Hierarchical Structure of Quantum Solution

This is a hierarchical map of my recent paper devoted to a fundamental problem of decaying hydrodynamic turbulence. Using a novel approach to fluid dynamics (loop equations) initiated by us in the 90s, we have recently found an exact solution to this problem in quadrature.

In this paper, we go step by step through this solution, starting from the loop equation itself. As is often the case with a microscopic solution to an empirical field (like in chemistry or astronomy), it presents some technical challenges related to new mathematical tools, specifically loop calculus, quantum mechanics, and number theory.

The turbulence problem has stood unsolved for centuries for a good reason — a direct approach is virtually impossible, as the vorticity in a turbulent fluid does not exist as a smooth vector field. The breakthrough described here is based on a new concept of duality, where the problem of strong fluctuations of the vector field can be exactly reduced to a dual problem of a weakly fluctuating field in another dimension.

This novelty necessarily takes some space to explain and time to understand. I have tried to make this paper easier to consume by introducing four levels of hierarchy, reflected in this interactive map. This map is intended to be browsed before a deep study of the paper (or perhaps as part of that study), allowing for back-and-forth navigation between the conventional PDF manuscript and this hierarchical map.

It may also be helpful for the reader to start with the slides of my recent talk at the CERN conference in Crete.

I. Prologue
- Complexity of vortex structures $\leftrightarrow$ simplicity of statistical theory.
- Statistical theory of turbulence $=$ fixed trajectory of the loop equation
  
  Hopf Equation (1952)
  
  This is a general equation for generating functional for stochastic Navier-Stokes velocity field.
  
  QCD loop equation (1983)
  
  Equation for Wilson loop (parallel transport factor) in the Gauge theory.
  
  Navier-Stokes loop equation (1993)
  
  Using the loop calculus from gauge theory, we applied that to the Navier-Stokes equation.
  
  Momentum loops (1995)
  
  Using the functional Fourier transform, we discovered the reduction of the loop equation to the one-dimensional singular PDE (momentum loop equation).
  
  Euler ensemble (2022)
  
  We found an exact solution of the momentum loop equation: an ensemble or $\mathbb{Z}_{2}$ numbers coupled to a random fraction (Euler ensemble).
  
  Fermi ring (2023)
  
  We used a representation of the average over Euler ensemble as a periodic Markov chain to derive an exact transformation of this ensemble as the set of Fermi particles on a ring.
  
  Instanton (2023)
  
  The turbulent limit of this Fermi ring corresponding to vanishing viscosity in Navier-Stokes equation is reduced to a single classical trajectory (instanton) plus quadratic fluctuations in its vicinity.
  
  Exact solution (2024)
  
  The remaining tedious technical work of computing the instanton and the functional determinant for the quadratic fluctuations in the turbulent limit led to analytic expression for the energy spectrum of decaying turbulence, matching experimental data and the DNS.
II. Definitions and notations
III. Summary.
- All the basic results summarized
  
  Log-log plot of $1/Energy(t)$ . Theory predicts slope $5/4=1.25$ in perfect agreement with experiments.
  
  The effective index (log-log derivative) for the energy spectrum as a function of the scaling variable $k\sqrt{t}$
  
  The deviation of our theory from the K41 ( $-5/3$ ) law for the energy spectrum in a log-log scale (acceptable match with DNS).
  
  The effective index $n(t)=-t\partial_{t}\log E(t)$ for the energy decay
IV. Introduction.
- The hierarchical structure of the paper and its messages for various readers
- Physical Introduction
  
  The Energy Flow and Random Vorticity Structures
- Mathematical Introduction.
  
  The loop equation and its solution
  
  Appendix A. Global Random rotation and Momentum Loop Space. An instructive example of the exact fixed point of the loop equation
  
  The big and small Euler ensembles.
  
  Appendix E. Small Euler ensemble in statistical limit. New formulas in the Number Theory: statistical distribution of fractions with large denominators.
V. The Fermi ring and its continuum limit.
- Deeper analysis of the mathematical structure of the solution
- Appendix B. The Markov chain and its Fermionic representation. Exact mathematical transformation of the Markov chain to the Fermionic trace.
- Appendix C. Path integral over Markov histories. From discrete Markov chain to the path integral using generating functions
- Appendix D. Matching path integral with combinatorial sums. Verifying the continuum results against exact combinatorial computations
VI. Instanton in the path integral.
- Appendix F. The velocity correlation in Fourier space. Transformation of the backtracking loop functional to the two-point vorticity correlation.
- All-in-depth technical calculations.
  
