On the connection between the Nekhoroshev theorem and Arnold Diffusion

The analytical techniques of the Nekhoroshev theorem are used to provide estimates on the coefficient of Arnold diffusion along a particular resonance in the Hamiltonian model of Froeschl\'{e} et al. (2000). A resonant normal form is constructed by a computer program and the size of its remainder $||R_{opt}||$ at the optimal order of normalization is calculated as a function of the small parameter $\epsilon$. We find that the diffusion coefficient scales as $D\propto||R_{opt}||^3$, while the size of the optimal remainder scales as $||R_{opt}|| \propto\exp(1/\epsilon^{0.21})$ in the range $10^{-4}\leq\epsilon \leq 10^{-2}$. A comparison is made with the numerical results of Lega et al. (2003) in the same model.Comment: Accepted in Celestial Mechanics and Dynamical Astronom

Non-Gaussian Velocity Distribution Function in a Vibrating Granular Bed

The simulation of granular particles in a quasi two-dimensional container under the vertical vibration as an experimental accessible model for granular gases is performed. The velocity distribution function obeys an exponential-like function during the vibration and deviates from the exponential function in free-cooling states. It is confirmed that this exponential-like distribution function is produced by Coulomb's friction force. A Langevin equation with Coulomb's friction is proposed to describe the motion of such the system.Comment: 4 pages, 4 figures. to be published in Journal of Physical Society of Japan Vol.73 No.

Continuum theory of partially fluidized granular flows

A continuum theory of partially fluidized granular flows is developed. The theory is based on a combination of the equations for the flow velocity and shear stresses coupled with the order parameter equation which describes the transition between flowing and static components of the granular system. We apply this theory to several important granular problems: avalanche flow in deep and shallow inclined layers, rotating drums and shear granular flows between two plates. We carry out quantitative comparisons between the theory and experiment.Comment: 28 pages, 23 figures, submitted to Phys. Rev.

Classical Mechanics and Renormalization Group

The KAM theory is discussed in detail from the point of view of the "renormalization group approach". We discuss also some aspects of the possible existence of universal structures in the chaotic transition. The quasi-periodic Schroedinger equation in one dimension is discussed as a special case