Publications (34)
Decaying Turbulence and the Riemann Hypothesis: The number theory behind the infinite-time singularity
Alexander Migdal
We derive a formal statistical solution of freely decaying incompressible turbulence in arbitrary dimension \(d>1\) using Navier--Stokes loop equations. The loop Fourier transform…
Scaling Index In Turbulent Area Law
Alexander Migdal
We analyze the Minimal Area solution to the Loop Equations in turbulence \cite{M93}. As it follows from the new derivation in the recent paper \cite{M19}, the vorticity is represen…
Confined Vortex Surface and Irreversibility. 2. Hyperbolic Sheets and Turbulent statistics
Alexander Migdal
We continue the study of Confined Vortex Surfaces (\CVS{}) that we introduced in the previous paper. We classify the solutions of the \CVS{} equation and find the analytical formul…
Geometric QCD III: Exact transition amplitudes and the glueball spectrum
Alexander Migdal
We complete the analysis of planar Makeenko--Migdal loop equations in the Lorentz-invariant continuum limit. Using our confining twistor-string representation, we compute the quant…
Duality of Navier-Stokes to a one-dimensional system
Alexander Migdal
The Navier--Stokes (NS) equations describe fluid dynamics through a high-dimensional, nonlinear system of partial differential equations (PDEs). Despite their fundamental importanc…
Integral Equation for CFT/String Duality
Alexander Migdal
We reinterpret and extend some old work on CFT/string duality. We consider some asymptotically conformal field theory in large N limit, with conformal symmetry broken by VEV's of i…
Geometric QCD I: The Hodge-Dual Surface and Quark Confinement
Alexander Migdal
This is the first of two papers presenting a geometric framework for Planar QCD (). In this part, we establish the kinematic foundation of the theory by constructin…
Quantum Solution of Classical Turbulence. Decaying Energy Spectrum
Alexander Migdal
This paper presents a recent advancement that transforms the problem of decaying turbulence in the Navier-Stokes equations in dimensions into a Number Theory challenge: findi…
Hidden Symmetries of Large N QCD
Alexander Migdal
The local SUSY symmetry of the loop dynamics of QCD is found. The remarkable thing is, there is no einbein-gravitino on this theory, which makes it a 1D topological supergravity, o…
Geometric QCD II: The Confining Twistor String and Meson Spectrum
Alexander Migdal
We present a local, asymptotically free solution of the planar Makeenko--Migdal loop equations in the continuum limit with full Lorentz invariance. The solution is constructed by q…
Asymmetric Vortex Sheet
Alexander Migdal
We present a steady analytical solution of the incompressible Navier-Stokes equation for arbitrary viscosity in an arbitrary dimension of space. It represents a dimension…
Dual theory of decaying turbulence. 2. Numerical simulation and continuum limit
Alexander Migdal
This is the second paper in a cycle investigating the exact solution of loop equations in decaying turbulence. We perform numerical simulations of the Euler ensemble, suggested in…
Superloop Equations and Minimal Surfaces I: Confining minimal surface in SYM
Alexander Migdal
We formulate the loop equations of pure super Yang--Mills (SYM) theory in a finite geometric form. The usual equal-point loop derivatives are singular because the order o…
Vortex Sheet Turbulence as Solvable String Theory
Alexander Migdal
We study steady vortex sheet solutions of the Navier-Stokes in the limit of vanishing viscosity at fixed energy flow. We refer to this as the turbulent limit. These steady flows co…
Ground State of 2D Quantum Gravity and Spectral Density of Random Matrices
Marek Karliner, Alexander Migdal, Boris Rusakov
We compute the exact spectral density of random matrices in the ground state of the quantum hamiltonian corresponding to the matrix model whose double scaling limit describes pure…
Dual Theory of MHD Turbulence
Alexander Migdal
We present an exact analytic solution for decaying incompressible magnetohydrodynamic (MHD) turbulence. Our solution reveals a dual formulation in terms of two interacting Euler en…
To the theory of decaying turbulence
Alexander Migdal
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 . This…
Geometric Solution of Turbulent Mixing
Alexander Migdal
