Loop functional and its general properties

The loop functional is defined as a phase factor associated with velocity circulation, averaged over the initial distribution $\vec{v}_0$ of the velocity field

We use viscosity $\nu$ as a unit of circulation. Both have the same dimension $L^2/T$ as the Planck's constant $\hbar$. The viscosity will play the same role in our theory as Planck's constant in quantum mechanics. The variable $\gamma$ with this definition is dimensionless.

This loop functional is the Fourier transform of the PDF for the circulation over fixed loop $C$

There is an implicit dependence of time, coming from the evolution of the velocity field by the $\NS$ equation

We restrict ourselves to three-dimensional Euclidean space, the most interesting case for physics applications. The generalization to arbitrary dimension is straightforward, as discussed in previous papers [1,2,3].

Loop equation

The first step is to write down the loop equation by projecting the Hopf equation to the loop space.

Before doing that, we have to specify certain boundary conditions which we assume in our fluid dynamics. Namely, we consider infinite space $\mathbb{R}_3$, with boundary condition of vanishing or constant velocity at infinity.

Vorticity can exist throughout the entire spatial domain, but it vanishes at infinity, as required by our boundary conditions, where velocity gradients diminish. We also assume the absence of internal boundaries, such as solid surfaces that the fluid flows around. However, we allow for the presence of singular regions of vorticity, such as vortex lines and sheets, in the inviscid limit, provided these regions are confined to a finite portion of the volume, consistent with our boundary conditions.

At finite viscosity, these singular regions acquire a finite thickness proportional to $\sqrt{\nu}$, leading to anomalous dissipation in turbulent flows.

An alternative, simpler mechanism for anomalous dissipation arises when velocity increases as viscosity vanishes, even in the absence of singular vortex structures like Burgers vortices. In this scenario, the turbulent energy grows as viscosity decreases, while the spatial distribution of vorticity remains homogeneous and non-singular.

As we will demonstrate below, this is the mechanism realized in our solution for decaying turbulence.

The computations leading to the loop equation were performed in the old papers [1,2]. For the reader's convenience, we repeat them here using another language, hopefully more clear for mathematicians.

The straightforward time derivative of the loop functional, assuming the constant loop $C$ and using time derivative of the velocity field in the circulation, yields

The averaging $\VEV{}_{sol}$ goes, as before, over time-dependent NS solutions $\vec{v}(\vec{r})$ with a given set of initial values $\vec{v}_0(\vec{r})$. It is implied that a probability measure (see examples below) is supplied for this set of initial velocity fields. The phase factor of circulation is averaged over initial data using this measure.

The gradient terms $\vec{\nabla} \left( p + \frac{\vec{v}^2}{2} \right)$ dropped in the time derivative of the circulation as the integral of a gradient of some single-valued function of coordinate $H(\vec{r}) = p(\vec{r}) + \frac{\vec{v}^2(\vec{r})}{2}$ around the closed loop:

The velocity field $\vec{v}$ is a solution of the Poisson equation, relating it to vorticity by incompressibility condition

This representation leaves vorticity as the main unknown variable in the time derivative of the loop functional.

To find the loop equation, we must replace the vorticity and its gradients with certain operators acting on the loop independently of the vorticity and velocity fields. As a result of such transformation, the vector function $\vec{L}(\vec{C}(\theta))$ will be replaced by a certain operator $\hat{L}(\theta)$ in loop space acting on $\Psi(\gamma,\vec{C})$

This operator $\hat{L}(\theta)$ depends on certain operators $\hat{\nabla}(\theta), \hat{\omega}(\theta), \hat{v}(\theta)$ instead of on the dynamical variables $\vec{v}(\vec{r}), \vec{\omega}(\vec{r})$, therefore it can be taken out of the averaging over trajectories starting from various initial data $\vec{v}_0(\vec{r})$ so that this operator acts on the loop average $\Psi(\gamma,\vec{C})$. Such is the plan of the proof of the loop equation. We define the loop operators and follow this plan in the next section.

The definitions of the loop operators and the proof of the loop equation

The operators in the loop equation were introduced in [1] and explained at length in my review paper [2].

In this paper, we do not assume any knowledge of the previous work; instead, we derive the loop operators from scratch using a simpler method.

First, we approximate the smooth loop $C(\theta)$ by a polygon with $N$ vertices $\vec{C}_k = \vec{C}(2\pi k/N)$ in the limit $N \to \8$. We postpone this local limit until we solve the discrete loop equation. This limit will define the continuum theory in the same way as in the QFT; the functional integral is discretized using a lattice with the lattice spacing going to zero at the end of the calculation. In this limit, the theory's parameters vary with the lattice spacing to provide a finite result for the physical observables.

The first observation is that with smooth velocity and vorticity fields, the discrete circulation around the polygon $\vec{C}_k, k = 0,\dots N-1, \vec{C}_N = \vec{C}_0$ converges to the circulation around the smooth loop

The finite difference becomes derivative for the smooth loop; the error will vanish as $\mathcal{O}(1/N)$.

