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Register a new userDirected chaos in a billiard chain with transversal magnetic field
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In generic Hamiltonian systems with a mixed phase space chaotic transport may be directed and ballistic rather than diffusive. We investigate one particular model showing this behaviour, namely a spatially periodic billiard chain in which electrons move under the influence of a perpendicular magnetic field. We analyze the phase-space structure and derive an explicit expression for the chaotic transport velocity. Unlike previous studies of directed chaos our model has a parameter regime in which the dispersion of an ensemble of chaotic trajectories around its moving center of mass is essentially diffusive. We explain how in this limit the deterministic chaos reduces to a biased random walk in a billiard with a rough surface. The diffusion constant for this simplified model is calculated analytically.
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We study the conductance of chaotic or disordered wires in a situation where equilibrium transport decomposes into biased diffusion and a counter-moving regular current. A possible realization is a semiconductor nanostructure with transversal magnetic field and suitably patterned surfaces. We find a non-trivial dependence of the conductance on the wire length which differs qualitatively from Ohms law by the existence of a characteristic length scale and a finite saturation value.
Billiard systems offer a simple setting to study regular and chaotic dynamics. Gravitational billiards are generalizations of these classical billiards which are amenable to both analytical and experimental investigations. Most previous work on gravitational billiards has been concerned with two dimensional boundaries. In particular the case of linear boundaries, also known as the wedge billiard, has been widely studied. In this work, we introduce a three dimensional version of the wedge; that is, we study the nonlinear dynamics of a billiard in a constant gravitational field colliding elastically with a linear cone of half angle $theta$. We derive a two-dimensional Poincar{e} map with two parameters, the half angle of the cone and $ell$, the $z$-component of the billiards angular momentum. Although this map is sufficient to determine the future motion of the billiard, the three-dimensional nature of the physical trajectory means that a periodic orbit of the mapping does not always correspond to a periodic trajectory in coordinate space. We demonstrate several integrable cases of the parameter values, and analytically compute the systems fixed point, analyzing the stability of this orbit as a function of the parameters as well as its relation to the physical trajectory of the billiard. Next, we explore the phase space of the system numerically. We find that for small values of $ell$ the conic billiard exhibits behavior characteristic of two-degree-of-freedom Hamiltonian systems with a discontinuity, and the dynamics is qualitatively similar to that of the wedge billiard, although the correspondence is not exact. As we increase $ell$ the dynamics becomes on the whole less chaotic, and the correspondence with the wedge billiard is lost.
The relation between the onset of chaos and critical phenomena, like Quantum Phase Transitions (QPT) and Excited-State Quantum Phase transitions (ESQPT), is analyzed for atom-field systems. While it has been speculated that the onset of hard chaos is associated with ESQPT based in the resonant case, the off-resonant cases show clearly that both phenomena, ESQPT and chaos, respond to different mechanisms. The results are supported in a detailed numerical study of the dynamics of the semiclassical Hamiltonian of the Dicke model. The appearance of chaos is quantified calculating the largest Lyapunov exponent for a wide sample of initial conditions in the whole available phase space for a given energy. The percentage of the available phase space with chaotic trajectories is evaluated as a function of energy and coupling between the qubit and bosonic part, allowing to obtain maps in the space of coupling and energy, where ergodic properties are observed in the model. Different sets of Hamiltonian parameters are considered, including resonant and off-resonant cases.
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.
Via evaluation of the Lyapunov exponent, we report the discovery of three prominent sets of phase space regimes of quasi-periodic orbits of charged particles trapped in a dipole magnetic field. Besides the low energy regime that has been studied extensively and covers more than 10% in each dimension of the phase space of trapped orbits, there are two sets of high energy regimes, the largest of which covers more than 4% in each dimension of the phase space of trapped orbits. Particles in these high energy orbits may be observed in space and be realized in plasma experiments on the Earth. It is well-known that there are quasi-periodic orbits around stable periodic orbits in Hamiltonian systems with 2 degrees of freedom and these quasi-periodic orbits are stable as well. Since periodic orbits appear to have a negligible measure in the phase space, they are difficult to realize in nature. Quasi-periodic orbits, on the other hand, may occupy a finite volume in the 4 dimensional (4D) phase space and be readily detectable. A chaotic orbit has at least one positive Lyapunov exponent. The Lyapunov exponents of quasi-periodic orbits, on the other hand, should be zero. Via calculation of the Lyapunov exponent of orbits of trapped charged particles in a dipole magnetic field, we scanned the corresponding phase space and found several prominent regimes of quasi-periodic orbits associated with stable periodic orbits in the equatorial plane. These regimes appear to be connected to some small regimes of quasi-periodic orbits associated with stable periodic orbits in the Meridian plane. Our numerical results also show a continuous spectrum of these orbits from stable periodic, to quasi-periodic with vanishing Lyapunov exponents, and eventually to chaotic ones with at least one positive Lyapunov exponent and there are unstable periodic orbits with a positive maximum Lyapunov exponent.
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