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This study explores fluid dynamics by perturbing a two-dimensional flow out of equilibrium. Numerical and rigorous proofs confirm that the work done during this process satisfies the Crooks relation, a key finding in statistical mechanics.

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Area of Science:

  • Physics of Fluids
  • Statistical Mechanics
  • Computational Physics

Background:

  • Two-dimensional inviscid and incompressible flows are fundamental in fluid dynamics.
  • Dynamical equilibrium in such systems can be characterized by conserved quantities like energy and enstrophy.
  • Understanding transitions out of equilibrium is crucial for statistical mechanics.

Purpose of the Study:

  • To investigate the statistical behavior of a two-dimensional inviscid flow perturbed from dynamical equilibrium.
  • To examine the work done during a domain deformation process and its statistical properties.
  • To verify the applicability of the Crooks relation to this specific fluid dynamics system.

Main Methods:

  • Utilized the two-dimensional vorticity equation with spectral truncation on a rectangular domain.
  • Perturbed the system from equilibrium by altering the domain shape, conserving enstrophy while changing kinetic energy.
  • Performed numerical simulations of forward and backward processes to analyze work distributions.

Main Results:

  • Numerical evidence demonstrated that the work done during the perturbation satisfies the Crooks relation.
  • The system's equilibrium statistics were accurately described by a canonical ensemble.
  • The change in domain shape was identified as a method for doing work on the system.

Conclusions:

  • The Crooks relation holds for the studied two-dimensional inviscid flow system.
  • Rigorous mathematical proof confirmed the numerical findings regarding the Crooks relation.
  • This work provides a bridge between fluid dynamics and non-equilibrium statistical mechanics.