The Wigner-Fokker-Planck equation: Stationary states and large-time behavior

We consider the linear Wigner-Fokker-Planck equation subject to confining potentials which are smooth perturbations of the harmonic oscillator potential. For a certain class of perturbations we prove that the equation admits a unique stationary solution in a weighted Sobolev space. A key ingredient of the proof is a new result on the existence of spectral gaps for Fokker-Planck type operators in certain weighted $L^2$-spaces. In addition we show that the steady state corresponds to a positive density matrix operator with unit trace and that the solutions of the time-dependent problem converge towards the steady state with an exponential rate.Comment: This manuscript essentially replaces (and corrects a mistake found in) the submission arXiv:0707.2445, by establishing a new functional framework and new spectral estimate

A phase-space study of jet formation in planetary-scale fluids

The interaction between planetary waves and an arbitrary zonal flow is studied from a phase-space viewpoint. Using the Wigner distribution, a planetary wave Vlasov equation is derived that includes the contribution of the mean flow to the zonal potential vorticity gradient. This equation is applied to the problem of planetary wave modulational instability, where it is used to predict a fastest growing mode of finite wavenumber. A wave-mean flow numerical model is used to test the analytical predictions, and an intuitive explanation of modulational instability and jet asymmetry is given via the motion of planetary wavepackets in phase space.Comment: 10 pages, 10 figure

Comparative study of semiclassical approaches to quantum dynamics

Quantum states can be described equivalently by density matrices, Wigner functions or quantum tomograms. We analyze the accuracy and performance of three related semiclassical approaches to quantum dynamics, in particular with respect to their numerical implementation. As test cases, we consider the time evolution of Gaussian wave packets in different one-dimensional geometries, whereby tunneling, resonance and anharmonicity effects are taken into account. The results and methods are benchmarked against an exact quantum mechanical treatment of the system, which is based on a highly efficient Chebyshev expansion technique of the time evolution operator.Comment: 32 pages, 8 figures, corrected typos and added references; version as publishe

Quantum to classical transition in a system with a mixed classical dynamics

We study how decoherence rules the quantum-classical transition of the Kicked Harmonic Oscillator (KHO). When the amplitude of the kick is changed the system presents a classical dynamics that range from regular to a strong chaotic behavior. We show that for regular and mixed classical dynamics, and in the presence of noise, the distance between the classical and the quantum phase space distributions is proportional to a single parameter $\chi\equiv K\hbar_{\rm eff}^2/4D^{3/2}$ which relates the effective Planck constant $\hbar_{\rm eff}$, the kick amplitude $K$ and the diffusion constant $D$. This is valid when $\chi < 1$, a case that is always attainable in the semiclassical regime independently of the value of the strength of noise given by $D$. Our results extend a recent study performed in the chaotic regime.Comment: 10 pages, 7 figure

Entropy production in quantum Yang-Mills mechanics in semi-classical approximation

We discuss thermalization of isolated quantum systems by using the Husimi-Wehrl entropy evaluated in the semiclassical treatment. The Husimi-Wehrl entropy is the Wehrl entropy obtained by using the Husimi function for the phase space distribution. The time evolution of the Husimi function is given by smearing the Wigner function, whose time evolution is obtained in the semiclassical approximation. We show the efficiency and usefullness of this semiclassical treatment in describing entropy production of a couple of quantum mechanical systems, whose classical counter systems are known to be chaotic. We propose two methods to evaluate the time evolution of the Husimi-Wehrl entropy, the test-particle method and the two-step Monte-Carlo method. We demonstrate the characteristics of the two methods by numerical calculations, and show that the simultaneous application of the two methods ensures the reliability of the results of the Husimi-Wehrl entropy at a given time.Comment: 11 pages, 8 figure