First passage and arrival time densities for L\'evy flights and the failure of the method of images

We discuss the first passage time problem in the semi-infinite interval, for homogeneous stochastic Markov processes with L{\'e}vy stable jump length distributions $\lambda(x)\sim\ell^{\alpha}/|x|^{1+\alpha}$ ($|x|\gg\ell$), namely, L{\'e}vy flights (LFs). In particular, we demonstrate that the method of images leads to a result, which violates a theorem due to Sparre Andersen, according to which an arbitrary continuous and symmetric jump length distribution produces a first passage time density (FPTD) governed by the universal long-time decay $\sim t^{-3/2}$. Conversely, we show that for LFs the direct definition known from Gaussian processes in fact defines the probability density of first arrival, which for LFs differs from the FPTD. Our findings are corroborated by numerical results.Comment: 8 pages, 3 figures, iopart.cls style, accepted to J. Phys. A (Lett

A geometric approach to time evolution operators of Lie quantum systems

Lie systems in Quantum Mechanics are studied from a geometric point of view. In particular, we develop methods to obtain time evolution operators of time-dependent Schrodinger equations of Lie type and we show how these methods explain certain ad hoc methods used in previous papers in order to obtain exact solutions. Finally, several instances of time-dependent quadratic Hamiltonian are solved.Comment: Accepted for publication in the International Journal of Theoretical Physic

Large scale dynamics in turbulent Rayleigh-Benard convection

The progress in our understanding of several aspects of turbulent Rayleigh-Benard convection is reviewed. The focus is on the question of how the Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the Prandtl number Pr, and on how the thicknesses of the thermal and the kinetic boundary layers scale with Ra and Pr. Non-Oberbeck-Boussinesq effects and the dynamics of the large-scale convection-roll are addressed as well. The review ends with a list of challenges for future research on the turbulent Rayleigh-Benard system.Comment: Review article, 34 pages, 13 figures, Rev. Mod. Phys. 81, in press (2009

Thermal Conductivity of Methane-Hydrate

The thermal conductivity of the methane hydrate CH4 (5.75 H2O) was measured in the interval 2-140 K using the steady-state technique. The thermal conductivity corresponding to a homogeneous substance was calculated from the measured effective thermal conductivity obtained in the experiment. The temperature dependence of the thermal conductivity is typical for the thermal conductivity of amorphous solids. It is shown that after separation of the hydrate into ice and methane, at 240 K, the thermal conductivity of the ice exhibits a dependence typical of heavily deformed fine-grain polycrystal. The reason for the glass-like behavior in the thermal conductivity of clathrate compounds has been discussed. The experimental results can be interpreted within the phenomenological soft-potential model with two fitting parameters.Comment: 13 pages, 3 figure