arXiv subjects

Alexei A. Mailybaev

Publications and source records attributed to Alexei A. Mailybaev.

At least 19 recordsLinked to original sources

arXiv2026

Renormalization Group on Wiener Space: Spectral Theory and Universality

We develop a rigorous renormalization group (RG) formalism on Wiener space. We introduce an RG operator $\mathcal R$ acting on probability measures over continuous paths, whose fixed point is the Wiener measure. We linearize $\mathcal{R}$ around the Brownian fixed point and analyze the spectral structure of the resulting operator $\mathscr{L}$. Its eigenvectors are too singular to be realized as honest measures on path space, and we therefore develop a generalized spectral theory within the framework of white noise analysis (Hida calculus). The eigenvectors of $\mathscr{L}$ are realized as Hida distributions, satisfying $\mathscr{L} U=λU$ in the weak sense. This analysis yields structural results such as a spectral gap and a diagonal-concentration property of the eigenvectors. Moreover, we identify a family $\{\mathfrak{U}_n\}_{n \geq 0}$ of eigenvectors with eigenvalues $λ_n = 2^{1-n/2}$, consisting of Wick polynomials of white noise formally expressed as $\mathfrak{U}_n = \int_0^1 {:}\dot W(t)^n{:}\, dt$, and rigorously constructed as Hida distributions. We then use this spectral structure to uncover a finer, second layer of universality in the Donsker invariance principle: not only is the convergence of random walks towards the Brownian scaling limit universal, but the entire hierarchy of leading corrections to the Brownian limit is universal as well, governed by the eigenpairs $(λ_n, \mathfrak{U}_n)$. We also show, at a formal level, that the framework developed here extends to Gibbs-type perturbations of the Brownian fixed point, in the spirit of self-interacting quantum field theories. In particular, we recover the standard irrelevant/marginal/relevant classification, usually obtained in the physics literature by power-counting, solely from the spectral analysis of $\mathscr{L}$.

math-ph↗

arXiv2026

Perturbative anomalous exponents from Kolmogorov multipliers

Intermittency, manifested through anomalous scaling, remains one of the central unresolved problems in turbulence theory, with few analytical approaches extending beyond idealized linear transport models. We introduce a perturbative framework for anomalous scaling in turbulent transport based on multiplier statistics, rather than zero-mode calculations. We demonstrate the approach using a shell model combining deterministic and Kraichnan-like stochastic components. We reduce the problem to the analysis of a stationary Fokker-Planck equation for Kolmogorov multipliers, defined as ratios of successive scalar amplitudes. Its solution yields the invariant measure through a perturbative expansion around a Gaussian distribution. Using the resulting multiplier statistics, we compute explicit anomalous scaling exponents for structure functions of arbitrary order, including odd, even, and non-integer moments. Although demonstrated here for a shell model, the framework suggests a systematic perturbative route toward analytical theories of intermittency in turbulence.

physics.flu-dyn↗

arXiv2026

First-principles perturbative theory of anomalous scaling in a stochastic shell model of turbulence

Deriving anomalous scaling exponents from the equations of motion remains a central problem in the statistical theory of turbulence. Here we obtain a first-principles perturbative solution for a nonlinear stochastic dyadic shell model. The model preserves the conservative cascade structure and exact scaling symmetry of the deterministic dynamics, while stochastic transfer fluctuations provide a perturbative setting in which the leading-order rescaled dynamics is Gaussian. Using the statistically restored hidden scaling symmetry of the inertial-range equations, we determine the stationary statistics of the rescaled variables. We then formulate anomalous scaling as a Perron--Frobenius eigenvalue problem for the multiplier statistics. The resulting perturbative expansion yields explicit analytical expressions for the scaling exponents of structure functions of arbitrary order in the weak-noise regime. Direct numerical simulations provide an independent verification of the theoretical predictions. The results demonstrate that the hidden-symmetry perturbation framework extends from linear random models to a genuinely nonlinear cascade system.

