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Register a new userDirected Chaotic Transport in Hamiltonian Ratchets
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We present a comprehensive account of directed transport in one-dimensional Hamiltonian systems with spatial and temporal periodicity. They can be considered as Hamiltonian ratchets in the sense that ensembles of particles can show directed ballistic transport in the absence of an average force. We discuss general conditions for such directed transport, like a mixed classical phase space, and elucidate a sum rule that relates the contributions of different phase-space components to transport with each other. We show that regular ratchet transport can be directed against an external potential gradient while chaotic ballistic transport is restricted to unbiased systems. For quantized Hamiltonian ratchets we study transport in terms of the evolution of wave packets and derive a semiclassical expression for the distribution of level velocities which encode the quantum transport in the Floquet band spectra. We discuss the role of dynamical tunneling between transporting islands and the chaotic sea and the breakdown of transport in quantum ratchets with broken spatial periodicity.
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In prl 84, 258 (2000), Mateos conjectured that current reversal in a classical deterministic ratchet is associated with bifurcations from chaotic to periodic regimes. This is based on the comparison of the current and the bifurcation diagram as a function of a given parameter for a periodic asymmetric potential. Barbi and Salerno, in pre 62, 1988 (2000), have further investigated this claim and argue that, contrary to Mateos claim, current reversals can occur also in the absence of bifurcations. Barbi and Salernos studies are based on the dynamics of one particle rather than the statistical mechanics of an ensemble of particles moving in the chaotic system. The behavior of ensembles can be quite different, depending upon their characteristics, which leaves their results open to question. In this paper we present results from studies showing how the current depends on the details of the ensemble used to generate it, as well as conditions for convergent behavior (that is, independent of the details of the ensemble). We are then able to present the converged current as a function of parameters, in the same system as Mateos as well as Barbi and Salerno. We show evidence for current reversal without bifurcation, as well as bifurcation without current reversal. We conjecture that it is appropriate to correlate abrupt changes in the current with bifurcation, rather than current reversals, and show numerical evidence for our claims.
Two deterministic models for Brownian motion are investigated by means of numerical simulations and kinetic theory arguments. The first model consists of a heavy hard disk immersed in a rarefied gas of smaller and lighter hard disks acting as a thermal bath. The second is the same except for the shape of the particles, which is now square. The basic difference of these two systems lies in the interaction: hard core elastic collisions make the dynamics of the disks chaotic whereas that of squares is not. Remarkably, this difference is not reflected in the transport properties of the two systems: simulations show that the diffusion coefficients, velocity correlations and response functions of the heavy impurity are in agreement with kinetic theory for both the chaotic and the non-chaotic model. The relaxation to equilibrium, however, is very sensitive to the kind of interaction. These observations are used to reconsider and discuss some issues connected to chaos, statistical mechanics and diffusion.
We study the quantum and classical scattering of Hamiltonian systems whose chaotic saddle is described by binary or ternary horseshoes. We are interested in parameters of the system for which a stable island, associated with the inner fundamental periodic orbit of the system exists and is large, but chaos around this island is well developed. In this situation, in classical systems, decay from the interaction region is algebraic, while in quantum systems it is exponential due to tunneling. In both cases, the most surprising effect is a periodic response to an incoming wave packet. The period of this self-pulsing effect or scattering echoes coincides with the mean period, by which the scattering trajectories rotate around the stable orbit. This period of rotation is directly related to the development stage of the underlying horseshoe. Therefore the predicted echoes will provide experimental access to topological information. We numerically test these results in kicked one dimensional models and in open billiards.
We study the chaotic dynamics of graphene structures, considering both a periodic, defect free, graphene sheet and graphene nanoribbons (GNRs) of various widths. By numerically calculating the maximum Lyapunov exponent, we quantify the chaoticity for a spectrum of energies in both systems. We find that for all cases, the chaotic strength increases with the energy density, and that the onset of chaos in graphene is slow, becoming evident after more than $10^4$ natural oscillations of the system. For the GNRs, we also investigate the impact of the width and chirality (armchair or zigzag edges) on their chaotic behavior. Our results suggest that due to the free edges the chaoticity of GNRs is stronger than the periodic graphene sheet, and decreases by increasing width, tending asymptotically to the bulk value. In addition, the chaotic strength of armchair GNRs is higher than a zigzag ribbon of the same width. Further, we show that the composition of ${}^{12}C$ and ${}^{13}C$ carbon isotopes in graphene has a minor impact on its chaotic strength.
The occurrence of an inverse current, where the sign of the induced current is opposite to the applied force, is a highly counterintuitive phenomenon. We show that inverse currents in coupled transport (ICC) of energy and particle can occur in a one-dimensional interacting Hamiltonian system when its equilibrium state is perturbed by coupled thermodynamic forces. This seemingly paradoxical result is possible due to the self-organization occurring in the system in response to the applied forces.
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