arXiv subjects

Carl P. Dettmann

Publications and source records attributed to Carl P. Dettmann.

79 records · Page 5Linked to original sources

arXiv2010

Transmission and Reflection in the Stadium Billiard: Time-dependent asymmetric transport

We investigate the transmission and reflection survival probabilities for the chaotic stadium billiard with two holes placed asymmetrically. Classically, these distributions are shown to have algebraic or exponential decays depending on the choice of injecting hole and exact expressions are given for the first time and confirmed numerically. As there is no reported quantum theoretical or experimental analogue we propose a model for experimental observation of the asymmetric transport using semiconductor nano-structures and comment on the relevant quantum time-scales.

nlin.CD↗

arXiv2010

Recent advances in open billiards with some open problems

Much recent interest has focused on "open" dynamical systems, in which a classical map or flow is considered only until the trajectory reaches a "hole", at which the dynamics is no longer considered. Here we consider questions pertaining to the survival probability as a function of time, given an initial measure on phase space. We focus on the case of billiard dynamics, namely that of a point particle moving with constant velocity except for mirror-like reflections at the boundary, and give a number of recent results, physical applications and open problems.

nlin.CD↗

arXiv2009

Survival Probability for the Stadium Billiard

We consider the open stadium billiard, consisting of two semicircles joined by parallel straight sides with one hole situated somewhere on one of the sides. Due to the hyperbolic nature of the stadium billiard, the initial decay of trajectories, due to loss through the hole, appears exponential. However, some trajectories (bouncing ball orbits) persist and survive for long times and therefore form the main contribution to the survival probability function at long times. Using both numerical and analytical methods, we concur with previous studies that the long-time survival probability for a reasonably small hole drops like Constant/time; here we obtain an explicit expression for the Constant.

nlin.CD↗

arXiv2006

Escape from a circle and Riemann hypotheses

We consider open circular billiards with one and with two holes. The exact formulas for escape are obtained which involve the Riemann zeta function and Dirichlet L-functions. It is shown that the problem of finding the exact asymptotics in the small hole limit for escape in some of these billiards is equivalent to the Riemann hypothesis.

math.DS↗

arXiv2003

Fractal asymptotics

Recent advances in the periodic orbit theory of stochastically perturbed systems have permitted a calculation of the escape rate of a noisy chaotic map to order 64 in the noise strength. Comparison with the usual asymptotic expansions obtained from integrals and with a previous calculation of the electrostatic potential of exactly selfsimilar fractal charge distributions, suggests a remarkably accurate form for the late terms in the expansion, with parameters determined independently from the fractal repeller and the critical point of the map. Two methods give a precise meaning to the asymptotic expansion, Borel summation and Shafer approximants. These can then be compared with the escape rate as computed by alternative methods.

nlin.CD↗

arXiv2002

Lyapunov spectra of periodic orbits for a many-particle system

The Lyapunov spectrum corresponding to a periodic orbit for a two dimensional many particle system with hard core interactions is discussed. Noting that the matrix to describe the tangent space dynamics has the block cyclic structure, the calculation of the Lyapunov spectrum is attributed to the eigenvalue problem of a 16x16 reduced matrices regardless of the number of particles. We show that there is the thermodynamic limit of the Lyapunov spectrum in this periodic orbit. The Lyapunov spectrum has a step structure, which is explained by using symmetries of the reduced matrices.

nlin.CD↗

arXiv1997

Computing the diffusion coefficient for intermittent maps: Resummation of stability ordered cycle expansions

We compute the diffusion coefficient and the Lyapunov exponent for a diffusive intermittent map by means of cycle expansion of dynamical zeta functions. The asymptotic power law decay of the coefficients of the relevant power series are known analytically. This information is used to resum these power series into generalized power series around the algebraic branch point whose immediate vicinity determines the desired quantities. In particular we consider a realistic situation where all orbits with instability up to a certain cutoff are known. This implies that only a few of the power series coefficients are known exactly and a lot of them are only approximately given. We develop methods to extract information from these stability ordered cycle expansions and compute accurate values for the diffusion coefficient and the Lyapunov exponent. The method works successfully all the way up to a phase transition of the map, beyond which the diffusion coefficient and Lyapunov exponent are both zero.

chao-dyn↗