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We describe particles in a potential by a special diffusion process, themaximal entropy random walk (MERW) on a lattice. Since MERW originates in avariational problem, it shares the linear algebra of Hilbert spaces withquantum mechanics. The Born rule appears from measurements between equilibriumstates in the past and the same equilibrium states in the future. Introducingpotentials by the observation that time, in a gravitational field running indifferent heights with a different speed, MERW respects the rule that alltrajectories of the same duration are counted with equal probability. In thisway, MERW allows us to derive the Schr\"odinger equation for a particle in apotential and the Darwin term of the nonrelativistic expansion of the Diracequation. Finally, we discuss why quantum mechanics cannot be simply a resultof MERW, but, due to the many analogies, MERW may pave the way for furtherunderstanding.

Stationary Schr\"odinger Equation and Darwin Term from Maximal Entropy Random Walk