The next property is also easy to prove using the Stokes theorem for a small triangle $\left(\vec{C}_{k-1}, \vec{C}_k, \vec{C}_{k+1}\right)$

This first derivative vanishes at $N \to \8$ as $\Delta \vec{C}_k \sim \mathcal{O}(1/N)$.

The second derivative, however, stays finite. We prefer to use another set of variables

The last relation follows from the chain rule

The vorticity can be represented as

The contour $C$ becomes an open line when we move all $\vec{s}_k$ independently, without restricting $\sum \vec{s}_k = 0$. However, the contribution to the time derivative of circulation from the extra gap between the endpoints $\Delta \partial_t \Gamma \propto H(\vec{C}_N) - H(\vec{C}_0)$ where $H(\vec{r}) = p(\vec{r}) + \frac{\vec{v}^2(\vec{r})}{2}$ is the enthalpy, which is supposed to be differentiable. Thus, this error term goes to zero as we reinstate the closure condition $\sum \vec{s}_k = \vec{C}_N - \vec{C}_0 = 0$.

Finally, the velocity field at the vertex $\vec{v}(\vec{C}_k)$ can be related to the vorticity through the Biot-Savart law

Let us verify this relation using the Biot-Savart integral formula for the inverse Laplace operator

At first glance, the loop in the new circulation $\tilde{\Gamma}(\vec{r})$ involves two long "wires": $(\vec{C}_{k-1}, \vec{C}_k + \vec{r})$ and $(\vec{C}_k + \vec{r}, \vec{C}_{k+1})$.

However, in the local limit, when the distance $|\vec{C}_{k+1}- \vec{C}_{k-1}| = \mathcal{O}(1/N)$, these two wires have zero area inside the arising thin triangle, so they effectively cancel in virtue of the Stokes theorem, assuming the Biot-Savart integral converges.

This produces the desired result in the Biot-Savart formula.

The convergence of the Biot-Savart integral follows from our boundary conditions, assuming no vorticity at infinity or even stronger requirement of finite support of vorticity. The phase factor $\exp{\I \tilde{\Gamma}(r)}$ does not influence the absolute convergence, so it can be set to $\exp{\frac{\I \gamma\Gamma}{\nu}}$ for that purpose and taken out of the integral, returning us to the convergence of the ordinary Biot-Savart integral.

Therefore, with $\mathcal{O}(1/N)$ accuracy, we can replace the right side by its discrete version with operators involving $\vec{\nabla}_k$

We restrict ourselves to the velocity vanishing at infinity and no internal boundaries in the physical domain. With this boundary condition, the harmonic potential is zero, and there is no zero mode to add to the inverse Laplace operator.

In the rest of the paper, we shall use the language of the continuum theory, implying the limit $N\to \8$ of a polygon $\vec{C}$ with $N$ sides. While the lengths of the sides of $\vec{C}$ vanish in the local limit $N \to \8$, the sides of $\vec{P}$ polygon are not at our disposal, so they may stay finite (this will happen in the decaying turbulence below).

Schrödinger equation in loop space

Before we investigate the solutions of the loop equation, let us consider its physical and mathematical meaning and its relation to the geometry of the incompressible flow.

By definition, the loop functional $\Psi(\gamma,\vec{C})$ is a superposition of the phase factors $\exp{\frac{\I \gamma}{\nu}\Gamma(\vec{v}, \vec{C})}$ with the circulation $\Gamma$ of a particular solution $\vec{v}(\vec{r}, t)$ of the $\NS$ equation. These solutions have initial values $\vec{v}(\vec{r}, 0) = \vec{v}_0(\vec{r})$, distributed by some distribution $P[\vec{v}]$ which we assume Gaussian with the mean given by some smooth initial field and some coordinate-independent variance $\sigma$.

In the turbulent scenario, the $\NS$ trajectories initiated from a narrow vicinity of some smooth velocity field eventually expand and cover some attractor, slowly varying with time and asymptotically converging to $\vec{v} = 0, \Psi = 1$.

The alternative smooth solution of the $\NS$ equation, sought after in numerous mathematical papers, would correspond to these trajectories staying close and converging to a single trajectory in the limit $\sigma \to 0$. This single trajectory would go along the unit circle, bounding our disk.

With this generalization of a definition of the Cauchy problem for the $\NS$ equation, we can address the existence of smooth, explosive, or stochastic (i.e., turbulent) solutions within the loop equation's framework.

The transformation from the $\NS$ equation to the loop equation is similar to that from the Newton equation of the particle in random media to the diffusion equation. We add dimension to the problem, switching to the probability distribution in $\mathbb{R}_d$, after which the particle's infinitesimal time steps translate into probability derivatives by coordinates.

There are two essential differences, however. Our loop space is not just higher-dimensional; in the local limit $N \to \8$, it is infinite-dimensional. The second difference is that in addition to diffusion terms $\nu \hat{\nabla} \times \hat{\omega}$, we have nonlocal advection terms $\hat{v} \times \hat{\omega}$ affecting the evolution of the distribution in loop space.

