The Thermodynamic Geometry of Controlled Stochastic Systems

Synopsis

At the scale of individual particles, fundamental physics has found powerful laws governing the dynamics of matter and energy. For systems composed of a great many particles, equilibrium statistical mechanics has—to a degree—explained the emergence of macroscopic “laws” from microscopic chaos. Between these limits, we find the basic constituents of life – neither simple enough for direct atomistic simulation nor large enough to neglect thermal fluctuations. Though these mesoscopic systems are as bound by the laws of physics as anything else, their enduring complexity offers new challenges to physicists.

One framework that can help us to navigate this complexity is optimal control theory. The problem of finding an optimal control protocol for a stochastic system arises in various contexts, from the design of experiments probing mesoscopic systems, to theoretical investigations of biological molecular motors, to the design of efficient nanotechnology. However, solving an optimal control problem exactly is seldom feasible. A widely used approximation recasts minimum-work control protocols as geodesics on a thermodynamic manifold whose metric tensor is a generalized friction. After a high-level and intuitive introduction to this geometric framework, I will describe ongoing theoretical efforts to advance the formalism. I will conclude with a broad outlook, speculating on future directions and cross-disciplinary implications.