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Bridge, Reverse Bridge, and Their Control.

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Summary
This summary is machine-generated.

We studied time-reversal symmetry in stochastic process bridges, finding that some non-equilibrium systems can exhibit perfect symmetry even when detailed balance is broken. This reveals new insights into out-of-equilibrium dynamics.

Keywords:
brownian gyratorsreverse diffusionstochastic bridgetime-reversal

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Area of Science:

  • Statistical physics
  • Non-equilibrium thermodynamics
  • Stochastic processes

Background:

  • The bridge problem in stochastic processes analyzes trajectories with fixed start and end points.
  • Time-reversal symmetry is a key property, questioning if statistical properties are invariant under time reversal (t → τ-t).
  • Investigating symmetry in systems violating detailed balance is crucial for understanding non-equilibrium dynamics.

Purpose of the Study:

  • To derive a local condition for time-reversal asymmetry specifically for stochastic bridges.
  • To explore conditions under which bridges can be time-reversal symmetric, even when the underlying process violates detailed balance.
  • To quantify bridge asymmetry and its utility in analyzing out-of-equilibrium dynamics.

Main Methods:

  • Comparison of stochastic differential equations for the bridge and its time-reversed counterpart, derived via Doob's transform or stochastic optimal control.
  • Derivation of a necessary condition for time-reversal symmetry involving current velocity and detailed balance.
  • Analysis of a minimal non-equilibrium model (Brownian Gyrators) and its transformation into polar coordinates.

Main Results:

  • A necessary condition for time-reversal symmetry was derived, confirming symmetry when detailed balance holds.
  • The study found that bridges can exhibit symmetry even if the underlying process breaks detailed balance.
  • An example of partial symmetry was presented in Brownian Gyrators, and perfect symmetry was demonstrated for the squared modulus of a transformed Brownian Gyrator model.

Conclusions:

  • Stochastic bridges can display time-reversal symmetry under conditions where the original process does not satisfy detailed balance.
  • The derived condition provides a framework for identifying and quantifying time-reversal asymmetry in bridges.
  • The findings highlight the nuanced behavior of non-equilibrium systems and offer methods for analyzing their dynamics.