BIRS Workshop Lecture Videos
Geometry of Noise Driven Dynamical Systems Neu, John
In their book on path integrals, Feyman and Hibbs formulated a "geometrical optics" of most probable paths in stochastic dynamical systems. The most probable path in state space between two given endpoints minimizes a stochastic action functional. Some significant features of this geometric optics: The speed along the most probable path equals the deterministic speed, and generally speaking, the "Snell’s law" that determines the geometry of the path is not invariant under orientation reversal. The paths from point \(a\) to point \(b\), and from \(b\) back to \(a\) may be different. We present a dramatic example demonstrating the breaking of orientation reversal. In the special cases with orientation reversal invariance, the stochastic dynamics is said to have detailed bal- ance. The local expression of detailed balance is the vanishing of a stochastic vorticity tensor which is determined from the velocity field and noise tensor of the dynamics. A more traditional notion of detailed balance is based upon the existence of equilibrium solutions to the Folker-Planck equation, in which the probability current vanishes identically. The condition for equilibrium solutions in this sense is vanishing vorticity. We can detect the breaking of detailed balance from direct measurements of stochastic trajectories: Project the dynamics down onto a two dimensional plane in state space, and look at the area swept out by the projected trajectory. If detailed balance is broken, there are planes of projection, so that the expected area grows linearly in time. We carry out this program for a simple RC network with "cold" and "hot" resistors. The breaking of detailed balance directly correlates with net heat transfer from the hot resistor to the cold resistor. One of the intriguing aspects of the area crtierion: We can implement it in practice, without knowledge of the velocity field and noise tensor of the stochastic dynamics.
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