From Maximal Entropy Random Walk to quantum thermodynamics

Surprisingly, the natural looking random walk leading to Brownian motion occurs to be often biased in a very subtle way: emphasizing some possibilities by only approximating maximal uncertainty principle. A new philosophy of stochastic modelling has been recently introduced, in which we use the only maximizing entropy choice of transition probabilities instead. Local behaviour of both approaches is similar, but they usually lead to completely different global situations. In contrast to Brownian motion leading to nearly uniform stationary density, this recent approach turns out in agreement with having strong localization properties, thermodynamical predictions of quantum mechanics, like thermalizing to dynamical equilibrium state of probability density as the quantum ground state: squares of coordinates of the lowest energy eigenvector of the Bose-Hubbard Hamiltonian for single particle in discrete case, or of the standard Schrödinger operator while including potential and making infinitesimal limit.