Stochastic control and non-equilibrium thermodynamics: fundamental limits.

Area of Science:

• Statistical Mechanics and Thermodynamics
• Stochastic Processes and Control Theory

Background:

• The second law of thermodynamics dictates minimum work for state transitions, achievable only via reversible, infinitely slow processes.
• Finite-time transitions in controlled stochastic systems deviate from this ideal, incurring additional work.
• Understanding these deviations is crucial for analyzing non-equilibrium dynamics.

Purpose of the Study:

• To introduce stochastic thermodynamics concepts into classical regulator theory.
• To analyze finite-time transitions between Gibbs-equilibrium states in damped stochastic systems.
• To derive fundamental relationships between work, control strategies, and thermodynamic quantities.

Main Methods:

• Consideration of damped stochastic systems within controlled, time-varying potentials.
• Application of optimal mass transport (Wasserstein-2 distance) to quantify state differences.
• Utilizing the Jordan-Kinderlehrer-Otto scheme to connect control protocols with entropy gradient flows.

Main Results:

• The minimal work gap for finite-time transitions is determined by the square of the Wasserstein-2 distance and the inverse transition time.
• This result establishes a direct link between non-equilibrium optimal control strategies and entropy functional gradient flows.
• Fundamental thermodynamic relations are derived from a control theory perspective in a multivariable setting.

Conclusions:

• Finite-time control strategies directly influence the energetic cost of transitioning between thermodynamic states.
• The study bridges stochastic thermodynamics and classical control theory, offering new insights into non-equilibrium systems.
• The derived relations provide a framework for understanding and optimizing work expenditure in controlled stochastic processes.