Carl P. Dettmann

Hard ball systems and the Lorentz gas

Part I. Mathematics: 1. D. Burago, S. Ferleger, A. Kononenko: A Geometric Approach to Semi-Dispersing Billiards.- 2. T. J. Murphy, E. G. D. Cohen: On the Sequences of Collisions Among Hard Spheres in Infinite Spacel- 3. N. Simanyi: Hard Ball Systems and Semi-Dispersive Billiards: Hyperbolicity and Ergodicity.- 4. N. Chernov, L.-S. Young: Decay of Correlations for Lorentz Gases and Hard Balls.- 5. N. Chernov: Entropy Values and Entropy Bounds.- 6. L. A. Bunimovich: Existence of Transport Coefficients.- 7. C. Liverani: Interacting Particles.- 8. J. L. Lebowitz, J. Piasecki and Ya. G. Sinai: Scaling Dynamics of a Massive Piston in an Ideal Gas .- Part II. Physics: 1. H. van Beijeren, R. van Zon, J. R. Dorfman: Kinetic Theory Estimates for the Kolmogorov-Sinai Entropy, and the Largest Lyapunov Exponents for Dilute, Hard-Ball Gases and for Dilute, Random Lorentz Gases.- 2. H. A. Posch and R. Hirschl: Simulation of Billiards and of Hard-Body Fluids.- 3. C. P. Dettmann: The Lorentz Gas: a Paradigm for Nonequilibrium Stationary States.- 4. T. Tl, J. Vollmer: Entropy Balance, Multibaker Maps, and the Dynamics of the Lorentz Gas.- Appendix: 1. D. Szasz: Boltzmanns Ergodic Hypothesis, a Conjecture for Centuries?

Thermostats for “Slow” Configurational Modes

Abstract Thermostats are dynamical equations used to model thermodynamic variables such as temperature and pressure in molecular simulations. For computationally intensive problems such as the simulation of biomolecules, we propose to average over fast momentum degrees of freedom and construct thermostat equations in configuration space. The equations of motion are deterministic analogues of the Smoluchowski dynamics in the method of stochastic differential equations.

Full connectivity: corners, edges and faces

We develop a cluster expansion for the probability of full connectivity of high density random networks in confined geometries. In contrast to percolation phenomena at lower densities, boundary effects, which have previously been largely neglected, are not only relevant but dominant. We derive general analytical formulas that show a persistence of universality in a different form to percolation theory, and provide numerical confirmation. We also demonstrate the simplicity of our approach in three simple but instructive examples and discuss the practical benefits of its application to different models.

Microscopic Chaos and Diffusion

We investigate the connections between microscopic chaos, defined on a dynamical level and arising from collisions between molecules, and diffusion, characterized by a mean square displacement proportional to the time. We use a number of models involving a single particle moving in two dimensions and colliding with fixed scatterers. We find that a number of microscopically nonchaotic models exhibit diffusion, and that the standard methods of chaotic time series analysis are ill suited to the problem of distinguishing between chaotic and nonchaotic microscopic dynamics. However, we show that periodic orbits play an important role in our models, in that their different properties in our chaotic and nonchaotic models can be used to distinguish them at the level of time series analysis, and in systems with absorbing boundaries. Our findings are relevant to experiments aimed at verifying the existence of chaoticity and related dynamical properties on a microscopic level in diffusive systems.

Unidirectional emission from circular dielectric microresonators with a point scatterer

Circular microresonators are micron-sized dielectric disks embedded in material of lower refractive index. They possess modes of extremely high Q-factors (low-lasing thresholds), which makes them ideal candidates for the realization of miniature laser sources. They have, however, the disadvantage of isotropic light emission caused by the rotational symmetry of the system. In order to obtain high directivity of the emission while retaining high Q-factors, we consider a microdisk with a pointlike scatterer placed off-center inside of the disk. We calculate the resulting resonant modes and show that some of them possess both of the desired characteristics. The emission is predominantly in the direction opposite to the scatterer. We show that classical ray optics is a useful guide to optimizing the design parameters of this system. We further find that exceptional points in the resonance spectrum influence how complex resonance wave numbers change if system parameters are varied.

Open Circular Billiards and the Riemann Hypothesis

A comparison of escape rates from one and from two holes in an experimental container (e.g., a laser trap) can be used to obtain information about the dynamics inside the container. If this dynamics is simple enough one can hope to obtain exact formulas. Here we obtain exact formulas for escape from a circular billiard with one and with two holes. The corresponding quantities are expressed as sums over zeros of the Riemann zeta function. Thus we demonstrate a direct connection between recent experiments and a major unsolved problem in mathematics, the Riemann hypothesis.

Impact of boundaries on fully connected random geometric networks

Many complex networks exhibit a percolation transition involving a macroscopic connected component, with universal features largely independent of the microscopic model and the macroscopic domain geometry. In contrast, we show that the transition to full connectivity is strongly influenced by details of the boundary, but observe an alternative form of universality. Our approach correctly distinguishes connectivity properties of networks in domains with equal bulk contributions. It also facilitates system design to promote or avoid full connectivity for diverse geometries in arbitrary dimension.

Microscopic chaos from Brownian motion

Gaspard et al. have analysed a time series of the positions of a brownian particle in a liquid, and claimed that it provides empirical evidence for microscopic chaos on a molecular scale. An accompanying comment emphasized the fundamental nature of the experiment. Here we show that virtually identical results can be obtained by analysing a corresponding numerical time series of a particle in a manifestly microscopically non-chaotic system.

Peeping at chaos: nondestructive monitoring of chaotic systems by measuring long-time escape rates

One or more small holes provide non-destructive windows to observe corresponding closed systems, for example by measuring long time escape rates of particles as a function of hole sizes and positions. To leading order, the escape rate of chaotic systems is proportional to the hole size and independent of position. Here we give exact formulas for the subsequent terms, as sums of correlation functions; these depend on hole size and position, hence yield information on the closed system dynamics. Conversely, the theory can be readily applied to experimental design, for example to control escape rates.

Survival probability for the stadium billiard

Abstract 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 C o n s t a n t × ( t i m e ) − 1 ; here we obtain an explicit expression for the Constant.