Universal response inequalities beyond steady states via trajectory information geometry

Area of Science:

• Statistical Mechanics
• Non-equilibrium Physics
• Information Geometry

Background:

• Fluctuation-dissipation relations describe near-equilibrium system responses.
• Extending response theory to non-equilibrium steady states has been achieved.
• A general response theory for systems far from steady states remained elusive.

Purpose of the Study:

• To generalize response theory for nonstationary Markov processes using a trajectory information geometric framework.
• To derive bounds for linear and nonlinear responses in far-from-equilibrium systems.
• To reveal connections between system dynamics, variance, and sensitivity.

Main Methods:

• Constructing the full trajectory probability manifold.
• Identifying a globally orthogonal coordinate system defined by transition rates.
• Deriving a diagonal Fisher information metric and a Cramér-Rao-type inequality.

Main Results:

• A diagonal Fisher information metric enabling explicit high-dimensional calculations.
• A Cramér-Rao-type inequality bounding the linear response of nonstationary observables.
• A universal nonperturbative (nonlinear) response inequality derived from manifold geometry.

Conclusions:

• The geometric framework generalizes response theory for nonstationary processes.
• Deep connections between dynamical activity, observable variance, and system sensitivity are revealed.
• The approach offers design principles for responsive behaviors in far-from-equilibrium systems.