arXiv subjects

Gregory Falkovich

Publications and source records attributed to Gregory Falkovich.

At least 19 recordsLinked to original sources

arXiv2025

Degrees of universality in wave turbulence

Turbulence of weakly interacting waves displays a great deal of universality: independence of the details of the interaction and of the pumping and dissipation scales. Here we study how inverse turbulent cascades (from small to large scales) transition from weak to strong. We find that while one-loop corrections can be dependent on excitation and dissipation scales, new types of universality appear in strong turbulence. We contrast turbulence of spin waves in ferromagnets with turbulent cascades in the Nonlinear Schr\"odinger Equation (NSE) and in an MMT-like model in higher dimensions having a multiplicative interaction vertex: vertex renormalization gives rise to dependence on the pumping (UV scale) in the former but not in the latter. As a result of this spectral nonlocality, spin-wave turbulence stops being weak if one is sufficiently far from the pumping scale, even when the interaction of waves with comparable wavenumbers is weak. We paraphrase this as: nonlocality enhances nonlinearity. We then describe strong turbulence in a multi-component version of these models with a large number of components. We argue that strong spin-wave turbulence is similar to turbulence of the focusing NSE, as it realizes a critical-balance state. However, UV nonlocality causes the level of spin-wave turbulence at large scales to decrease with increasing pumping level, culminating in a state that is independent of the level of pumping.

cond-mat.stat-mech↗

arXiv2025

Universal regimes of strong turbulence in the multi-component Gross-Pitaevskii model

The Gross-Pitaevskii (GP) model, also known as the nonlinear Schr\"odinger equation, is arguably the most universal model in classical and quantum physics, describing spectrally narrow or long-wavelength distributions of interacting waves or particles. Modern applications -- from oceanic and atmospheric flows to photonics and cold atoms -- predominantly involve states that are far from equilibrium, culminating in the regime of fully developed turbulence. To date, a consistent theoretical description of such states has only existed for weakly interacting quasiparticles. Here we present a theory of strong turbulence in the two-dimensional $N$-component Gross-Pitaevskii model for both repulsive and attractive interactions, corresponding to the defocusing and focusing cases, respectively. In the focusing case, we show that attraction is enhanced by multi-wave effects, leading to a critical-balance state independent of the pumping level. In the defocusing case, repulsion is suppressed by collective effects, giving rise to another type of universality in strong turbulence -- independence from the bare coupling constant. The theory is confirmed by analytical results in the many-component limit and by direct numerical simulations of the single-component GP model.

cond-mat.quant-gas↗

arXiv2025

Weak and strong turbulence in self-focusing and defocusing media

While the focusing and defocusing Nonlinear Schrodinger Equations have similar behavior in the weak turbulence regime, they must differ dramatically in the strong turbulence regime. Here, we show that this difference is already present at next-to-leading order in the nonlinearity in the weak turbulence regime: The one-loop correction to the interaction vertex suppresses repulsion (like screening in electrodynamics), leading to a steeper spectrum in the defocusing case. In contrast, attraction enhancement (like antiscreening in chromodynamics) makes the spectrum less steep in the focusing case. To describe strong turbulence, we consider a vector model in the limit of a large number of components. A large-N kinetic equation, valid at all scales, can be derived analytically. It has an inverse-cascade solution whose two asymptotics, at high and low wavenumbers, describe weak and strong turbulence, respectively. We find two forms of universality in the strong turbulence spectrum: in focusing media it is independent of the flux magnitude, while in defocusing media it is independent of the bare coupling constant, with the largest scale appearing instead.

physics.flu-dyn↗

arXiv2024

Interplay between two mechanisms of resistivity

Mechanisms of resistivity can be divided into two basic classes: one is dissipative (like scattering on phonons) and another is quasi-elastic (like scattering on static impurities). They are often treated by the empirical Matthiessen rule, which says that total resistivity is just the sum of these two contributions, which are computed separately. This is quite misleading for two reasons. First, the two mechanisms are generally correlated. Second, computing the elastic resistivity alone masks the fundamental fact that the linear-response approximation has a vanishing validity interval at vanishing dissipation. Limits of zero electric field and zero dissipation do not commute for the simple reason that one needs to absorb the Joule heat quadratic in the applied field. Here, we present a simple model that illustrates these two points. The model also illuminates the role of variational principles for non-equilibrium steady states.

cond-mat.stat-mech↗

arXiv2024

Interaction renormalization and validity of kinetic equations for turbulent states

