arXiv subjects

R. Schubert

Publications and source records attributed to R. Schubert.

5 recordsLinked to original sources

arXiv2025

Derivation of the Monokinetic Vlasov-Stokes equations

We consider a microscopic model of spherical particles with inertia in a Stokes flow. As the particle number grows to infinity and their size goes to zero we derive the monokinetic Vlasov-Stokes equations as mean-field limit. We do this under the assumption that the particles have initial velocities given by a Lipschitz velocity profile and prove the mean-field limit for times of the order of the inverse Lipschitz constant. Notably this is not a perturbative result. In particular, we do not require the inertia of the particles to vanish in the limit. Thereby the result improves upon the perturbative derivation in [HS23] in the case of a monokinetic limit density.

math.AP↗

arXiv2007

Fractional $\hbar$-scaling for quantum kicked rotors without cantori

Previous studies of quantum delta-kicked rotors have found momentum probability distributions with a typical width (localization length $L$) characterized by fractional $\hbar$-scaling, ie $L \sim \hbar^{2/3}$ in regimes and phase-space regions close to `golden-ratio' cantori. In contrast, in typical chaotic regimes, the scaling is integer, $L \sim \hbar^{-1}$. Here we consider a generic variant of the kicked rotor, the random-pair-kicked particle (RP-KP), obtained by randomizing the phases every second kick; it has no KAM mixed phase-space structures, like golden-ratio cantori, at all. Our unexpected finding is that, over comparable phase-space regions, it also has fractional scaling, but $L \sim \hbar^{-2/3}$. A semiclassical analysis indicates that the $\hbar^{2/3}$ scaling here is of quantum origin and is not a signature of classical cantori.

cond-mat.stat-mech↗

arXiv2003

Poincaré Husimi representation of eigenstates in quantum billiards

For the representation of eigenstates on a Poincaré section at the boundary of a billiard different variants have been proposed. We compare these Poincaré Husimi functions, discuss their properties and based on this select one particularly suited definition. For the mean behaviour of these Poincaré Husimi functions an asymptotic expression is derived, including a uniform approximation. We establish the relation between the Poincaré Husimi functions and the Husimi function in phase space from which a direct physical interpretation follows. Using this, a quantum ergodicity theorem for the Poincaré Husimi functions in the case of ergodic systems is shown.

nlin.CD↗

arXiv2002

Behaviour of boundary functions for quantum billiards

We study the behaviour of the normal derivative of eigenfunctions of the Helmholtz equation inside billiards with Dirichlet boundary condition. These boundary functions are of particular importance because they uniquely determine the eigenfunctions inside the billiard and also other physical quantities of interest. Therefore they form a reduced representation of the quantum system, analogous to the Poincaré section of the classical system. For the normal derivatives we introduce an equivalent to the standard Green function and derive an integral equation on the boundary. Based on this integral equation we compute the first two terms of the mean asymptotic behaviour of the boundary functions for large energies. The first term is universal and independent of the shape of the billiard. The second one is proportional to the curvature of the boundary. The asymptotic behaviour is compared with numerical results for the stadium billiard, different limacon billiards and the circle billiard, and good agreement is found. Furthermore we derive an asymptotic completeness relation for the boundary functions.

nlin.CD↗

arXiv1997

On the rate of quantum ergodicity in Euclidean billiards

For a large class of quantized ergodic flows the quantum ergodicity theorem due to Shnirelman, Zelditch, Colin de Verdière and others states that almost all eigenfunctions become equidistributed in the semiclassical limit. In this work we first give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the Wigner function in the case of ergodic systems. Of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value. This is studied numerically for three Euclidean billiards (stadium, cosine and cardioid billiard) using up to 6000 eigenfunctions. We find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions like bouncing ball modes or scarred eigenfunctions. We give a detailed discussion and explanation of these effects using a simple but powerful model. For the rate of quantum ergodicity in momentum space we observe a slower decay. We also study the suitably normalized fluctuations of the expectation values around their mean, and find good agreement with a Gaussian distribution.

chao-dyn↗