The effect of Lagrangian chaos on locking bifurcations in shear flows

The effect of an externally imposed perturbation on an unstable or weakly stable shear flow is investigated, with a focus on the role of Lagrangian chaos in the bifurcations that occur. The external perturbation is at rest in the laboratory frame and can form a chain of resonances or cat's eyes where the initial velocity v(x0)(y) vanishes. If in addition the shear profile is unstable or weakly stable to a Kelvin-Helmholtz instability, for a certain amplitude of the external perturbation there can be an unlocking bifurcation to a nonlinear wave resonant around a different value of y, with nonzero phase velocity. The interaction of the propagating nonlinear wave with the external perturbation leads to Lagrangian chaos. We discuss results based on numerical simulations for different amplitudes of the external perturbation. The response to the external perturbation is strong, apparently because of non-normality of the linear operator, and the unlocking bifurcation is hysteretic. The results indicate that the observed Lagrangian chaos is responsible for a second bifurcation occurring at larger external perturbation, locking the wave to the wall. This bifurcation is nonhysteretic. The mechanism by which the chaos leads to locking in this second bifurcation is by means of chaotic advective transport of momentum from one chain of resonances to the other (Reynolds stress) and momentum transport to the vicinity of the wall via chaotic scattering. These results suggest that locking of waves in rotating tank experiments in the presence of two unstable modes is due to a similar process. (c) 2002 American Institute of Physics.     

Resonant scattering of a lepton by a lepton in the pulsed light field

The process of resonant scattering of a lepton by a lepton in the field of pulsed light wave in general relativistic case is studied theoretically. The approximation when a pulsewidth is considerably greater than the characteristic time of wave oscillation is considered. The analytical expression for the resonant differential cross-section of concerned process is derived within the framework of the first order of the perturbation theory with respect to external laser field; it contains the resonant peak, the altitude and the width of which are defined by the external pulsed wave characteristics. It is demonstrated, that the resonant differential cross-section of scattering of an electron by an electron (an electron by a positron, an electron by a muon) in the pulsed light field may be several orders of magnitude greater than the corresponding one in the external field absence. The derived results may be verified experimentally for example by scientific facility at SLAC National Accelerator Laboratory.     

Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems

The dynamics in weakly chaotic Hamiltonian systems strongly depends on initial conditions (ICs) and little can be affirmed about generic behaviors. Using two distinct Hamiltonian systems, namely one particle in an open rectangular billiard and four particles globally coupled on a discrete lattice, we show that in these models, the transition from integrable motion to weak chaos emerges via chaotic stripes as the nonlinear parameter is increased. The stripes represent intervals of initial conditions which generate chaotic trajectories and increase with the nonlinear parameter of the system. In the billiard case, the initial conditions are the injection angles. For higher-dimensional systems and small nonlinearities, the chaotic stripes are the initial condition inside which Arnold diffusion occurs.     

Fermi Acceleration in driven relativistic billiards

We show numerical experiments of driven billiards using special relativity. We have the remarkable fact that for the relativistic driven circular and annular concentric billiards, depending on initial conditions and parameters, we observe Fermi Acceleration, absent in the Newtonian case. The velocity for these cases tends to the speed of light very quickly. We find that for the annular eccentric billiard the initial velocity grows for a much longer time than the concentric annular billiard until it asymptotically reach c.     

Three particles on a ring

We study the ergodic properties of three classical particles of unequal mass moving on a ring and colliding like hard points.Although it is generally believed that this system is ergodic, we show that extensive numerical calculations do not support this belief.This may be connected with our discovery that for vanishing total momentum, any initial state with one particle at rest is periodic, independent of the mass values.The analogous periodicity for two unequal mass particles bouncing in a one-dimensional box can be understood on the basis of a remarkable property of a billiard in the form of a rectangular triangle.     

Classical dynamics and particle transport in kicked billiards

We study nonlinear dynamics of the kicked particle whose motion is confined by square billiard. The kick source is considered as localized at the center of a square with central symmetric spatial distribution. It is found that ensemble averaged energy of the particle diffusively grows as a function of time. This growth is much more extensive than that of kicked rotor energy. It is shown that momentum transfer distribution in a kicked billiard is considerably different than that for kicked free particle. Time-dependence of the ensemble averaged energy for different localizations of the kick source is also explored. It is found that changing of localization does not lead to crucial changes in the time-dependence of the energy. Also, escape and transport of particles are studied by considering a kicked open billiard with one and three holes, respectively. It is found that for the open billiard with one hole the number of (non-interacting) billiard particles decreases according to exponential law.     

