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Optimal protocols for driving nonequilibrium systems are geodesics of the friction tensor. This study links thermodynamic geometry to optimal transport, enabling computation of exact protocols beyond the slow-driving limit.

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Area of Science:

  • Thermodynamics
  • Statistical Mechanics
  • Non-equilibrium Systems

Background:

  • Thermodynamic geometry provides a framework for understanding the geometry of thermodynamic state spaces.
  • Optimal control protocols in the slow-driving limit are known to follow geodesics of the friction tensor.
  • Extending these optimal protocols beyond the linear response regime remains a significant challenge.

Purpose of the Study:

  • To demonstrate the equivalence between thermodynamic geometry and L^2 optimal transport geometry for overdamped dynamics.
  • To develop a computationally tractable method for obtaining optimal protocols beyond the slow-driving limit.
  • To explain observed phenomena in optimal protocols, such as nonmonotonic behavior and discontinuous jumps.

Main Methods:

  • Equating thermodynamic geometry with L^2 optimal transport geometry for overdamped systems.
  • Formulating optimal protocols as a sum of friction tensor geodesics and a counterdiabatic term.
  • Utilizing the Fisher information metric to define the counterdiabatic term.

Main Results:

  • Thermodynamic geometry is shown to be equivalent to L^2 optimal transport geometry for overdamped dynamics.
  • A method is presented for computing optimal protocols beyond the slow-driving limit, involving geodesic paths and a counterdiabatic correction.
  • The derived geodesic-counterdiabatic protocols accurately describe parametric harmonic potentials and reproduce complex behaviors in double-well systems.

Conclusions:

  • The connection between thermodynamic geometry and optimal transport provides a powerful tool for analyzing non-equilibrium systems.
  • The developed method offers a computationally feasible approach to finding optimal driving protocols in various regimes.
  • This work elucidates the origins of nonmonotonicity and abrupt changes in optimal control protocols.