Prologue: The intrinsic Fokker-Planck equation

Abstract

Examination of equation (3.10) leads us to an interesting conclusion about the classes of FPE (3.1) with non singular diffusion that will represent exactly the same physical content in spite of the fact that they may be written in terms of quite different drifts f^ν (q) and diffusion tensor D ^µν (q), related among themselves by general gross variables transformations q′= q′(q). Variations in the phenomenological drift f ^ν (q) or in the non singular diffusion matrix D ^µν(q) will not necessarily imply changes in the physical content of the system described by (3.1). However, the classes of FPE that have the same physical content will be represented by exactly the same intrinsic FPE presented in (3.10). This is precisely the advantage of using intrinsic languages in physical theories. As an example we should mention that the class of exactly solvable FPE's presented by San Miguel ⁴ have, when written in covariant notation, very different expressions for their drifts and diffusion tensors; however, this whole class of equations would be represented by a single intrinsic. equation corresponding to the Orstein-Uhlenbeck process with a flat metric and drift linear in the grass variables.

Following the viewpoint represented by the intrinsic equation (3.10) it will be interesting to classify all the diffusion matrices that will yield all classes of FPE with the same physical content.