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.     

Three unequal masses on a ring and soft triangular billiards

The dynamics of three soft interacting particles on a ring is shown to correspond to the motion of one particle inside a soft triangular billiard. The dynamics inside the soft billiard depends only on the masses ratio between particles and softness ratio of the particles interaction. The transition from soft to hard interactions can be appropriately explored using potentials for which the corresponding equations of motion are well defined in the hard wall limit. Numerical examples are shown for the soft Toda-like interaction and the error function.     

Tunable fermi acceleration in the driven elliptical billiard

We explore the dynamical evolution of an ensemble of noninteracting particles propagating freely in an elliptical billiard with harmonically driven boundaries. The existence of Fermi acceleration is shown thereby refuting the established assumption that smoothly driven billiards whose static counterparts are integrable do not exhibit acceleration dynamics. The underlying mechanism based on intermittent phases of laminar and stochastic behavior of the strongly correlated angular momentum and velocity motion is identified and studied with varying parameters. The diffusion process in velocity space is shown to be anomalous and we find that the corresponding characteristic exponent depends monotonically on the breathing amplitude of the billiard boundaries. Thus it is possible to tune the acceleration law in a straightforwardly controllable manner.     

Thermal rectification in billiardlike systems

We study the thermal rectification phenomenon in billiard systems with interacting particles. This interaction induces a local dynamical response of the billiard to an external thermodynamic gradient. To explain this dynamical effect we study the steady state of an asymmetric billiard in terms of the particle and energy reflection coefficients. This allows us to obtain expressions for the region in parameter space where large thermal rectifications are expected. Our results are confirmed by extensive numerical simulations.     

Transport in polygonal billiards

We present in this work a numerical study of the dynamics of ensembles of point particles within a polygonal billiard chain. This billiard is a system with no exponential instability. Our numerical results suggest that some members of the family exhibit normal diffusive behavior while others present anomalous diffusion. Our conclusions are drawn from the numerical evaluation of the mean square displacement, the velocity autocorrelation function and its spectral analysis. Furthermore we analyze the properties of the incoherent scattering function. The super Burnett coefficient seems to be ill defined in all systems. The multifractal analysis of the spectrum of the velocity autocorrelation functions is also reported. Finally, we study the heat conduction in our polygonal chain.     

Power law decay of correlations in a billiard problem

A billiard problem with boundary arcs that meet tangentially is studied both analytically and numerically. It is shown that the presence of tangential vertices leads to velocity correlations which decay like 1/n wheren is the number of collisions. This result contrasts with related billiard and Lorentz models where velocity correlations decay exponentially.     

三角形反射系统动力学模型研究

建立了三角形反射系统的动力学模型,在此基础上考察了系统轨迹的稳定性.结果表明,三角形反射系统运行轨迹对初始条件不敏感,它既不收敛也不发散.并对三角形反射系统中的一些非线性动力学现象进行了讨论.     

LiNbO3晶体-熔体界面的失稳及向胞状界面演化的实验研究

利用周期旋转生长条纹作为时标(timemarker),研究了直拉法掺钇LiNbO₃的各向异性生长系统中晶体-熔体界面的失稳及向胞状界面的演化。测得平界面失稳的临界条件,观测到失稳初期的界面上存在的两种干扰,即正弦式干扰和正弦式行波干扰。实验结果表明,平界面失稳后,经历正弦干扰、干扰振幅的增长、干扰的小面化、干扰的合并,最后演化为稳态胞状界面。实验观测还表明,稳态胞状界面的波长甚大于界面初始干扰的波长,且为初始干扰波长的整数倍;以及同一系统中平界面向胞状界面转变的临界生长速度小于胞状界面向平界面转变的临界速度,这表明小面化的胞比非小面的平面更为稳定。 关键词：     