physics.flu-dyn↗

arXiv2026

RG theory of spontaneous stochasticity for Sabra model of turbulence

We consider fluctuating Sabra models of turbulence, which exhibit the phenomenon of spontaneous stochasticity: their solutions converge to a stochastic process in the ideal limit, when both viscosity and small-scale noise vanish. In this paper, we develop a renormalization group (RG) approach to explain this phenomenon. Here, RG is understood as an exact relation between the stochastic properties of systems with different dissipative and noise terms, in contrast to the Kadanoff-Wilson coarse-graining procedure, which involves small-scale integration. We argue that the stochastic process in the ideal limit is represented as a fixed point of the RG operator. The existence of such a fixed point confirms not only the convergence in the ideal limit, but also the universality of the spontaneously stochastic process, i.e. its independence from the type of dissipation and noise. The dominant eigenmode of the linearized RG operator determines the leading correction in the convergence process. The RG eigenvalue $ρ\approx 0.84 \exp(2.28i)$ is universal and it turns out to be complex, which explains the rather slow and oscillatory convergence in the ideal limit. These universality predictions are accurately confirmed by numerical data.

nlin.CD↗

arXiv2026

Renormalization-group perspective on spontaneous stochasticity

We present a renormalization-group perspective on spontaneous stochasticity in hydrodynamic turbulence, viewed through the lens of multiscale dynamical systems. Building on previously established results for a solvable multiscale Arnold's cat model, we show that spontaneous stochasticity emerges as a universal fixed point of an RG transformation acting on Markov kernels, independent of the microscopic regularization. Classical examples - including the Feigenbaum equation, the central limit theorem, and hierarchical spin models - are reinterpreted within the same framework, placing spontaneous stochasticity alongside other universality phenomena.

nlin.CD↗

arXiv2026

On the importance of stochasticity in closures of turbulence

Deterministic closures for coarse-grained turbulence models help reproduce mean statistics, but often fail to capture the finite-time growth of uncertainty. Using the framework of shell models as a quantitative multi-scale testbed, we compare fully resolved simulations with large-eddy simulations using either stochastic or deterministic subgrid closures. While in the fully resolved system a single microscopic perturbation is rapidly amplified by strongly chaotic dynamics, truncation produces a strong delay and suppression of variance growth when uncertainty is introduced through initial condition perturbations only. We show that a data-driven Langevin-type stochastic closure restores the correct timing and magnitude of variance growth across scales, demonstrating that sustained stochasticity is essential for predictability in reduced turbulent dynamics.

physics.flu-dyn↗

arXiv2025

Spontaneous stochasticity in the fluctuating Navier-Stokes equations on a logarithmic lattice

The predictability of turbulent flows remains a challenging problem for mathematicians, physicists, and meteorologists. In this context, we consider the 3D incompressible Navier-Stokes equations with small-scale random forcing on logarithmic lattices in Fourier space. Our goal is to probe the phenomenon of spontaneous stochasticity in this system, which means that its solutions remain stochastic in the limit of vanishing viscosity and noise. For this, we consider numerical simulations with increasing Reynolds numbers and vanishing noise amplitudes. Through measurements of statistics of individual large-scale Fourier modes, we verify the spontaneous stochasticity in two different setups: from rough initial data, and after a finite-time blowup of a strong solution. The convergence of probability density functions for distinct parameters suggests that the limiting solution is a universal stochastic process.

physics.flu-dyn↗

arXiv2025

RG approach to the inviscid limit for shell models of turbulence

We consider an initial value problem for shell models that mimic turbulent velocity fluctuations over a geometric sequence of scales. Our goal is to study the convergence of solutions in the inviscid (more generally, vanishing regularization) limit and explain the universality of both the limiting solutions and the convergence process. We develop a renormalization group (RG) formalism representing this limit as dynamics in a space of flow maps. For the dyadic shell model, the RG dynamics has a fixed-point attractor, which determines universal limiting solutions. Deviations from the limiting solutions are also universal and given by a leading eigenmode (eigenvalue and eigenvector) of the linearized RG operator. Application to the Gledzer shell model reveals the RG attractor in the form of a closed invariant curve, while the Sabra shell model yields chaotic RG dynamics. An important consequence of the RG formalism is the understanding of the different roles of symmetry-preserving (canonical) and symmetry-breaking (e.g. viscous) regularizations.

