Project R-6299

Geometric phases in stochastic processes (Research)

Suppose one perturbs a system very slowly, and such as to bring the control parameters back to their original values at the end of the perturbation. The surprising feature is that the system need not return to its original state. A difference can appear under the form of a "geometric phase". A simple example: suppose you carry a compass from the North pole down along a meridian but coming back to the pole via another meridian. There will be a difference in orientation of the needle, corresponding to a geometric phase. Other famous examples are the Foucault's pendulum, and, in the realm of quantum mechanics, the so-called Berry phase. The concept of geometric phase also appears in the context of stochastic processes. Consider for example a Brownian particle in an on average horizontal periodic potential landscape with maxima and minima. It turns out that one can not systematically move particles if one modulates the maxima of the potential alone, but transport becomes possible if one independently modulates both maxima and minima. This is again an example of a geometric phase. Our main purpose is to develop a general theory for geometric phases in stochastic processes. We will do that mainly by transposing existing techniques from quantum mechanics and statistical mechanics.

16 September 2015 - 15 September 2018