Possibility of electrical conduction in filled bands

It is shown that a moving neutral particle interacting with electrons may cause an “electron drag” within a filled band. The calculation uses perturbation theory and periodic boundary conditions and is based on the one-electron model. WithN being the number and ¯v the average velocity of the electrons, one finds that for largeN the electronic velocity sumN¯v induced by the motion of the neutral particle is independent ofN, i.e. of the size of the system. The lowest-order contributions toN¯v that do not necessarily vanish are seen to be those of second order in the interaction potential. These second-order contributions are studied. In a simple one-dimensional model they are found to be, in fact, not necessarily zero and to be proportional to the velocity of the neutral particle. An order-of-magnitude formula forN¯v is derived for this case. The calculation suggests that mobile neutral particles may act as charge carriers, their effective charge possibly being much smaller than the elementary charge. In real systems, neutral particles which interact with electrons might be represented by phonons and excitons.     

Decay of energy and suppression of Fermi acceleration in a dissipative driven stadium-like billiard

The behavior of the average energy for an ensemble of non-interacting particles is studied using scaling arguments in a dissipative time-dependent stadium-like billiard. The dynamics of the system is described by a four dimensional nonlinear mapping. The dissipation is introduced via inelastic collisions between the particles and the moving boundary. For different combinations of initial velocities and damping coefficients, the long time dynamics of the particles leads them to reach different states of final energy and to visit different attractors, which change as the dissipation is varied. The decay of the average energy of the particles, which is observed for a large range of restitution coefficients and different initial velocities, is described using scaling arguments. Since this system exhibits unlimited energy growth in the absence of dissipation, our results for the dissipative case give support to the principle that Fermi acceleration seems not to be a robust phenomenon.     

Charge exchange of a polar molecule at its cation

The Landau-Herring method is used to derive an analytic expression for the one-electron exchange interaction of a polar molecule with its positively charged ion, induced by a σ-electron. Analogously to the classical Van der Pole method, the exchange interaction potential is averaged over the rotational states of colliding particles. The resonant charge-transfer cross section is calculated, and the effect of the dipole moments of the core on the cross section is analyzed. It is shown that allowance for the dependence of the exchange potential on the orientation of the dipole moments relative to the molecular axis may change the dependence of the cross section on the velocity of colliding particles, which is typical of the resonant charge exchange, from the resonance to the quasi-resonance dependence.     

Anomalous drift of resonant particles in a buffer medium under pressure of light

We study theoretically the drift of resonant particles in a buffer medium when a traveling light wave impinges on the medium, with allowance for the velocity dependence of the transport collision rate. When the pressure of light dominates over the light-induced drift (low pressure of the buffer gas or the drift of conduction electrons in semiconductors), we discover a new sudden transformation of the spectral dependence of the drift velocity of the resonant particles: Instead of the ordinary bell-shaped function representing the velocity spectrum we have a double-humped curve with deep dip at the center of the absorption line. We show that the largest transformation of the drift velocity spectrum occurs in the atmosphere of a heavy buffer gas in the case of Coulomb interaction between the resonant and buffer particles. The transformation effect is caused by the variation of the transport rate of the collisions of the resonant and buffer particles due to the recoil effect in the absorption of radiation. Zh. éksp. Teor. Fiz. 112, 856–868 (September 1997)     

开放量子台球体系散射的赝路径半经典近似

本文讨论正方形量子台球的输运性质，考虑电子以费米能量穿过台球区域，在台球出口和入口处对入射和出射波函数采用基尔霍夫散射．采用微扰论的Dyson方程得到半经典格林函数，并把赝路径半经典近似作微扰展开得到体系的传输矩阵元．比较了传输矩阵元的傅立叶变换谱的峰位置与腔内自由电子经典轨道长度，发现在精度允许范围内它们符合的很好．     

Separatrices splitting for Birkhoff's billiard in symmetric convex domain,closed to an ellipse

An example of a convex domain on the plane with the phenomenon of the transversal intersection of separatrices of the corresponding billiard mapping is presented. This example is constructed as an analytic global symmetric perturbation of an ellipse and we investigate the global symmetric analytic perturbation of the integrable billiard mapping in the ellipse. We establish a theorem on the separatrices splitting of the perturbed billiard mapping and derive the asymptotic formulas for a homoclinic invariant as well as for a "principal" splitting angle of separatrices, arising from the hyperbolic fixed point of the mapping. (c) 1994 American Institute of Physics.