Noise equals control

Abstract

Stochastic systems have a control-theoretic interpretation in which noise plays the role of control. In the weak-noise limit, relevant at low temperatures or in large populations, this leads to a precise mathematical mapping: the most probable trajectory between two states minimizes an action functional and corresponds to an optimal control strategy. In Langevin dynamics, the noise term itself serves as the control. For general Markov jump processes, such as chemical reaction networks or electronic circuits, we use the Doi-Peliti formalism to identify the `response' (or `momentum') field $\pi$ as the control variable. This resolves a long-standing interpretational problem in the field-theoretic description of stochastic systems: although $\pi$ evolves backward in time, it has a clear physical role as the control that steers the system along rare trajectories. This implies that Nature is constantly sampling control strategies. We illustrate the mapping on multistable chemical reaction networks, systems with unstable fixed points, and specifically on stochastic resonance and Brownian ratchets. The noise-control mapping justifies agential descriptions of these phenomena, and builds intuition for otherwise puzzling phenomena of stochastic systems: why probabilities are generically non-smooth functions of state out of thermal equilibrium; why biological mechanisms can work better in the presence of noise; and how agential behavior emerges naturally without recourse to mysticism.

10 pages + 7 pages SI. v3: expanded to full-length article; clarified discussion throughout, and many examples added (stochastic resonance, Brownian ratchet); v4: discuss control-stabilized fixed point of the double well in main text

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