We consider turbulence of waves that interact weakly via four-wave scattering (sea waves, plasma waves, spin waves, and many others). In the first non-vanishing order in the interaction, the occupation number of waves satisfy a closed kinetic equation which has stationary solutions describing turbulent cascades. We show that a straightforward perturbation theory beyond the kinetic equation gives terms that generally diverge both at small (IR) and large (UV) wavenumbers for a direct cascade. The analysis up to the third order identifies the most UV-divergent terms. In order to gain qualitative analytic control, we sum a subset of the most UV divergent term, to all orders, giving a perturbation theory which is generally free from UV divergence, showing that turbulence becomes independent of the dissipation scale when it goes to zero. On the contrary, the ever-present IR divergence means that the effective coupling is parametrically larger than the naive estimate and grows with the pumping scale (similar to anomalous scaling in fluid turbulence). This suggests that the kinetic equation does not describe wave turbulence even of arbitrarily small level if the cascade is sufficiently long. We show that the character of strong turbulence is determined by the sign of the coupling, that is, whether the effective four-wave interaction is enhanced or suppressed by collective effects. The enhancement possibly signals that strong turbulence is dominated by multi-wave bound states (solitons, shocks, cusps), similar to confinement in quantum chromodynamics.

hep-th↗

arXiv2023

Multi-mode correlations and the entropy of turbulence

We suggest a new focus for turbulence studies -- multi-mode correlations -- which reveal the hitherto hidden nature of turbulent state. We apply this approach to shell models describing basic properties of turbulence. The family of such models allows one to study turbulence close to thermal equilibrium, which happens when the interaction time weakly depends on the mode number. As the number of modes increases, the one-mode statistics approaches Gaussian (like in weak turbulence), the occupation numbers grow, while the three-mode cumulant describing the energy flux stays constant. Yet we find that higher multi-mode cumulants grow with the order. We derive analytically and confirm numerically the scaling law of such growth. The sum of all squared dimensionless cumulants is equal to the relative entropy between the full multi-mode distribution and the Gaussian approximation of independent modes; we argue that the relative entropy could grow as the logarithm of the number of modes, similar to the %mutual information and entanglement entropy in critical phenomena. Therefore, the multi-mode correlations give a new way to characterize turbulence states and possibly divide them into universality classes.

nlin.CD↗

arXiv2022

Sum-of-squares bounds on correlation functions in a minimal model of turbulence

We suggest a new computer-assisted approach to the development of turbulence theory. It allows one to impose lower and upper bounds on correlation functions using sum-of-squares polynomials. We demonstrate it on the minimal cascade model of two resonantly interacting modes, when one is pumped and the other dissipates. We show how to present correlation functions of interest as part of a sum-of-squares polynomial using the stationarity of the statistics. That allows us to find how the moments of the mode amplitudes depend on the degree of non-equilibrium (analog of the Reynolds number), which reveals some properties of marginal statistical distributions. By combining scaling dependence with the results of direct numerical simulations, we obtain the probability densities of both modes in a highly intermittent inverse cascade. We also show that the relative phase between modes tends to $π/2$ and $-π/2$ in the direct and inverse cascades as the Reynolds number tends to infinity, and derive bounds on the phase variance. Our approach combines computer-aided analytical proofs with a numerical algorithm applied to high-degree polynomials.

physics.flu-dyn↗

arXiv2022

Direct observation of vortices in an electron fluid

Vortices are the hallmarks of hydrodynamic flow. Recent studies indicate that strongly-interacting electrons in ultrapure conductors can display signatures of hydrodynamic behavior including negative nonlocal resistance, Poiseuille flow in narrow channels, and a violation of the Wiedemann-Franz law. Here we provide the first visualization of whirlpools in an electron fluid. By utilizing a nanoscale scanning superconducting quantum interference device on a tip (SQUID-on-tip) we image the current distribution in a circular chamber connected through a small aperture to an adjacent narrow current carrying strip in high-purity type-II Weyl semimetal WTe2. In this geometry, the Gurzhi momentum diffusion length and the size of the aperture determine the vortex stability phase diagram. We find that the vortices are present only for small apertures, whereas the flow is laminar (non-vortical) for larger apertures, consistent with the theoretical analysis of the hydrodynamic regime and in contrast to the expectations of ballistic transport in WTe2 at low temperatures. Moreover, near the vortical-to-laminar transition, we observe a single vortex in the chamber splitting into two vortices, a behavior that can occur only in the hydrodynamic regime and cannot be sustained by ballistic transport. These findings suggest a novel mechanism of hydrodynamic flow: instead of the commonly considered electron-electron scattering at the bulk, which becomes extremely weak at low temperatures, the spatial diffusion of charge carriers' momenta is enabled by small-angle scattering at the planar surfaces of thin pure crystals. This surface-induced para-hydrodynamics opens new avenues for exploring and utilizing electron fluidics in high-mobility electron systems.

