One-particle and few-particle billiards

We study the dynamics of one-particle and few-particle billiard systems in containers of various shapes. In few-particle systems, the particles collide elastically both against the boundary and against each other. In the one-particle case, we investigate the formation and destruction of resonance islands in (generalized) mushroom billiards, which are a recently discovered class of Hamiltonian systems with mixed regular-chaotic dynamics. In the few-particle case, we compare the dynamics in container geometries whose counterpart one-particle billiards are integrable, chaotic, and mixed. One of our findings is that two-, three-, and four-particle billiards confined to containers with integrable one-particle counterparts inherit some integrals of motion and exhibit a regular partition of phase space into ergodic components of positive measure. Therefore, the shape of a container matters not only for noninteracting particles but also for interacting particles.     

Nonequilibrium Gas and Generalized Billiards

Generalized billiards describe nonequilibrium gas, consisting of finitely many particles, that move in a container, whose walls heat up or cool down. Generalized billiards can be considered both in the framework of the Newtonian mechanics and of the relativity theory. In the Newtonian case, a generalized billiard may possess an invariant measure; the Gibbs entropy with respect to this measure is constant. On the contrary, generalized relativistic billiards are always dissipative,and the Gibbs entropy with respect to the same measure grows under some natural conditions. In this article, we find the necessary and sufficient conditions for a generalized Newtonian billiard to possess a smooth invariant measure, which is independent of the boundary action: the corresponding classical billiard should have an additional first integral of special type. In particular,the generalized Sinai billiards do not possess a smooth invariant measure. We then consider generalized billiards inside a ball, which is one of the main examples of the Newtonian generalized billiards which does have an invariant measure. We construct explicitly the invariant measure, and find the conditions for the Gibbs entropy growth for the corresponding relativistic billiard both formonotone and periodic action of the boundary.     

The Boundary Condition in the Classical Derivation of the BBR

The classical derivation of the black body radiation (BBR) spectrum by Boyer was based on an equilibrium mechanism such that in the absence of thermal radiation particles in a container can gain kinetic energy from the random electromagnetic zero point field (ZPF) radiation. Their loss of that energy was to be by means of their collisions with the walls of the container. Theoretically, energy dissipation through collisions with the walls might lead to a divergent kinetic energy value for the particles. This is because the box can be taken large enough to minimize the collisions probability, and that can lead to a particle’s indefinite growth in energy. Furthermore, a derivation of a general phenomenon such as the BBR should be possible without relying on the walls boundary of a box. Therefore, a new boundary condition is proposed here which is related to relativistic effects. It is shown that with the new boundary condition one may still recover the BBR spectrum. A discussion is presented that shows how the new boundary condition is indeed responsible for energy dissipations.     

Hyperbolic billiards with nearly flat focusing boundaries, I

The standard Wojtkowski-Markarian-Donnay-Bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary. This is due to the physical principle that is used in the proofs, the so-called defocusing mechanism of geometrical optics. In this paper we construct examples of hyperbolic billiards with a focusing boundary component of arbitrarily small curvature whose diameter is bounded by a constant independent of that curvature. Our proof employs a nonstandard cone bundle that does not solely use the familiar dispersing and defocusing mechanisms.     

Improved Estimates for Correlations in Billiards

We consider several classes of chaotic billiards with slow (polynomial) mixing rates, which include Bunimovich’s stadium and dispersing billiards with cusps. In recent papers by Markarian and the present authors, estimates on the decay of correlations were obtained that were sub-optimal (they contained a redundant logarithmic factor). We sharpen those estimates by removing that factor.     

The role of dissipation in time-dependent non-integrable focusing billiards

In this study, we compare the dynamical properties of chaotic and nearly integrable time-dependent focusing billiards with elastic and dissipative boundaries. We show that in the system without dissipation the average velocity of particles scales with the number of collisions as ?V∝n(α). In the fully chaotic case, this scaling corresponds to a diffusion process with α≈1/2, whereas in the nearly integrable case, this dependence has a crossover; slow particles accelerate in a slow subdiffusive manner with α<1/2, while acceleration of fast particles is much stronger and their average velocity grows super-diffusively, i.e., α>1/2. Assuming ?V∝n(α) for a non-dissipative system, we obtain that in its dissipative counterpart the average velocity approaches to ?V(fin)∝1/δ(α), where δ is the damping coefficient. So that ?V(fin)∝√1/δ in the fully chaotic billiards, and the characteristics exponents α changes with δ from α(1)>1/2 to α(2)<1/2 in the nearly integrable systems. We conjecture that in the limit of moderate dissipation the chaotic time-depended billiards can accelerate the particles more efficiently. By contrast, in the limit of small dissipations, the nearly integrable billiards can become the most efficient accelerator. Furthermore, due to the presence of attractors in this system, the particles trajectories will be focused in narrow beams with a discrete velocity spectrum.     

