History of Area Law in Turbulence

The Loop Equation and the Area Law

We derived a functional equation for the so-called loop average or Wilson loop in turbulence in the early nineties. All references to our previous works can be found in a recent review paper [10].

Loop Equation
The history of the loop equation in turbulence

The path to an exact solution by a dimensional reduction in this equation was proposed in the 1993 paper but has just been explored in the last two years.

Recent Developments

The situation has changed over the last decades. No alternative microscopic theory based on the Navier-Stokes equation emerged, but our understanding of strong turbulence phenomena grew significantly.

The latest DNS studies [5] and other numerical experiments revealed and quantified violations of K41 scaling laws. These numerical results align with multifractal scaling laws [12].

Verification of the Area Law

Theoretically, we studied the loop equation in the confinement region and justified the Area Law suggested in '93. This law states that the tails of velocity circulation PDF in the confinement region are functions of the minimal area inside this loop.

It was verified in DNS a few years ago [5], which triggered the further development of the geometric theory of turbulence [2], [10].

References

  1. Peter D. Anderson and Martin Kruczenski, Loop equations and bootstrap methods in the lattice, Nuclear Physics B, 2017.
  2. Gabriel Apolinario, Luca Moriconi, Rodrigo Pereira, and Victor Valadão, Vortex gas modeling of turbulent circulation statistics, PHYSICAL REVIEW E, 2020.
  3. Abhay Ashtekar, New variables for classical and quantum gravity, Physical Review Letters, 1986.
  4. Kartik P Iyer, Sachin S Bharadwaj, and Katepalli R Sreenivasan, The area rule for circulation in three-dimensional turbulence, PNAS, 2021.
  5. Kartik P. Iyer, Katepalli R. Sreenivasan, and P. K. Yeung, Circulation in high Reynolds number isotropic turbulence is a bifractal, Phys. Rev. X, 2019.
  6. Vladimir Kazakov and Zechuan Zheng, Bootstrap for lattice Yang-Mills theory, Phys. Rev. D, 2023.
  7. Vladimir Kazakov and Zechuan Zheng, Bootstrap for finite n lattice Yang-Mills theory, 2024.
  8. A.A. Migdal, Momentum loop dynamics and random surfaces in QCD, Nuclear Physics B, 1986.
  9. A.A. Migdal, Second quantization of the Wilson loop, Nuclear Physics B - Proceedings Supplements, 1995.
  10. Alexander Migdal, Statistical equilibrium of circulating fluids, Physics Reports, 2023.
  11. Nicolás P. Müller, Juan Ignacio Polanco, and Giorgio Krstulovic, Intermittency of velocity circulation in quantum turbulence, Phys. Rev. X, 2021.
  12. G Parisi and U Frisch, On the singularity structure of fully developed turbulence, 1985.
  13. Carlo Rovelli and Lee Smolin, Knot theory and quantum gravity, Phys. Rev. Lett., 1988.