arXiv subjects

Amirali Hannani

Publications and source records attributed to Amirali Hannani.

12 recordsLinked to original sources

arXiv2026

Two-Dimensional $β$-plane Turbulence: Dual Cascade and Zonal Jets

We derive an exact and novel expression for an averaged two-point correlation function in the statistically stationary, forced-dissipative two-dimensional Navier-Stokes equations subject to the Coriolis force under the beta-plane approximation. This identity is related to the so-called geostrophic balance: it connects the effect of the Coriolis force to the pressure gradient through a two-point correlation function. Additionally, we provide sufficient conditions under which the asymptotics of the averaged third-order structure function at large spatial scales follow the universal third-order law of two-dimensional turbulence in the absence of the Coriolis force. This complements our previous results on small spatial scales. Together, our results provide a clear picture of the role of the Coriolis force in beta-plane turbulence. On the one hand, the spherically averaged rates of enstrophy and energy transfer are not affected by the Coriolis force. On the other hand, the Coriolis force contributes to anisotropic large-scale organization by altering the spatial distribution of energy and promoting the formation of zonal structures. The proof relies on a new formulation of the Karman-Howarth-Monin relation. For the geostrophic balance, we use a novel antisymmetric projection of the KHM relation under which only the pressure and Coriolis terms survive. For the cascade laws, we show that the Coriolis contribution to the averaged classical KHM relation vanishes identically at any scale.

physics.flu-dyn↗

arXiv2026

Many-body Localization and Poisson statistics in the Quantum Sun model

The Quantum Sun model is a many-body Hamiltonian model of interacting spins arranged on the half-line (or the line). Spins at distance $n$ from the origin are coupled to the rest of the system via a term of strength $α^n$, with $α\in (0,1)$. From theoretical and numerical considerations, it is believed that this model undergoes a localization-delocalization transition at the critical value $α=\frac{1}{\sqrt{2}}$. We prove that, for $α\ll \frac{1}{\sqrt{2}}$, the model is localized and that its spectral statistics are Poissonian. The main novelty of this result is that the model considered here is a genuine many-body model, in which the interaction has a profound effect.

math-ph↗

arXiv2026

Stark localization of interacting particles

We consider N interacting quantum particles on a one-dimensional lattice, and subjected to an external linear potential. For N = 1, the corresponding Hamiltonian is explicitly diagonalizable, with superexponentially localized eigenstates. This is called Stark localization. We prove that superexponential spectral localization persists for arbitrary N and every interaction strength.

math-ph↗

arXiv2025

On the effect of the Coriolis force on the enstrophy cascade

We study the direct enstrophy cascade at small spatial scales in statistically stationary forced-dissipated 2D Navier-Stokes equations subject to the Coriolis force in the $β$-plane approximation. We provide sufficient conditions inspired by [6,63] to prove that at small scales, in the presence of the Coriolis force, the so-called third-order structure function's asymptotics follows the third-order universal law of 2D turbulence without the Coriolis force. Our result indicates that at small scales, the enstrophy flux from larger to smaller scales is not affected by the Coriolis force, confirming experimental and numerical observations. To the best of our knowledge, this is the first mathematically rigorous study of the above equations.

math.AP↗

arXiv2025

Sub-Power Law Decay of the Wave Packet Maximum in Disordered Anharmonic Chains

We show that the peak of an initially localized wave packet in one-dimensional nonlinear disordered chains decays more slowly than any power law of time. The systems under investigation are Klein-Gordon and nonlinear disordered Schrödinger-type chains, characterized by a harmonic onsite disordered potential and quartic nearest-neighbor coupling. Our results apply in the long-time limit, hold almost surely, and are valid for arbitrary finite energy values.

math-ph↗

arXiv2025

Controlling the Rates of a Chain of Harmonic Oscillators with a Point Langevin Thermostat

We consider the control problem of controlling the rates of an infinite chain of coupled harmonic oscillators with a Langevin thermostat at the origin. We study the effect of two types of open-loop boundary controls, impulsive control and linear memory-feedback control, in the high frequency limit. We investigate their action on the reflection-transmission coefficients for the wave energy for the scattering of the thermostat. Our study shows that the impulsive boundary controls have no impact on the rates and are thus not appropriate to act on the system, despite their physical meaning and relevance. In contrast, the second kind of control that we propose, which is less standard and uses the past of the state solution of the system, is adequate and relevant. We prove that any triple of rates satisfying appropriate assumptions is asymptotically reachable thanks to the linear memory-feedback controls that we design explicitly.

