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Register a new userTransport Properties of the Lorentz Gas in Terms of Periodic Orbits
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Added by
Jean-Pierre Eckmann
Publication date
1993
fields
Physics
and research's language is
English
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We establish a formula relating global diffusion in a space periodic dynamical system to cycles in the elementary cell which tiles the space under translations.
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Analytic methods to investigate periodic orbits in galactic potentials. To evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms. The solutions of the equations of motion corresponding to periodic orbits are obtained as series expansions computed by inverting the normalizing canonical transformation. To improve the convergence of the series a resummation based on a continued fraction may be performed. This method is analogous to that looking for approximate rational solutions (Prendergast method). It is shown that with a normal form truncated at the lowest order incorporating the relevant resonance it is possible to construct quite accurate solutions both for normal modes and periodic orbits in general position.
Gutzwillers trace formula and Bogomolnys formula are applied to a non--specific, non--scalable Hamiltonian system, a two--dimensional anharmonic oscillator. These semiclassical theories reproduce well the exact quantal results over a large spatial and energy range.
We investigate analytically and numerically the spatial structure of the non-equilibrium stationary states (NESS) of a point particle moving in a two dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a constant external electric field E as well as a Gaussian thermostat which keeps the speed |v| constant. We show that despite the singular nature of the SRB measure its projections on the space coordinates are absolutely continuous. We further show that these projections satisfy linear response laws for small E. Some of them are computed numerically. We compare these results with those obtained from simple models in which the collisions with the obstacles are replaced by random collisions.Similarities and differences are noted.
The chaotic properties of some subshift maps are investigated. These subshifts are the orbit closures of certain non-periodic recurrent points of a shift map. We first provide a review of basic concepts for dynamics of continuous maps in metric spaces. These concepts include nonwandering point, recurrent point, eventually periodic point, scrambled set, sensitive dependence on initial conditions, Robinson chaos, and topological entropy. Next we review the notion of shift maps and subshifts. Then we show that the one-sided subshifts generated by a non-periodic recurrent point are chaotic in the sense of Robinson. Moreover, we show that such a subshift has an infinite scrambled set if it has a periodic point. Finally, we give some examples and discuss the topological entropy of these subshifts, and present two open problems on the dynamics of subshifts.
We present recent results on a systematic method to calculate transport coefficients for a meson gas (in particular, we analyze a pion gas) at low temperatures in the context of Chiral Perturbation Theory. Our method is based on the study of Feynman diagrams with a power counting which takes into account collisions in the plasma by means of a non-zero particle width. In this way, we obtain results compatible with analysis of Kinetic Theory with just the leading order diagram. We show the behavior with temperature of electrical and thermal conductivities and shear and bulk viscosities, and we discuss the fundamental role played by unitarity. We obtain that bulk viscosity is negligible against shear viscosity near the chiral phase transition. Relations between the different transport coefficients and bounds on them based on different theoretical approximations are also discussed. We also comment on some applications to heavy-ion collisions.
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