Transmission and reflection in the stadium billiard: time-dependent asymmetric transport

Characterizing a transition from limited to unlimited diffusion in energy for a time-dependent stochastic billiard

We explore Fermi acceleration in a stochastic oval billiard which shows unlimited to limited diffusion in energy when passing from the free to the dissipative case. We provide evidence for a transition from limited to unlimited energy growth taking place while detuning the corresponding restitution coefficient responsible for the degree of dissipation. A corresponding order parameter is suggested, and its susceptibility is shown to diverge at the critical point. We show that this order parameter is also be applicable to the periodically driven oval billiard and discuss the elementary excitation of the controlled diffusion process.

Spherical billiards with almost complete escape

A dynamical billiard consists of a point particle moving uniformly except for mirror-like collisions with the boundary. Recent work has described the escape of the particle through a hole in the boundary of a circular or spherical billiard, making connections with the Riemann Hypothesis. Unlike the circular case, the sphere with a single hole leads to a non-zero probability of never escaping. Here, we study variants in which almost all initial conditions escape, with multiple small holes or a thin strip. We show that equal spacing of holes around the equator is an efficient means of ensuring almost complete escape and study the long time survival probability for small holes analytically and numerically. We find that it approaches a universal function of a single parameter, hole area multiplied by time.

Effectiveness of glass beads for plating cell cultures

Cell plating, the spreading out of a liquid suspension of cells on a surface followed by colony growth, is a common laboratory procedure in microbiology. Despite this, the exact impact of its parameters on colony growth has not been extensively studied. A common protocol involves the shaking of glass beads within a Petri dish containing solid growth media. We investigated the effects of multiple parameters in this protocol: the number of beads, the shape of movement, and the number of movements. Standard suspensions of Escherichia coli were spread while varying these parameters to assess their impact on colony growth. Results were assessed by a variety of metrics: the number of colonies, the mean distance between closest colonies, and the variability and uniformity of their spatial distribution. Finally, we devised a mathematical model of shifting billiard to explain the heterogeneities in the observed spatial patterns. Exploring the parameters that affect the most fundamental techniques in microbiology allows us to better understand their function, giving us the ability to precisely control their outputs for our exact needs.

Internal chaotic sea structure and chaos-chaos intermittency in Hamiltonian systems

In this paper, we study the inhomogeneity of chaotic sea properties far from islands in billiardlike systems and its influence on distributions of particle's return times. A visibly homogeneous chaotic sea at certain parameters has a nontrivial internal structure, in particular, being divided into two chaotic phases with different properties. These phases are not separated by any obstacles, neither in phase nor in configuration spaces, and are partially overlaying. The emergence of a chaotic sea structure may be explained by the existence of remnants of integrable behavior, like sites of regular trajectories of broken islands of stability built into the chaotic sea. In the case of such chaotic seas, we find distributions of return times with two main sites of exponential decay.

Thin structure of the transit time distributions of open billiards

It is known that typical open billiards distribution of transit times is an exponentially decaying function, possibly with a power-law tail. In the paper we show that on small scales some of such distributions change their appearance. These distributions contain a quasiperiodic thin structure, which carries a significant amount of information about the system. Origin and properties of this structure are discussed.

Escape dynamics through a continuously growing leak

We formulate a model that describes the escape dynamics in a leaky chaotic system in which the size of the leak depends on the number of the in-falling particles. The basic motivation of this work is the astrophysical process, which describes the planetary accretion. In order to study the dynamics generally, the standard map is investigated in two cases when the dynamics is fully hyperbolic and in the presence of Kolmogorov-Arnold-Moser islands. In addition to the numerical calculations, an analytic solution to the temporal behavior of the model is also derived. We show that in the early phase of the leak expansion, as long as there are enough particles in the system, the number of survivors deviates from the well-known exponential decay. Furthermore, the analytic solution returns the classical result in the limiting case when the number of particles does not affect the leak size.

Thermodynamics of a time-dependent and dissipative oval billiard: A heat transfer and billiard approach

We study some statistical properties for the behavior of the average squared velocity-hence the temperature-for an ensemble of classical particles moving in a billiard whose boundary is time dependent. We assume the collisions of the particles with the boundary of the billiard are inelastic, leading the average squared velocity to reach a steady-state dynamics for large enough time. The description of the stationary state is made by using two different approaches: (i) heat transfer motivated by the Fourier law and (ii) billiard dynamics using either numerical simulations and theoretical description.

Rotating leaks in the stadium billiard

The open stadium billiard has a survival probability, P(t), that depends on the rate of escape of particles through the leak. It is known that the decay of P(t) is exponential early in time while for long times the decay follows a power law. In this work, we investigate an open stadium billiard in which the leak is free to rotate around the boundary of the stadium at a constant velocity, ω. It is found that P(t) is very sensitive to ω. For certain ω values P(t) is purely exponential while for other values the power law behaviour at long times persists. We identify three ranges of ω values corresponding to three different responses of P(t). It is shown that these variations in P(t) are due to the interaction of the moving leak with Marginally Unstable Periodic Orbits (MUPOs).

Billiard with a handle

We consider an open billiard with two holes, connected by a handle. The central billiard is chosen so that its closed form's islands of stability occupy a significant part of the phase space. Holes destroy these islands, which leads to almost all trajectories of the system being interleaved. We also study the unbalanced flow of billiard particles through the handle, which appears only after a small border site of nonspecular reflection is added to the system. With this site our system is rather a ratchet of a different type, since the site does not produce an explicitly acting force or violate the reversibility of trajectories.