math.OC↗

arXiv2024

Well-Posedness and regularity properties of 2d $β$-plane stochastic Navier-Stokes equations in a periodic channel

We consider the 2d $β$-plane stochastic Navier-Stokes equations in a periodic channel. We prove the well-posedness and existence of the stationary measure, as well as certain regularity estimates concerning the support of the stationary measure. The mentioned estimates are crucial for the rigorous study of the cascade phenomena in this equation [8]. To the best of our knowledge, this is the first mathematically rigorous treatment of these equations involving both the stochastic noise and the Coriolis force.

math.AP↗

arXiv2024

Internal Control of The Transition Kernel for Stochastic Lattice Dynamics

In [5], we have designed impulsive and feedback controls for harmonic chains with a point thermostat. In this work, we study the internal control for stochastic lattice dynamics, with the goal of controlling the transition kernel of the kinetic equation in the limit. A major novelty of the work is the introduction of a new geometric combinatorial argument, used to establish paths for the controls.

math.OC↗

arXiv2022

On the wave turbulence theory for a stochastic KdV type equation -- Generalization for the inhomogeneous kinetic limit

Starting from a stochastic Zakharov-Kuznetsov (ZK) equation on a lattice, the previous work [ST21] by the last two authors gave a derivation of the homogeneous 3-wave kinetic equation at the kinetic limit under very general assumptions: the initial condition is out of equilibrium, the dimension $d\ge 2$, the smallness of the nonlinearity $λ$ is allowed to be independent of the size of the lattice, the weak noise is chosen not to compete with the weak nonlinearity and not to inject energy into the equation. In the present work, we build on the framework of [ST21], following the formal derivation of Spohn [Spo06] and inspired by previous work [HO21] of the first author and Olla, so that the inhomogeneous 3-wave kinetic equation can also be obtained at the kinetic limit under analogous assumptions. Similar to the homogeneous case -- and unlike the cubic nonlinear Schrödinger equation -- the inhomogeneous kinetic description of the deterministic lattice ZK equation is unlikely to happen due to the vanishing of the dispersion relation on a certain singular manifold on which not only $3$-wave interactions but also all $n$-wave interactions ($n\ge3$) are allowed to happen, a phenomenon first observed by Lukkarinen [Luk07]. To the best of our knowledge, our work provides the first rigorous derivation of a nonlinear inhomogeneous wave kinetic equation in the kinetic limit.

math.AP↗

arXiv2022

Derivation of Euler equations from quantum and classical microscopic dynamics

We derive Euler equations from a Hamiltonian microscopic dynamics. The microscopic system is a one-dimensional disordered harmonic chain, and the dynamics is either quantum or classical. This chain is an Anderson insulator with a symmetry protected mode: Thermal fluctuations are frozen while the low modes ensure the transport of elongation, momentum and mechanical energy, that evolve according to Euler equations in an hyperbolic scaling limit. In this paper, we strengthen considerably our previous results, where we established a limit in mean starting from a local Gibbs state: We now control the second moment of the fluctuations around the average, yielding a limit in probability, and we enlarge the class of admissible initial states.

math-ph↗

arXiv2021

A stochastic thermalization of the Discrete Nonlinear Schrödinger Equation

We introduce a mass conserving stochastic perturbation of the discrete nonlinear Schrödinger equation that models the action of a heat bath at a given temperature. We prove that the corresponding canonical Gibbs distribution is the unique invariant measure. In the one-dimensional cubic focusing case on the torus, we prove that in the limit for large time, continuous approximation, and low temperature, the solution converges to the steady wave of the continuous equation that minimizes the energy for a given mass.

math-ph↗

arXiv2021

Hydrodynamic limit for a disordered quantum harmonic chain

In this note, we study the hydrodynamic limit, in the hyperbolic space-time scaling, for a one-dimensional unpinned chain of quantum harmonic oscillators with random masses. To the best of our knowledge, this is among the first examples, where one can prove the hydrodynamic limit for a quantum system rigorously. In fact, we prove that after hyperbolic rescaling of time and space the distribution of the elongation, momentum, and energy averaged under the proper Gibbs state, converges to the solution of the Euler equation. There are two main phenomena in this chain which enable us to deduce this result. First is the Anderson localization which decouples the mechanical and thermal energy, providing the closure of the equation for energy and indicating that the temperature profile will be frozen. The second phenomena is similar to some sort of decay of correlation phenomena which let us circumvent the difficulties arising from the fact that our Gibbs state is not a product state due to the quantum nature of the system.

math-ph↗