Solution to the Fokker-Planck equation for slowly driven Brownian motion: Emergent geometry and a formula for the corresponding thermodynamic metric

Abstract

Considerable progress has recently been made with geometrical approaches to understanding and controlling small out-of-equilibrium systems, but a mathematically rigorous foundation for these methods has been lacking. Towards this end, we develop a perturbative solution to the Fokker-Planck equation for one-dimensional driven Brownian motion in the overdamped limit enabled by the spectral properties of the corresponding single-particle Schrödinger operator. The perturbation theory is in powers of the inverse characteristic timescale of variation of the fastest varying control parameter, measured in units of the system timescale, which is set by the smallest eigenvalue of the corresponding Schrödinger operator. It applies to any Brownian system for which the Schrödinger operator has a confining potential. We use the theory to rigorously derive an exact formula for a Riemannian "thermodynamic" metric in the space of control parameters of the system. We show that up to second-order terms in the perturbation theory, optimal dissipation-minimizing driving protocols minimize the length defined by this metric. We also show that a previously proposed metric is calculable from our exact formula with corrections that are exponentially suppressed in a characteristic length scale. We illustrate our formula using the two-dimensional example of a harmonic oscillator with time-dependent spring constant in a time-dependent electric field. Lastly, we demonstrate that the Riemannian geometric structure of the optimal control problem is emergent; it derives from the form of the perturbative expansion for the probability density and persists to all orders of the expansion.

14 pages; edited for clarity, matches published version

References in corpus (15)

• Efficiency at maximum power: An analytically solvable model for stochastic heat engines
• High-precision test of Landauer's principle in a feedback trap
• Optimal finite-time processes in stochastic thermodynamics
• The unlikely Carnot efficiency
• Optimal protocols for minimal work processes in underdamped stochastic thermodynamics
• The geometry of thermodynamic control
• Efficiency statistics at all times: Carnot limit at finite power
• Work and heat probability distribution of an optically driven Brownian particle: Theory and experiments
• Adiabatic processes realized with a trapped Brownian particle
• Erasure without work in an asymmetric, double-well potential
• A geometric approach to optimal nonequilibrium control: Minimizing dissipation in nanomagnetic spin systems
• Brownian duet: A novel tale of thermodynamic efficiency
• Thermodynamic geometry of minimum-dissipation driven barrier crossing
• Work distribution for the driven harmonic oscillator with time-dependent strength: Exact solution and slow driving
• Fluctuations and response in a non-equilibrium micron-sized system

Cited by in corpus (8)

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• Limited-control optimal protocols arbitrarily far from equilibrium
• Thermal brachistochrone for harmonically confined Brownian particles
• Performance of optimal linear-response processes in driven Brownian motion far from equilibrium
• Beyond Linear Response: Equivalence between Thermodynamic Geometry and Optimal Transport
• Failure of the geometric approach prediction of excess work scaling for open and isolated quantum systems
• Perturbative expansion of irreversible works in symmetric and asymmetric processes