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Euler fluid in two dimensions: Statistical approach.

Calvin A F Farias1, Renato Pakter1, Yan Levin1

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This study challenges the maximum entropy principle for 2D fluid relaxation. A new core-halo distribution model accurately predicts fluid equilibrium states, differing from previous mixing assumptions.

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Area of Science:

  • Fluid dynamics
  • Statistical mechanics
  • Computational physics

Background:

  • The equilibrium state of 2D incompressible fluids is often assumed to follow Lynden-Bell's theory of collisionless relaxation, which posits maximum Boltzmann entropy under conserved Casimir invariants.
  • Kirchhoff's vortex formulation describes 2D fluid dynamics using Hamiltonian mechanics, where vortex positions act as conjugate variables.

Purpose of the Study:

  • To investigate the equilibrium state of 2D incompressible fluids from arbitrary initial conditions.
  • To test the validity of the maximum entropy principle in collisionless fluid relaxation.
  • To develop an accurate predictive model for fluid equilibrium states.

Main Methods:

  • Utilizing Kirchhoff's vortex formulation of 2D Euler fluid equations.
  • Implementing a Monte Carlo method to find maximum entropy states.
  • Performing molecular dynamics simulations for comparison.
  • Developing a novel core-halo distribution model.

Main Results:

  • The Monte Carlo method, based on maximum entropy, failed to predict the observed fluid equilibrium state.
  • Molecular dynamics simulations showed that the final fluid state significantly deviates from the maximum entropy prediction.
  • The assumption of mixing dynamics in fluid relaxation was found to be incorrect.
  • The proposed core-halo distribution model accurately predicts the fluid's final equilibrium state.

Conclusions:

  • The maximum entropy principle, under the assumption of mixing, is not a correct predictor for the equilibrium states of 2D incompressible fluids.
  • A new core-halo distribution approach provides an accurate method for determining fluid relaxation outcomes from arbitrary initial conditions.
  • This work necessitates a re-evaluation of theoretical models for collisionless fluid dynamics.