  Appendix G. Turbulent viscosity and the local limit. Deep technical details of the instanton calculation. Eigenvalues of the quadratic functional of fluctuations.
  
  Appendix H. Linearized classical trajectory. The harmonic fluctuations around instanton in the turbulent limit.
  
  Appendix I. Functional determinant in the path integral. The beautiful part. The zeta regularization of functional determinant and exact topological formula for the critical index
  
  Appendix J. The fluctuation term in $\alpha^{\prime}(\xi_{1})\alpha^{\prime}(\xi_{2})$ . Adding a term into the instanton contribution
VII. The decaying energy in a finite system.
- The energy spectrum in a finite system, with energy spectrum matched by k2 law at small k
- Computation of the energy dissipation
  
  We elaborate on the internal representation for the correlation function in the WKB limit for the Fermionic ring
- The energy normalization problem,
  
  The limit $\nu\to 0$ can only be taken in the universal functions depending on the circulation ratio to viscosity.
VIII. Comparing our theory with the DNS.
- The moment of truth: how does our theory compare with DNS?
- Digging in the archives from the DNS simulation for decaying turbulence.
- Selecting the turbulent interval of time where the equilibrium is already reached but the decaying Reynolds number is still large.
- Perfect match between theory and DNS for the leading scaling indexes.
- Effective length as a function of time vs theoretical $\sqrt{t}$ .
- Energy as a function of effective length
- universal function for the effective index of the second moment of velocity differences.
IX. The spectrum of scaling dimensions.
- Mathematical definition of scaling laws via Mellin transform and infinite spectrum of scaling indexes.
- Analytical formulas for the spectrum, – the first precedent in critical phenomena above two dimensions.
- Appendix K. Mellin integral for the energy spectrum and energy decay. Putting together instanton contribution to the energy spectrum in a form of Mellin integral.
- A. The energy spectrum and replacement of $k^{-5/3}$ by $k^{-7/2}$ .
- B. The energy decay and $t^{-11/4}$ correction to $t^{-5/4}$
- C. The velocity correlation growth index spectrum.
X. Discussion. This theory contradicts accepted paradigms about energy cascades, scaling laws, etc. Let us discuss it.
- A. Myth and Reality of Turbulent Scaling Laws.
  
  Most of the phenomenological theories of decaying turbulence in the past were myths based on stretching the data and wishful thinking.
- B. Stochastic solution of the Navier-Stokes equation and ergodic hypothesis.
  
  Our theory is an exact stochastic solution of the Navier-Stokes equation. What is the analog of the ergodic hypothesis?
- C. The physical meaning of the loop equation and dimensional reduction.
  
  How is the 3D classical turbulence reduced to a one-dimensional system?
- D. Classical Flow and Quantum Mechanics.
  
  The most important section. Spontaneous quantization of nonlinear classical system
- E. Renormalizability of the inviscid Limit of the Loop Equation.
  
  Analogy with QCD.
  
  How the parameters in the loop equation must be tuned to get a nontrivial continuum limit?
  
  Turbulent viscosity (universal or not?)
- F. Relation of Our Solution to Weak Turbulence.
  
  In the Navier-Stokes equation, Wylde diagrams can represent the terms of the expansion in inverse viscosity powers.
  
  Our solution can be regarded as an exact summation of that expansion with extrapolation to zero viscosity.
XI. Remaining problems.
XII. Conclusion. The big picture of what was accomplished in this work.
- We have unveiled a duality between decaying classical turbulence in 3+1 dimensions and the one-dimensional quantum theory of $N\to\infty$ Fermi particles on a ring.
- The density fluctuations of these Fermi particles disappear in the turbulent limit $\nu\sim 1/N^{2}\to 0$ , leading to the exact WKB (instanton) solution for this density.
- This establishes a new relation between classical nonlinear dynamics and quantum theory: the Fourier transform of classical probability adheres to the QM evolution with the interference of alternative histories.
- The spectrum of the turbulence scaling dimensions is related to the zeros of the Riemann $\zeta$ function .
- The grid-turbulence experiments and recent DNS rule out K41, multifractal, and Heisenberg scaling laws but confirm this theory.
Acknowledgments
Data Availability
References