW), also generalized to L{\'e}vy flights. Path ensembles generally have much stronger Anderson-like localization, MERW has stationary distribution exactly as quantum ground state. Proposed novel MERW in phase space has some slight differences, which might be distinguished experimentally. For example for 1D infinite potential well: $\rho=1$ stationary distribution for step ensemble, $\rho\sim \sin^2$ for path ensemble (as in QM), and $\rho\sim \sin$ for proposed smooth path ensembles - more frequently approaching the barriers due to randomly gained velocity.