cond-mat.mes-hall↗

arXiv2021

Fibonacci turbulence

Never is the difference between thermal equilibrium and turbulence so dramatic, as when a quadratic invariant makes the equilibrium statistics exactly Gaussian with independently fluctuating modes. That happens in two very different yet deeply connected classes of systems: incompressible hydrodynamics and resonantly interacting waves. This work presents the first case of a detailed information-theoretic analysis of turbulence in such strongly interacting systems. The analysis elucidates the fundamental roles of space and time in setting the cascade direction and the changes of the statistics along it. We introduce a beautifully simple yet rich family of discrete models with neighboring triplet interactions and show that it has families of quadratic conservation laws defined by the Fibonacci numbers. Depending on the single model parameter, three types of turbulence were found: single direct cascade, double cascade, and the first ever case of a single inverse cascade. We describe quantitatively how deviation from thermal equilibrium all the way to turbulent cascades makes statistics increasingly non-Gaussian and find the self-similar form of the one-mode probability distribution. We reveal where the information (entropy deficit) is encoded and disentangle the communication channels between modes, as quantified by the mutual information in pairs and the interaction information inside triplets.

nlin.CD↗

arXiv2020

Singular measures and information capacity of turbulent cascades

Is there really such a thing as weak turbulence? Here we analyze turbulence of weakly interacting waves using the tools of information theory. It offers a unique perspective for comparing thermal equilibrium and turbulence: the mutual information between modes is shown to be stationary and small in equilibrium but grows linearly with time in weak turbulence. We trace this growth to the concentration of probability on the resonance surfaces, which can go all the way to a singular measure. The surprising conclusion is that no matter how small is the nonlinearity and how close to Gaussian is the statistics of any single amplitude, a stationary phase-space measure is far from Gaussian, as manifested by a large relative entropy. Though it might be upsetting to practitioners of weak turbulence approach, this is a rare piece of good news for turbulence modeling: the resolved scales carry significant information about the unresolved scales. The mutual information between large and small scales is the information capacity of turbulent cascade, setting the limit on the representation of subgrid scales in turbulence modeling.

physics.flu-dyn↗

arXiv2019

Light transport and vortex-supported wave-guiding in micro-structured optical fibres

In hydrodynamics, vortex generation upon the transition from smooth laminar flows to turbulence is generally accompanied by increased dissipation. However, plane vortices can provide transport barriers and decrease losses, as it happens in numerous geophysical, astrophysical flows and in tokamaks. Photon interactions with matter can affectlight transportin waveguides in unexpected and somewhat counterintuitive ways resembling fluid dynamics. Here, we demonstrate dramatic impact of light vortex formation in micro-structured optical fibres on the energy dissipation. We show possibility of vortices formationin both solid core and hollow core fibres on the zero amplitude lines in the cladding. We find that vortices reduce light leakage by three orders of magnitude, effectively improving wave guiding. A strong light localization based on the same principle can also be achieved in the negative curvature hollow core fibres.

physics.optics↗

arXiv2018

Freely flowing currents and electric field expulsion in viscous electronics

Electronic fluids bring into hydrodynamics a new setting: equipotential flow sources embedded inside the fluid. Here we show that nonlocal relation between current and electric field due to momentum-conserving inter-particle collisions leads to a total or partial field expulsion from such flows. That results in freely flowing currents in the bulk and boundary jump in electric potential at current-injecting electrodes. We derive the appropriate boundary conditions, analyze current distribution in free flows, discuss how the field expulsion depends upon geometry of the electrode, and link the phenomenon to breakdown of conformal invariance.

cond-mat.mes-hall↗

arXiv2018

Particle Collisions and Negative Nonlocal Response of Ballistic Electrons

An electric field that builds in the direction against current, known as negative nonlocal resistance, arises naturally in viscous flows and is thus often taken as a telltale of this regime. Here we predict negative resistance for the ballistic regime, wherein the ee collision mean free path is greater than the length scale at which the system is being probed. Therefore, negative resistance alone does not provide strong evidence for the occurrence of the hydrodynamic regime; it must thus be demoted from the rank of a smoking gun to that of a mere forerunner. Furthermore, we find that negative response is log-enhanced in the ballistic regime by the physics related to the seminal Dorfman-Cohen log divergence due to memory effects in the kinetics of dilute gases. The ballistic regime therefore offers a unique setting for exploring these interesting effects due to electron interactions.

cond-mat.mes-hall↗

arXiv2018

Fluidity Onset in Graphene

Viscous electron fluids have emerged recently as a new paradigm of strongly-correlated electron transport in solids. Here we report on a direct observation of the transition to this long-sought-for state of matter in a high-mobility electron system in graphene. Unexpectedly, the electron flow is found to be interaction-dominated but non-hydrodynamic (quasiballistic) in a wide temperature range, showing signatures of viscous flows only at relatively high temperatures. The transition between the two regimes is characterized by a sharp maximum of negative resistance, probed in proximity to the current injector. The resistance decreases as the system goes deeper into the hydrodynamic regime. In a perfect darkness-before-daybreak manner, the interaction-dominated negative response is strongest at the transition to the quasiballistic regime. Our work provides the first demonstration of how the viscous fluid behavior emerges in an interacting electron system.