math.AP↗

arXiv2025

RG analysis of spontaneous stochasticity on a fractal lattice: stability and bifurcations

In this paper, we study the stability and bifurcations of spontaneous stochasticity using an approach reminiscent of the Feigenbaum renormalization group (RG). We consider dynamical models on a self-similar space-time lattice as toy models for multiscale motion in hydrodynamic turbulence. Here an ill-posed ideal system is regularized at small scales and the vanishing regularization (inviscid) limit is considered. By relating the inviscid limit to the dynamics of the RG operator acting on the flow maps, we explain the existence and universality (regularization independence) of the limiting solutions as a consequence of the fixed-point RG attractor. Considering the local linearized dynamics, we show that the convergence to the inviscid limit is governed by the universal RG eigenmode. We also demonstrate that the RG attractor undergoes a period-doubling bifurcation with parameter variation, thereby changing the nature of the inviscid limit. In the case of chaotic RG dynamics, we introduce the stochastic RG operator acting on Markov kernels. Then the RG attractor becomes stochastic, which explains the existence and universality of spontaneously stochastic solutions in the limit of vanishing noise. We study a linearized structure (RG eigenmode) of the stochastic RG attractor and its period-doubling bifurcation. Viewed as prototypes of Eulerian spontaneous stochasticity, our models explain its mechanism, universality and potential diversity.

math-ph↗

arXiv2025

Hidden symmetry in passive scalar advected by 2D Navier-Stokes turbulence

Here we show that passive scalars possess a hidden scaling symmetry when considering suitably rescaled fields. Such a symmetry implies (i) universal probability distribution for scalar multipliers and (ii) Perron-Frobenius scenario for the anomalous scaling of structure functions. We verify these predictions with high resolution simulations of a passive scalar advected by a 2D turbulent flow in inverse cascade.

physics.flu-dyn↗

arXiv2025

Multi-time scale-invariance of turbulence in a shell model

When time and velocities are dynamically rescaled relative to the instantaneous turnover time, the Sabra shell model acquires another (hidden) form of scaling symmetry. It has been previously shown that this symmetry is statistically restored in the inertial interval of developed turbulence, thereby establishing a self-similarity property derived from first principles and replacing the broken $1/3$-scaling of the K41 theory. Multifractal intermittency follows from the restored hidden symmetry, in which the anomalous scaling exponents $ζ_p$ are identified as Perron-Frobenius eigenvalues. In this paper, we use the hypothesis of restored hidden symmetry to address the multi-time statistics of turbulent fluctuations. The central result is the self-similarity rule stating that any observable that is time-scale homogeneous of degree $p$ is self-similar with the Hölder exponent $h = ζ_p/p$. As a particular case, it yields the scaling laws for decorrelation times of fluctuations obtained previously within the phenomenological multifractal approach. As further applications, we formulate self-similarity rules for multi-time structure functions and multi-time Kolmogorov multipliers and verify them numerically.

