From Maximal Entropy Random Walk to quantum thermodynamics

Surprisingly the looking natural random walk leading to Brownian motion occurs to be often biased in a very subtle way: usually refers to only approximate fulfillment of thermodynamical principles like maximizing uncertainty. Recently, a new philosophy of stochastic modeling was introduced, which by being mathematically similar to euclidean path integrals, finally fulfills these principles exactly. Their local behavior is usually similar, but may lead to completely different global properties. 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 having 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 Schrodinger operator while including potential and making infinitesimal limit. It also provides a natural intuition of the amplitudes' squares relating to probabilities. The present paper gathers, formalizes and extends these results. There are also introduced and discussed some new generalizations, like considering multiple particles with thermodynamical analogue of Pauli exclusion principle or time dependent cases, which allowed to introduce thermodynamical analogues of momentum operator, Ehrenfest equation and Heisenberg uncertainty principle.