We derive an analytical solution for the one-point distribution of a passive scalar in decaying homogeneous turbulence, in the limit of strong turbulence (high Re, fixed Schmidt nu…
Topological vortexes, asymptotic freedom, and multifractals
Alexander Migdal
We study the Kelvinons: monopole ring solutions to the Euler equations, regularized as the Burgers vortex in the viscous core. There is finite anomalous dissipation in the inviscid…
Meromorphization of Large N QFT
Alexander Migdal
We find general relations between RG equations and planar unitarity-analyticity. These relations are summarized in meromorphization procedure, generalizing the Padé approximation…
Exact Confining Solution of the Planar QCD Loop Equation via a Matrix Ensemble
Alexander Migdal
We construct an exact solution to the planar QCD loop equation in four-dimensional Euclidean space using a novel matrix-valued momentum loop formalism. Central to this construction…
Exact Area Law for Planar Loops in Turbulence in Two and Three Dimensions
Alexander Migdal
We study properties of the minimal surface in the Area Law Solution \cite{M93}, \cite{M19a}, \cite{M19b}. We find out that Area Law holds exactly for 2D turbulence as well as for a…
Probability Distribution of Velocity Circulation in Three Dimensional Turbulence
Alexander Migdal
We elaborate the statistical field theory of Turbulence suggested in the previous paper \cite{M20a}. We clarify and simplify the basic Energy pumping equation of that theory and st…
Analytic and Numerical Study of Navier-Stokes Loop Equation in Turbulence
Alexander Migdal
We developed analytic approach to the non-planar loop equation, which we derived in previous papers \cite{M19a},\cite{M19b},\cite{M19c}. We found quadratic integral equation for th…
Confined Vortex Surface and Irreversibility. 3. Nested Tubes and Energy Cascade
Alexander Migdal
We find a new family of exact solutions of the Confined Vortex Surface equations (The Euler equations with extra boundary conditions coming from the stability of the Navier-Stokes…
Statistical Equilibrium of Circulating Fluids
Alexander Migdal
We are investigating the inviscid limit of the Navier-Stokes equation, and we find previously unknown anomalous terms in Hamiltonian, Dissipation, and Helicity, which survive this…
Towards Field Theory of Turbulence
Alexander Migdal
We revisit the problem of stationary distribution of vorticity in three-dimensional turbulence. Using Clebsch variables we construct an explicit invariant measure on stationary sol…
Euler Ensemble as Decaying Turbulence Attractor: Universality, Stability and Parity Classes
Alexander Migdal
The paper analyzes the local Lyapunov stability of self‑similar Euler ensemble solutions in freely decaying Navier‑Stokes turbulence, showing that only the odd‑N planar sector is a…
Spontaneous quantization of the Yang--Mills gradient flow
Alexander Migdal
We formulate a nonsingular loop-space calculus for Yang-Mills (YM) gradient flow directly in terms of Wilson loops. Variations act within the manifold of smooth loops via finite, r…
Confined Vortex Surface and Irreversibility. 1. Properties of Exact solution
Alexander Migdal
We revise the steady vortex surface theory following the recent finding of asymmetric vortex sheets (AM,2021). These surfaces avoid the Kelvin-Helmholtz instability by adjusting th…
Universal Area Law in Turbulence
Alexander Migdal
We re-visit the Area Law in Turbulence discovered many years ago \cite{M93} and verified recently in numerical experiments\cite{S19}. We derive this law in a simpler way, at the sa…
Clebsch Confinement and Instantons in Turbulence
Alexander Migdal
We introduce a concept of Clebsch confinement related to unbroken gauge invariance and study Clebsch instantons: singular vorticity sheets with nontrivial helicity. This is realiza…
Geometric Solution of Turbulence as Diffusion in Loop Space
Alexander Migdal
Strongly nonlinear dynamics, from fluid turbulence to quantum chromodynamics, have long constituted some of the most challenging problems in theoretical physics. This review descri…
Turbulence, String Theory and Ising Model
Alexander Migdal
We advance the vortex cell approach to turbulence \cite{TSVS} by elaborating the Clebsch field dynamics on the surface of vortex cells. We argue that resulting statistical system c…