Transport of chaotic trajectories from regions distant from or near to structures of regular motion of the Fermi-Ulam model

The chaotic portion of phase space of the simplified Fermi-Ulam model is studied under the context of transport of trajectories in two scenarios: (i) the trajectories are originated from a region distant from the islands of regular motion and are transported to a region located at a high portion of phase space and (ii) the trajectories are originated from chaotic regions around the islands of regular motion and are transported to other regions around islands of regular motion. The transport is investigated in terms of the observables histogram of transport and survival probability. We show that the histogram curves are scaling invariant and we organize the survival probability curves in four kinds of behavior, namely (a) transition from exponential decay to power law decay, (b) transition from exponential decay to stretched exponential decay, (c) transition from an initial fast exponential decay to a slower exponential decay, and (d) a single exponential decay. We show that, depending on choice of the regions of origin and destination, the transport process is weakly affected by the stickiness of trajectories around islands of regular motion.

Scalar wave scattering in spherical cavity resonator with conical channels

We study the scalar wave scattering off the spherical cavity resonator with two finite-length conical channels attached. We use the boundary wall method to explore the response of the system to changes in control parameters, such as the size of the structure and the angular width of the input and output channels, as well as their relative angular position. We found that the system is more sensitive to changes in the input channel, and a standing wave phase distribution occurs within the cavity for nontransmitting values of the incident wave number. We also saw that an optical vortex can travel unaffected through the system with aligned channels.

Equation of state of an ideal gas with nonergodic behavior in two connected vessels

We consider a two-dimensional collisionless ideal gas in the two vessels connected through a small hole. One of them is a well-behaved chaotic billiard, another one is known to be nonergodic. A significant part of the second vessel's phase space is occupied by an island of stability. In the works of Zaslavsky and coauthors, distribution of Poincaré recurrence times in similar systems was considered. We study the gas pressure in the vessels; it is uniform in the first vessel and not uniform in second one. An equation of the gas state in the first vessel is obtained. Despite the very different phase-space structure, behavior of the second vessel is found to be very close to the behavior of a good ergodic billiard but of different volume. The equation of state differs from the ordinary equation of ideal gas state by an amendment to the vessel's volume. Correlation of this amendment with a share of the phase space under remaining intact islands of stability is shown.

New developments in classical chaotic scattering

Classical chaotic scattering is a topic of fundamental interest in nonlinear physics due to the numerous existing applications in fields such as celestial mechanics, atomic and nuclear physics and fluid mechanics, among others. Many new advances in chaotic scattering have been achieved in the last few decades. This work provides a current overview of the field, where our attention has been mainly focused on the most important contributions related to the theoretical framework of chaotic scattering, the fractal dimension, the basins boundaries and new applications, among others. Numerical techniques and algorithms, as well as analytical tools used for its analysis, are also included. We also show some of the experimental setups that have been implemented to study diverse manifestations of chaotic scattering. Furthermore, new theoretical aspects such as the study of this phenomenon in time-dependent systems, different transitions and bifurcations to chaotic scattering and a classification of boundaries in different types according to symbolic dynamics are also shown. Finally, some recent progress on chaotic scattering in higher dimensions is also described.

Faster than expected escape for a class of fully chaotic maps

We investigate the dependence of the escape rate on the position of a hole placed in uniformly hyperbolic systems admitting a finite Markov partition. We derive an exact periodic orbit formula for finite size Markov holes which differs from other periodic expansions in the literature and can account for additional distortion to maps with piecewise constant expansion rate. Using asymptotic expansions in powers of hole size we show that for systems conjugate to the binary shift, the average escape rate is always larger than the expectation based on the hole size. Moreover, we show that in the small hole limit the difference between the two decays like a known constant times the square of the hole size. Finally, we relate this problem to the random choice of hole positions and we discuss possible extensions of our results to non-Markov holes as well as applications to leaky dynamical networks.

Quantifying intermittency in the open drivebelt billiard

A "drivebelt" stadium billiard with boundary consisting of circular arcs of differing radius connected by their common tangents shares many properties with the conventional "straight" stadium, including hyperbolicity and mixing, as well as intermittency due to marginally unstable periodic orbits (MUPOs). Interestingly, the roles of the straight and curved sides are reversed. Here, we discuss intermittent properties of the chaotic trajectories from the point of view of escape through a hole in the billiard, giving the exact leading order coefficient lim(t→∞)tP(t) of the survival probability P(t) which is algebraic for fixed hole size. However, in the natural scaling limit of small hole size inversely proportional to time, the decay remains exponential. The big distinction between the straight and drivebelt stadia is that in the drivebelt case, there are multiple families of MUPOs leading to qualitatively new effects. A further difference is that most marginal periodic orbits in this system are oblique to the boundary, thus permitting applications that utilise total internal reflection such as microlasers.

Dependence of chaotic diffusion on the size and position of holes

A particle driven by deterministic chaos and moving in a spatially extended environment can exhibit normal diffusion, with its mean square displacement growing proportional to the time. Here, we consider the dependence of the diffusion coefficient on the size and the position of areas of phase space linking spatial regions ('holes') in a class of simple one-dimensional, periodically lifted maps. The parameter dependent diffusion coefficient can be obtained analytically via a Taylor-Green-Kubo formula in terms of a functional recursion relation. We find that the diffusion coefficient varies non-monotonically with the size of a hole and its position, which implies that a diffusion coefficient can increase by making the hole smaller. We derive analytic formulas for small holes in terms of periodic orbits covered by the holes. The asymptotic regimes that we observe show deviations from the standard stochastic random walk approximation. The escape rate of the corresponding open system is also calculated. The resulting parameter dependencies are compared with the ones for the diffusion coefficient and explained in terms of periodic orbits.