cond-mat.mes-hall↗

arXiv2018

Turbulence appearance and non-appearance in thin fluid layers

Flows in fluid layers are ubiquitous in industry, geophysics and astrophysics. Large-scale flows in thin layers can be considered two-dimensional (2d) with bottom friction added. Here we find that the properties of such flows depend dramatically on the way they are driven. We argue that wall-driven (Couette) flow cannot sustain turbulence at however small viscosity and friction. Direct numerical simulations (DNS) up to the Reynolds number $Re=10^6$ confirm that all perturbations die in a plane Couette flow. On the contrary, for sufficiently small viscosity and friction, we show that finite perturbations destroy the pressure-driven laminar (Poiseuille) flow. What appears instead is a traveling wave in the form of a jet slithering between wall vortices. For $10^4<Re<5\cdot10^4$, the mean flow has remarkably simple structure: the jet is sinusoidal with a parabolic velocity profile, vorticity is constant inside vortices, while the fluctuations are small. At higher $Re$ strong fluctuations appear, yet the mean traveling wave survives. Considering the momentum flux barrier in such a flow, we derive a new scaling law for the $Re$-dependence of the friction factor and confirm it by DNS.

physics.flu-dyn↗

arXiv2018

Precise measurements of torque in von Karman swirling flow driven by a bladed disk

Scrupulous measurements and detailed data analysis of the torque in a swirling turbulent flow driven by counter-rotating bladed disks reveals an apparent breaking of the law of similarity. Potentially, such breakdown could arise from several possible factors, including dependence on dimensionless numbers other that $Re$ or velocity coupling to other fields such as temperature. However, careful redesign and calibration of the experiment showed that this unexpected result was due to background errorscaused by minute misalignments which lead to a noisy and irreproducible torque signal at low rotation speeds and prevented correct background subtraction normally ascribed to frictional losses. An important lesson to be learnt is that multiple minute misalignments can nonlinearly couple to the torque signal and provide a dc offset that cannot be removed by averaging. That offset can cause the observed divergence of the friction coefficient C_f from its constant value observed in the turbulent regime. To minimize the friction and misalignments, we significantly modified the experimental setup and carried out the experiment with one bladed disk where the disk, torque meter and motor shaft axes can be aligned with significantly smaller error, close to the torque meter resolution. As a result we made precise measurements with high resolution and sensitivity of the small torques produced for low rotation speeds for several water-glycerin solutions of different viscosities and confirmed the similarity law in a wide range of Re in particular in low viscosity fluids.

physics.flu-dyn↗

arXiv2017

Alternating currents and shear waves in viscous electronics

Strong interaction among charge carriers can make them move like viscous fluid. Here we explore alternating current (AC) effects in viscous electronics. In the Ohmic case, incompressible current distribution in a sample adjusts fast to a time-dependent voltage on the electrodes, while in the viscous case, momentum diffusion makes for retardation and for the possibility of propagating slow shear waves. We focus on specific geometries that showcase interesting aspects of such waves: current parallel to a one-dimensional defect and current applied across a long strip. We find that the phase velocity of the wave propagating along the strip respectively increases/decreases with the frequency for no-slip/no-stress boundary conditions. This is so because when the frequency or strip width goes to zero (alternatively, viscosity go to infinity), the wavelength of the current pattern tends to infinity in the no-stress case and to a finite value in a general case. We also show that for DC current across a strip with no-stress boundary, there only one pair of vortices, while there is an infinite vortex chain for all other types of boundary conditions.

cond-mat.mes-hall↗

arXiv2017

How vortices and shocks provide for a flux loop in two-dimensional compressible turbulence

Large-scale turbulence in fluid layers and other quasi-two-dimensional compressible systems consists of planar vortices and waves. Separately, wave turbulence usually produces a direct energy cascade, while solenoidal planar turbulence transports energy to large scales by an inverse cascade. Here, we consider turbulence at finite Mach numbers when the interaction between acoustic waves and vortices is substantial. We employ solenoidal pumping at intermediate scales and show how both direct and inverse energy cascades are formed starting from the pumping scale. We show that there is an inverse cascade of kinetic energy up to a scale $\ell$, where a typical velocity reaches the speed of sound; this creates shock waves, which provide for a compensating direct cascade. When the system size is less than $\ell$, the steady state contains a system-size pair of long-living condensate vortices connected by a system of shocks. Thus turbulence in fluid layers processes energy via a loop: Most energy first goes to large scales via vortices and is then transported by waves to small-scale dissipation.

physics.flu-dyn↗