physics.flu-dyn↗

arXiv2025

Spontaneous stochasticity in a 3d Weierstrass-ABC flow

Chaotic systems are characterised by exponential separation between close-by trajectories, which in particular leads to deterministic unpredictability over an infinite time-window. It is now believed, that such butterfly effect is not fully relevant to account for the type of randomness observed in turbulence. For example, tracers in homogeneous isotropic flows are observed to separate algebraically, following a universal cubic growth, independent from the initial separation. This regime, known as Richardon's regime, suggests that at the level of trajectories, and unlike in chaos theory, randomness may in fact emerge in finite-time. This phenomenon called 'spontaneous stochasticity' originates from the singular nature of the underlying dynamics, and provides a candidate framework for turbulent randomness and transport. While spontaneous stochasticity has been mathematically formalised in simplified turbulence models, a precise and systematic tool for quantifying the various facets of this phenomenon is to this day missing. In particular, it is still unclear whether chaos is important for that behaviour to appear. In this paper we introduce a 3d rough flow that can be tuned to present Lagrangian chaos. The flow is inspired by the Weierstrass function and is entitled 'the WABC model'. After analysing its properties, we define what is spontaneous stochasticity in this context. The provided formal definition is then adapted to better suit for numerical analysis. We present the results from Monte-Carlo simulations of Lagrangian particles in this flow. Within the numerical precision, we quantitatively observe the appearance of spontaneous stochasticity in this model. We investigate the influence of noise type and find that the observed spontaneous stochasticity does not depend on the chosen stochastic regularisations.

nlin.CD↗

arXiv2024

From zero-mode intermittency to hidden symmetry in random scalar advection

The statistical behavior of scalars passively advected by random flows exhibits intermittency in the form of anomalous multiscaling, in many ways similar to the patterns commonly observed in incompressible high-Reynolds fluids. This similarity suggests a generic dynamical mechanism underlying intermittency, though its specific nature remains unclear. Scalar turbulence is framed in a linear setting that points towards a zero-mode scenario connecting anomalous scaling to the presence of statistical conservation laws; the duality is fully substantiated within Kraichnan theory of random flows. However, extending the zero-mode scenario to nonlinear settings faces formidable technical challenges. Here, we revisit the scalar problem in the light of a hidden symmetry scenario introduced in recent deterministic turbulence studies addressing the Sabra shell model and the Navier-Stokes equations. Hidden symmetry uses a rescaling strategy based entirely on symmetry considerations, transforming the original dynamics into a rescaled (hidden) system; It ultimately identifies the scaling exponents as the eigenvalues of a Perron-Frobenius operator acting on invariant measures of the rescaled equations. Considering a minimal shell model of scalar advection of the Kraichnan type that was previously studied by Biferale & Wirth, the present work extends the hidden symmetry approach to a stochastic setting, in order to explicitly contrast it with the zero-mode scenario. Our study indicates that the zero-mode and the multiplicative scenarios are intrinsically related. For systems of the Kraichnan type, the first approach provides a quantitative chararacterization of intermittency, while the second approach highlights the universal connection between the scalar case and a larger class of hydrodynamic models.

physics.flu-dyn↗

arXiv2024

Extreme statistics and extreme events in dynamical models of turbulence

We present a study of the intermittent properties of a shell model of turbulence with unprecedented statistics, about $\sim 10^7$ eddy turn over time, achieved thanks to an implementation on a large-scale parallel GPU factory. This allows us to quantify the inertial range anomalous scaling properties of the velocity fluctuations up to the 24th order moment. Through a careful assessment of the statistical and systematic uncertainties, we show that none of the phenomenological and theoretical models previously proposed in the literature to predict the anomalous power-law exponents in the inertial range is in agreement with our high-precision numerical measurements. We find that at asymptotically high order moments, the anomalous exponents tend towards a linear scaling, suggesting that extreme turbulent events are dominated by one leading singularity. We found that systematic corrections to scaling induced by the infrared and ultraviolet (viscous) cut-offs are the main limitations to precision for low-order moments, while high orders are mainly affected by the finite statistical samples. The unprecedentedly high fidelity numerical results reported in this work offer an ideal benchmark for the development of future theoretical models of intermittency in dynamical systems for either extreme events (high-order moments) or typical fluctuations (low-order moments). For the latter, we show that we achieve a precision in the determination of the inertial range scaling exponents of the order of one part over ten thousand (5th significant digit), which must be considered a record for out-of-equilibrium fluid-mechanics systems and models.

