The Berry phase in stochastic kinetics

From Ilya Nemenman: Theoretical Biophysics @ Emory
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Berry phase is a quantum mechanical phenomena that exhibits itself as a phase change of the wave function when the system is evolved adiabatically slowly by a periodic perturbation, coming to the ooriginal state. Related phenomenology has been observed in chemical kinetics systems


  1. P. Ao. Potential in stochastic differential equations: novel construction. J. Phys. A: Math. Gen. 37, L25, 2004. PDF.
    There is a whole range of emergent phenomena in a complex network such as robustness, adaptiveness, multiple-equilibrium, hysteresis, oscillation and feedback. Those non-equilibrium behaviours can often be described by a set of stochastic differential equations. One persistent important question is the existence of a potential function. Here we demonstrate that a dynamical structure built into stochastic differential equation allows us to construct such a global optimization potential function. We present an explicit construction procedure to obtain the potential and relevant quantities. In the procedure no reference to the Fokker–Planck equation is needed. The availability of the potential suggests that powerful statistical mechanics tools can be used in nonequilibrium situations.
    This is interesting work. Author considers the Langeving equation in space with continuous coordinates and some force fields. General force field cannot be written just as a gradient of potential, but author still tries to derive such formulation. He finds that it is possible to rewrite Langevin equation in the form where instead of force there is a gradient of a potential. The price is new terms, which resemble Berry curvature terms from wave packet equations in quantum mechanics. This indicates that some Berry phase theory can be useful, though this question is not studied. Drawback of this work is that in new form noise also becomes modified and Einstein relation becomes very complicated, so it is still unclear whether the found potential is useful. Also it seems that the paper only presents a proof, and the author in fact does not know what transformation can bring original Langevin equation to the form he needs.