Optical billiards for atoms

One of the central paradigms for classical and quantum chaos in conservative systems is the two-dimensional billiard in which particles are confined to a closed region in the plane, undergoing elastic collisions with the walls and free motion in between. We report the first realization of billiards using ultracold atoms bouncing off beams of light. These beams create the desired spatial pattern, forming an "optical billiard." We find excellent agreement between theory and our experimental demonstration of chaotic and stable motion in optical billiards, establishing a new testing ground for classical and quantum chaos.     

垂直振动颗粒混合气体的振荡现象研究

运用事件驱动算法研究颗粒混合物在垂直振动容器中的振荡现象.容器被具有一定高度的隔板分成相等大小的两个小室,并采用半径相差一倍的两种颗粒.研究结果表明,颗粒振荡周期随两种颗粒密度比的减少而急剧增加.通过计算在垂直方向上两种颗粒高度之比随颗粒密度的变化关系,说明决定颗粒振荡与否的主要因素并不是"巴西坚果效应"或"反巴西坚果效应".通过计算颗粒温度,发现颗粒振荡取决于颗粒混合气体的小颗粒温度.当小颗粒温度大于一定值时,颗粒混合气体发生颗粒振荡现象.根据Viridi等提出的流体动力学模型,文中对该模型做出相应的修改,加入密度因子,从而可以解释颗粒振荡周期与颗粒密度比的关系.     

Generation of large-amplitude solitons in the extended Korteweg-de Vries equation

We study the extended Korteweg-de Vries equation, that is, the usual Korteweg-de Vries equation but with the inclusion of an extra cubic nonlinear term, for the case when the coefficient of the cubic nonlinear term has an opposite polarity to that of the coefficient of the linear dispersive term. As this equation is integrable, the number and type of solitons formed can be determined from an appropriate spectral problem. For initial disturbances of small amplitude, the number and type of solitons generated is similar to the well-known situation for the Korteweg-de Vries equation. However, our interest here is in initial disturbances of larger amplitude, for which there is the possibility of the generation of large-amplitude "table-top" solitons as well as small-amplitude solitons similar to the solitons of the Korteweg-de Vries equation. For this case, and in contrast to some earlier results which assumed that an initial disturbance in the shape of a rectangular box would be typical, we show that the number and type of solitons formed depend crucially on the disturbance shape, and change drastically when the initial disturbance is changed from a rectangular box to a "sech"-profile. (c) 2002 American Institute of Physics.     

Relaxation of ideal classical particles in a one‐dimensional box

We study the deterministic dynamics of non‐interacting classical gas particles confined to a one‐dimensional box as a pedagogical toy model for the relaxation of the Boltzmann distribution towards equilibrium. Hard container walls alone induce a uniform distribution of the gas particles at large times. For the relaxation of the velocity distribution we model the dynamical walls by independent scatterers. The Markov property guarantees a stationary but not necessarily thermal velocity distribution for the gas particles at large times. We identify the conditions for physical walls where the stationary velocity distribution is the Maxwell distribution. For our numerical simulation we represent the wall particles by independent harmonic oscillators. The corresponding dynamical map for oscillators with a fixed phase (Fermi–Ulam accelerator) is chaotic for mesoscopic box dimensions.     

New mechanism of chaos in triangular billiards

A new mechanism of weak chaos in triangular billiards has been proposed owing to the effect of cutting of beams of rays. A similar mechanism is also implemented in other polygonal billiards. Cutting of beams results in the separation of initially close rays at a finite angle by jumps in the process of reflections of beams at the vertices of a billiard. The opposite effect of joining of beams of rays occurs in any triangular billiard along with cutting. It has been shown that the cutting of beams has an absolute character and is independent of the form of a triangular billiard or the parameters of a beam. On the contrary, joining has a relative character and depends on the commensurability of the angles of the triangle with π. Joining always suppresses cutting in triangular billiards whose angles are commensurable with π. For this reason, their dynamics cannot be chaotic. In triangular billiards whose angles are rationally incommensurable with π, cutting always dominates, leading to weak chaos. The revealed properties are confirmed by numerical experiments on the phase portraits of typical triangular billiards.     

Separation of particles in time-dependent focusing billiards

We consider a family of stadium-like billiards with time-dependent boundaries. Two different cases of time dependence are studied: (i) the fixed boundary approximation and (ii) the exact model which takes into account the motion of the boundary. It is shown that when the billiards possess strong chaotic properties, the sequence of their boundary perturbations is the Fermi acceleration phenomenon which is three times larger than in the case of the fixed boundary approximation. However, weak mixing in such billiards leads to particle separation. Depending on the initial velocity three different things occur: (i) the particle ensemble may accelerate; (ii) the average velocity may stay constant or (iii) it may even decrease.     