physics.flu-dyn↗

arXiv2024

Logarithmic lattice models for flows with boundaries

Many fundamental problems in fluid dynamics are related to the effects of solid boundaries. In general, they install sharp gradients and contribute to the developement of small-scale structures, which are computationally expensive to resolve with numerical simulations. A way to access extremely fine scales with a reduced number of degrees of freedom is to consider the equations on logarithmic lattices in Fourier space. Here we introduce new toy models for flows with walls, by showing how to add boundaries to the logarithmic lattice framework. The resulting equations retain many important properties of the original systems, such as the conserved quantities, the symmetries and the boundary effects. We apply this technique to many flows, with emphasis on the inviscid limit of the Navier-Stokes equations. For this setup, simulations reach impressively large Reynolds numbers and disclose interesting insights about the original problem.

physics.flu-dyn↗

arXiv2023

Data-based approach for time-correlated closures of turbulence models

Developed turbulent motion of fluid still lacks an analytical description despite more than a century of active research. Nowadays phenomenological ideas are widely used in practical applications, such as small-scale closures for numerical simulations of turbulent flows. In the present work, we use a shell model of turbulence to construct a closure intended to have a solid theoretical background and to capture intrinsic probabilistic features of turbulence. Shell models of turbulence are dynamical deterministic systems used to model energy cascade and other key aspects of the Navier-Stokes such as intermittency. We rescale the variables of the Sabra model in a way which leads to hidden symmetries and universal distributions. We then use such fine distributions to write closures, i.e., missing expressions for some of the Sabra variables. Our closures rely on approximating probability density functions using a Gaussian Mixture Model, which makes them probabilistic by nature and allows us to write time-correlated closures. We also provide a framework where other Machine Learning tools can be employed with reduced black-box aspects.

physics.flu-dyn↗

arXiv2023

Hidden scale invariance of turbulence in a shell model: from forcing to dissipation scales

Intermittency is one of central obstacles for understanding small-scale dynamics in the fully developed hydrodynamic turbulence. The modern approach is largely based on the multifractal theory of Parisi and Frisch which is, however, phenomenological. It was shown recently that the intermittency can be related to the hidden scale invariance. The latter is a new statistical scaling symmetry unbroken in a rescaled (projected) formulation of equations of motion. In the present work, we consider a shell model of turbulence and describe how the hidden symmetry manifests itself through all scales, both in the inertial interval and in the transition to forcing and dissipation ranges. In the inertial interval, we derive anomalous scaling laws from the hidden symmetry. Then, we show how a complicated form of the dissipation range is controlled by intermittent rescaled Reynolds numbers within a large range of dissipation scales. This dissipative intermittency can be removed by using a special class of dissipation models. For such models, the hidden scale invariance is restored both in the inertial interval and the dissipation range. Overall, the presented approach deduces the multifractal theory and some of its basic conclusions from the hidden scaling symmetry of equations of motion.

physics.flu-dyn↗

arXiv2023

Post-blowup dynamics for the nonlinear Schrödinger equation

In this work we present a systematic numerical study of the post-blowup dynamics of singular solutions of the 1D focusing critical NLS equation in the framework of a nonlinear damped perturbation. The first part of this study shows that initially the post-blowup is described by the adiabatic approximation, in which the collapsing core approaches an universal profile and the solution width is governed by a system of ODEs (reduced system). After that, a non-adiabatic regime is observed soon after the maximum of the solution, in which our direct numerical simulations show a clear deviation from the dynamics based on the reduced system. Our study suggests that such non-adiabatic regime is caused by the increasing influx of mass into the collapsing core of the solution, which is not considered in the derivation of the reduced system. Also, adiabatic theoretical predictions related to the wave-maximum and wave-dissipation are compared with our numerical simulations. The second part of this work describes the non-adiabatic dynamics. Here, numerical simulations reveal a dominant quasi linear regime, caused by the rapid defocusing process. The collapsing core approaches the universal profile, after removing some oscillations resulting from the interference with the tail. Finally, our numerical study suggests that in the limit of vanishing dissipation, and in a free-space domain, the critical mass is radiated to infinity instantly at the collapse time.

nlin.PS↗