1206.2798 (Matteo Polettini)

Generally covariant state-dependent diffusion    [PDF]

Matteo Polettini

We show that Wiener increments are invariant under internal gauge transformations, analogous to rotations of frames in General Relativity, and assume that the stochastic dynamics of open systems is both coordinate covariant and gauge invariant. We derive a second-order Langevin equation with state-dependent diffusion matrix and vanishing environmental forces. It differs from previous proposals but nevertheless entails the Einstein relation and a Maxwellian conditional steady state for the velocities. Both a first-principle derivation and the over-damping limit lead to a stochastic differential equation in state space that cannot be interpreted as a pure differential (Ito, Stratonovich or else). At odds with the latter interpretations, the corresponding Fokker-Planck equation admits an equilibrium steady state. Experimentally testable predictions of the theory are: The equilibrium nature of the spatial density; Its non-uniform profile; A non-Gaussian velocity distribution.

View original: http://arxiv.org/abs/1206.2798