Title: Exact Solution Of Decaying Turbulence
Abstract: We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This solution family is equivalent to a fractal curve in complex space $\mathbb C^d$ with random steps parametrized by $N$ Ising variables $\sigma_i=\pm 1$, in addition to a rational number $\frac{p}{q}$ and an integer winding number $r$, related by $\sum \sigma_i = q r$. This equivalence provides a \textbf{dual} theory describing a strong turbulent phase of the Navier-Stokes flow in $\mathbb R_d$ space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. This is a \textbf{quantum} statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size $N \rightarrow \infty$ or its chemical potential $\mu \rightarrow 0$. The system with fixed $N$ has different asymptotics at odd and even $N\rightarrow \infty$, but the limit $\mu \rightarrow 0$ is well defined. The energy dissipation rate is analytically calculated as a function of $\mu$ using methods of number theory. It grows as $\nu/\mu^2$ in the continuum limit $\mu \rightarrow 0$, leading to anomalous dissipation at $\mu \propto \sqrt{\nu} \rightarrow 0$. The same method applies to other observables.
About the speaker: Alexander A. Midgal is a Research Professor of Physics at the New York University. He received his Ph.D. in 1969 from Landau University. He has significantly contributed to the theory of critical phenomena, quantum chromodynamics, and conformal field theory. With his colleague Alexander Polyakov, he independently worked out the theory of the dynamical mass generation in gauge theories, commonly known as the Higgs mechanism. He discovered the conformal bootstrap, which had a profound influence and helped progress in the theory of critical phenomena and strong interactions. Migdal made important progress in quantum chromodynamics by introducing a novel form of the renormalization group with Yu. Makeenko, he derived the equation for the Wilson loops. Along with Volodya Kazakov, he provided an exact solution of 2D Quantum Gravity. During 1993-1995 he initiated nonperturbative approach to Turbulence by discovering the loop equations for fluid dynamics. The area law predicted by him from this equation was verified in 2019 in numerical simulations by Sreenivasan. In 1996, he left academia to become an inventor and entrepreneur, where he directed several companies and founded four, the latest being Migdal Research LLC, which is dedicated to stock market prediction. He has recently moved back to academia and studies various aspects of turbulence.
This lecture is part of the program "Field Theory and Turbulence".