Our definition of the loop functional already by construction has superficial similarities with quantum mechanics. We are summing phase factor over a manifold of solutions of the $\NS$ equations. The circulation plays the role of classical Action, and viscosity plays the role of Planck's constant.

This analogy becomes a complete equivalence when the time derivative of the loop functional is represented as an operator $\vec{L}(\vec{C}(\theta)) \Rightarrow \hat{L}(\theta)$ in the loop space acting on this functional.

Now we have quantum mechanics in loop space, with the Hamiltonian

The operator $\hat{L}(\theta)$ depends of functional derivatives $\delta/\delta\vec{C}(\theta)$, as was determined, and discussed in previous works [1,2,3]. Our polygonal approximation has no functional derivatives, just ordinary derivatives $\vec{\nabla}_k = \partial/\partial\vec{C}_k, \vec{\eta}_k = \partial/\partial\Delta\vec{C}_k$. Thus, our quantum-mechanical system has $3N$ continuum degrees of freedom $\vec{C}_1, \dots \vec{C}_N$ with periodicity constraint $\vec{C}_0 = \vec{C}_N$.

This Hamiltonian is not Hermitian, which reflects the dissipation phenomena. The time reversal leads to complex conjugation of the loop functional, a nontrivial transformation, as there is no symmetry for the reflection of velocity field $\vec{v}(\vec{r},t) \to -\vec{v}(\vec{r}, t)$.

The loop in our theory is a periodic function of the angular variable $\theta$. Geometrically, this is a map of the unit circle into Euclidean space $\mathbb{S}_1 \mapsto \mathbb{R}_d$. In particular, there could be several smaller periods, in which case this loop becomes a set of several closed loops connected by backtracking wires like in Figure.

Also, this map could have an arbitrary winding number $n$ corresponding to the same geometric loop in $\mathbb{R}_d$ traversed $n$ times.

The linearity of the loop equation is the most important property of this transformation from $\NS$ equation to the quantum mechanics in loop space.

This transformation exemplifies how the nonlinear PDE reduces to the linear problem projected from high dimensional space. In our case, this space is the loop space, which is infinite-dimensional.

As a consequence of linearity, the generic solution of the loop equation is a superposition of particular solutions with various parameters. More generally, this is an integral (or sum, in discrete case) over the space $\mathcal{S}$ of solutions of the loop equation.

In the case of the Cauchy problem in loop space, the measure for this integration over space $\mathcal{S}$ is determined by the initial distribution of the velocity field. The asymptotic turbulent solution [3] uniformly covers the Euler ensemble, like the microcanonical distribution in Newton's mechanics covers the energy surface.

This turbulent solution does not solve a Cauchy problem; it rather solves the loop equation with the boundary condition at infinite time $\Psi_{t= \8} = 1$.

In the next section, we simplify the loop equation using Fourier space; this will be the foundation for the subsequent analysis.

Momentum Loop Equation

The loop operator, $\hat{L}$ in equation (LoopEq), dramatically simplifies in the functional Fourier space, which we call momentum loop space. In our discrete approximation, the momentum loop will also be a polygon with $N$ sides.

The origin of this simplification is the lack of direct dependence of the loop operator $\hat{L}(\theta)$ on the loop $C$ itself. Only derivatives $\vec{\nabla}_k, \vec{\eta}_k$ enter this operator.

From the point of view of quantum mechanics in loop space, our Hamiltonian only depends on the canonical momenta but not on the canonical coordinates. This property is exact as long as we do not add external forces.

This remarkable symmetry property (translational invariance in loop space) allows us to look for the "superposition of plane waves" Ansatz:

Here the averaging $\VEV{\dots}_{init}$ goes over all trajectories $P_{k}(t)$ passing through random initial data $\vec{P}_k(0)$ distributed with the corresponding probability to reproduce initial value $\Psi(\gamma,C|0)$. We discuss this initial distribution in the next sections.

The operators $\vec{\nabla}_k, \vec{\eta}_k$ become ordinary vectors when applied to $\psi_p$:

The velocity circulation can be rewritten up to $\mathcal{O}(1/N)$ corrections as a symmetric sum

We did not assume here anything about the continuity of $\vec{P}_k$; we only assumed that $|\Delta C_{k+1} - \Delta \vec{C}_k| \ll |\Delta \vec{C}_k|$ which is true for smooth loop.

The discrete loop equation with our Ansatz after some algebraic transformations using the above identities reduces to the following momentum loop equation (MLE) [1,2]:

In the local limit $N \to \8$, the momentum loop will have a discontinuity $\Delta \vec{P}(\theta)$ at every parameter $0 < \theta \le 2\pi$, making it a fractal curve in complex space $\mathbb{C}_d$. Such a curve can only be defined using a limit like a polygonal approximation.

You can regard this curve as a periodic random process hopping around the circle (more about this process below, in the context of the decaying turbulence).

The details can be found in [1,2]. We will skip the arguments $t, k$ in these loop equations, as there is no explicit dependence of these equations on either of these parameters.