Mixing of wavefunctions in rectangular microwave billiards

A set-up is described allowing the automatic registration of wavefunctions of quasi-two-dimensional microwave billiards of arbitrary shape. Tests of the apparatus with rectangular shaped billiards showed that a precision of some percent in the wavefunction amplitudes can be obtained, as far as isolated resonances are considered. For the case of overlapping resonances, however, the measurement yields wavefunctions which are close to a symmetric and an antisymmetric linear combination of the original rectangle eigenfunctions. The cause for this at first sight surprising result is discussed. Received 21 January 2000     

Quantum transport and classical dynamics in open billiards

We report numerical results of an investigation of quantum transport for a weakly opened integrable circle and chaotic stadium billiards with a pair of conducting leads. While the statistics of spacings of resonance energies commonly follow the Wigner (GOE)-like distribution, the electric conductance as a function of the Fermi wavenumber shows characteristic noisy fluctuations associated with a typical set of classical orbits unique for both billiards. The wavenumber autocorrelation for the conductance is stronger in the stadium than the circle billiard, which we show is related to the length spectrum of classical short orbits. We propose an explanation of these contrasts in terms of the effect of phase decoherence due to the underlying chaotic dynamics.     

Release potential of single-wall carbon nanotubes produced by super-growth method during manufacturing and handling

We investigated the release potential of single-wall carbon nanotubes (CNTs) produced by the super-growth method during their manufacturing and handling processes at a research facility. We generally sampled air at points both outside and inside of protective enclosures such as a glove box and fume hood. Sampling the air outside of the enclosures was intended to evaluate the actual exposure of workers to CNTs, while sampling the air inside the enclosures was performed to quantify the release of CNTs to the air in order to estimate the potential exposure of workers without protection. The results revealed that airborne CNTs were generated when (1) CNTs were separated from the substrates using a spatula and placed in a container in a glove box; (2) an air gun was used to clean the air filters (containing dust that included CNTs) of a vacuum cleaner; (3) a vacuum cleaner was used to collect CNTs (emission with exhaust air from the cleaner); (4) the container of CNTs was opened; and (5) CNTs in the bin of the cleaner were transferred to a container. In these processes, airborne CNTs were only found inside the enclosures, except for a small amount of CNTs released from the glove box when it was opened. Electron microscopic observations of aerosol particles found CNT clusters, which were fragments of CNT forests, with sizes ranging from submicrometers to tens of micrometers.     

Integrability and ergodicity of classical billiards in a magnetic field

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not trivially integrable can be described by a bouncing map. We compute a general expression for the Jacobian matrix of this map, which allows us to determine stability and bifurcation values of specific periodic orbits. In some cases, the bouncing map is a twist map and admits a generating function. We give a general form for this function which is useful to do perturbative calculations and to classify periodic orbits. We prove that billiards in convex domains with sufficiently smooth boundaries possess invariant tori corresponding to skipping trajectories. Moreover, in strong field we construct adiabatic invariants over exponentially large times. To some extent, these results remain true for a class of nonconvex billiards. On the other hand, we present evidence that the billiard in a square is ergodic for some large enough values of the magnetic field. A numerical study reveals that the scattering on two circles is essentially chaotic.     

气泡室中"胚胎"气泡的联并成长为可见气泡的理论计算

借助岛的联并理论,可以很好地解决气泡室中"胚胎"气泡成长为可见气泡问题.理论计算表明,联并后的大"胚胎"气泡在成长为可见气泡的过程中,气泡的半径不仅与工作物质如液体的表面张力系数、饱和蒸汽压和流体的沸点有关,而且还与"胚胎 "气泡从其周围吸收热量和"胚胎"气泡联并的个数有关.理论上可以合理解释能量相同的中子和质子入射到气泡室所产生的径迹粗短;也可以合理解释电荷数较多的入射粒子较能量相同但电荷数不同的入射粒子,其在气泡室中径迹上气泡的半径要大.     

Gaussian pre-filtering for uncertainty minimization in digital image correlation using numerically-designed speckle patterns

This paper discusses the effect of pre-processing image blurring on the uncertainty of two-dimensional digital image correlation (DIC) measurements for the specific case of numerically-designed speckle patterns having particles with well-defined and consistent shape, size and spacing. Such patterns are more suitable for large measurement surfaces on large-scale specimens than traditional spray-painted random patterns without well-defined particles. The methodology consists of numerical simulations where Gaussian digital filters with varying standard deviation are applied to a reference speckle pattern. To simplify the pattern application process for large areas and increase contrast to reduce measurement uncertainty, the speckle shape, mean size and on-center spacing were selected to be representative of numerically-designed patterns that can be applied on large surfaces through different techniques (e.g., spray-painting through stencils). Such “designer patterns” are characterized by well-defined regions of non-zero frequency content and non-zero peaks, and are fundamentally different from typical spray-painted patterns whose frequency content exhibits near-zero peaks. The effect of blurring filters is examined for constant, linear, quadratic and cubic displacement fields. Maximum strains between ±250 and ±20,000 µε are simulated, thus covering a relevant range for structural materials subjected to service and ultimate stresses. The robustness of the simulation procedure is verified experimentally using a physical speckle pattern subjected to constant displacements. The stability of the relation between standard deviation of the Gaussian filter and measurement uncertainty is assessed for linear displacement fields at varying image noise levels, subset size, and frequency content of the speckle pattern. It is shown that bias error as well as measurement uncertainty are minimized through Gaussian pre-filtering. This finding does not apply to typical spray-painted patterns without well-defined particles, for which image blurring is only beneficial in reducing bias errors.