Laszlo Matyas

Spiral modes in the diffusion of a single granular particle on a vibrating surface

We consider a particle that is subject to a constant force and scatters inelastically on a vibrating periodically corrugated floor. At small friction and for small scatterers the dynamics is dominated by resonances forming spiral structures in phase space. These spiral modes lead to pronounced maxima and minima in the diffusion coefficient as a function of the vibration frequency, as is shown in computer simulations. Our theoretical predictions may be verified experimentally by studying transport of single granular particles on vibratory conveyors.

Thermodynamic cross effects from dynamical systems

We give a thermodynamically consistent description of simultaneous heat and particle transport, as well as of the associated cross effects, in the framework of a chaotic dynamical system, a generalized multibaker map. Besides the density, a second field with appropriate source terms is included in order to mimic, after coarse graining, a spatial temperature distribution and its time evolution. An expression is derived for the irreversible entropy production in a steady state, as the average of the growth rate of the relative density, a unique combination of the two fields.

Multibaker map for thermodynamic cross effects in dynamical systems

A consistent description of simultaneous heat and particle transport, including cross effects, and the associated entropy balance is given in the framework of a deterministic dynamical system. This is achieved by a multibaker map where, in addition to the phase-space density of the multibaker, a second field with appropriate source terms is included in order to mimic a spatial temperature distribution and its time evolution. Conditions are given to ensure consistency in an appropriately defined continuum limit with the thermodynamic entropy balance. They leave as the only free parameter of the model the entropy flux let directly into the surroundings. If it vanishes in the bulk, the transport properties of the model are described by the thermodynamic transport equations. Another choice leads to a uniform temperature distribution. It represents transport problems treated by means of a thermostating algorithm, similar to the one considered in nonequilibrium molecular dynamics.

Analytic solutions for the three-dimensional compressible Navier–Stokes equation

We investigate the three-dimensional compressible Navier–Stokes (NS) and the continuity equations in Cartesian coordinates for Newtonian fluids. The problem has an importance in different fields of science and engineering like fluid, aerospace dynamics or transfer processes. Finding an analytic solution may bring considerable progress in understanding the transport phenomena and in the design of different equipments where the NS equation is applicable. For solving the equation the polytropic equation of state is used as a closing condition. The key idea is the three-dimensional generalization of the well-known self-similar ansatz which was already used for non-compressible viscous flow in our former study. The geometrical interpretations of the trial function is also discussed. We compared our recent results to the former non-compressible ones.

Analytic self-similar solutions of the Oberbeck–Boussinesq equations

Abstract In this article we will present pure two-dimensional analytic solutions for the coupled non-compressible Newtonian–Navier–Stokes — with Boussinesq approximation — and the heat conduction equation. The system was investigated from E.N. Lorenz half a century ago with Fourier series and pioneered the way to the paradigm of chaos. We present a novel analysis of the same system where the key idea is the two-dimensional generalization of the well-known self-similar Ansatz of Barenblatt which will be interpreted in a geometrical way. The results, the pressure, temperature and velocity fields are all analytic and can be expressed with the help of the error functions. The temperature field shows a strongly damped single periodic oscillation which can mimic the appearance of Rayleigh–Benard convection cells. Finally, it is discussed how our result may be related to nonlinear or chaotic dynamical regimes.

Geometrical origin of chaoticity in the bouncing ball billiard

We present a study of the chaotic behaviour of the bouncing ball billiard. The work is realised on the purpose of finding at least certain causes of separation of the neighbouring trajectories. Having in view the geometrical construction of the system, we report a clear origin of chaoticity of the bouncing ball billiard. By this we claim that in case when the floor is made of arc of circles – in a certain interval of frequencies – one can give semi-analytical estimates on chaotic behaviour.

Analytic Solutions of the Madelung Equation

We present analytic self-similar solutions for the one, two and three dimensional Madelung hydrodynamical equation for a free particle. There is a direct connection between the zeros of the Madelung fluid density and the magnitude of the quantum potential.

An Analytical Study of Bifurcations Generated by Some Iterated Function Systems

Abstract In a recent paper Bahar [Chaos, Solitons + Fractals, 1996, 7(1), 41] described bifurcation from a fixed point generated by iterated function systems. An analytical study of it, by using Banach theorem, was proposed by us in Chaos, Solitons + Fractals, 1998, 9(3), 449. In this paper we present an extension of our previous study and we prove that by a special transformation, the considered two-dimensional map can be reduced to two distinctive one-dimensional maps, such that each one determines the behavior of the entire system.© 1999 Elsevier Science Ltd. All rights reserved.