Integrated covariances as excess observables weighted by currents and activities

Area of Science:

• Statistical physics
• Non-equilibrium thermodynamics
• Stochastic processes

Background:

• Near equilibrium, fluctuation-dissipation theorem governs symmetric covariances, while antisymmetric parts vanish due to Onsager reciprocity.
• These principles fail far from equilibrium, necessitating new theoretical frameworks.

Purpose of the Study:

• Develop a unified formalism for both symmetric and antisymmetric components of integrated covariances in nonequilibrium steady states.
• Derive exact, computationally tractable expressions for these quantities.
• Establish thermodynamic bounds and connections to excess observables and activity.

Main Methods:

• Developed a unified formalism for nonequilibrium steady states.
• Derived exact expressions for integrated covariances in Markov jump processes and Fokker-Planck equations.
• Expressed components in terms of excess observables and established thermodynamic bounds.

Main Results:

• Obtained exact, computationally tractable expressions for symmetric and antisymmetric integrated covariances.
• Expressed these components using excess observables, linking statistical physics and reinforcement learning.
• Established thermodynamic upper bounds for antisymmetric covariances in terms of entropy production and cycle affinities.
• Showed activity-driven self-averaging speed-up is bounded by cycle affinities.

Conclusions:

• The developed formalism provides a unified approach to integrated covariances in nonequilibrium steady states.
• Excess observables and thermodynamic bounds offer new insights into nonequilibrium systems.
• The findings have implications for understanding stochastic processes and reinforcement learning.