A Generalized Smoluchowsky Equation: The Hydrodynamical and Thermodynamical Picture of Brownian Motion

We present a systematic expansion of Kramers equation in the high friction limit. The latter is expanded within an operator continued fraction scheme. The relevant operators include both temporal and spatial derivatives and a covariant derivate or gauge like operator associated to the potential energy. Trivially, the first order term yields the Smoluchowsky equation. The second order term is readily obtained, known as the corrected Smoluchowsky equation. Further terms are computed in compact and straightforward fashion. As an application, the nonequilibrium thermodynamics and hydrodynamical schemes for the one dimensional Brownian motion is presented.