Dispersionless motion in a periodically rocked periodic potential

Origin of dispersionless transport in spite of thermal noise

The "dispersionless transport" of a weakly damped Brownian particle in a tilted periodic potential is defined by (i) a plateau of the particle's coordinate dispersion extending over a very broad time interval and (ii) by the impossibility to measure the diffusion coefficient within this plateau region. While the first part of this definition has been explained in the literature, the second part has been thought to follow from (i). Here, the impossibility to measure the diffusion coefficient is shown to be actually due to the wild fluctuations of the dispersion itself in the plateau region. An expression for the timescale over which a reliable determination of the diffusion coefficient is possible is derived. A procedure that allows accurate determination of the diffusion coefficient by observing the particle trajectory only within a small part of the plateau region is suggested and shown to be feasible by numerical simulations of a weakly damped Brownian particle in a tilted washboard potential.

Enhanced diffusion with abnormal temperature dependence in underdamped space-periodic systems subject to time-periodic driving

We present a study of the diffusion enhancement of underdamped Brownian particles in a one-dimensional symmetric space-periodic potential due to external symmetric time-periodic driving with zero mean. We show that the diffusivity can be enhanced by many orders of magnitude at an appropriate choice of the driving amplitude and frequency. The diffusivity demonstrates abnormal (decreasing) temperature dependence at the driving amplitudes exceeding a certain value. At any fixed driving frequency Ω normal temperature dependence of the diffusivity is restored at low enough temperatures, T<T_{TAD}(Ω)-in contrast with the problem with constant external driving. At fixed temperature at small driving frequency the diffusivity either slowly decreases with Ω, or (at stronger driving) goes through a maximum near Ω_{2}, the reciprocal superdiffusion regime termination time. At high frequencies, between Ω_{2} and a fraction of the oscillation frequency at the potential minimum, the diffusivity is shown to decrease with Ω according to a power law, with the exponent related to the transient superdiffusion exponent. This behavior is found similar for the cases of sinusoidal in time and piecewise constant periodic ("square") driving.

Vibrational resonance in an inhomogeneous medium with periodic dissipation

The role of nonlinear dissipation in vibrational resonance (VR) is investigated in an inhomogeneous system characterized by a symmetric and spatially periodic potential and subjected to nonuniform state-dependent damping and a biharmonic driving force. The contributions of the parameters of the high-frequency signal to the system's effective dissipation are examined theoretically in comparison to linearly damped systems, for which the parameter of interest is the effective stiffness in the equation of slow vibration. We show that the VR effect can be enhanced by varying the nonlinear dissipation parameters and that it can be induced by a parameter that is shared by the damping inhomogeneity and the system potential. Furthermore, we have apparently identified the origin of the nonlinear-dissipation-enhanced response: We provide evidence of its connection to a Hopf bifurcation, accompanied by monotonic attractor enlargement in the VR regime.

Diffusion of a massive particle in a periodic potential: Application to adiabatic ratchets

We generalize a theory of diffusion of a massive particle by the way in which transport characteristics are described by analytical expressions that formally coincide with those for the overdamped massless case but contain a factor comprising the particle mass which can be calculated in terms of Risken's matrix continued fraction method (MCFM). Using this generalization, we aim to elucidate how large gradients of a periodic potential affect the current in a tilted periodic potential and the average current of adiabatically driven on-off flashing ratchets. For this reason, we perform calculations for a sawtooth potential of the period L with an arbitrary sawtooth length (l<L) instead of the smooth potentials typically considered in MCFM-solvable problems. We find nonanalytic behavior of the transport characteristics calculated for the sharp extremely asymmetric sawtooth potential at l→0 which appears due to the inertial effect. Analysis of the temperature dependences of the quantities under study reveals the dominant role of inertia in the high-temperature region. In particular, we show, by the analytical strong-inertia approach developed for this region, that the temperature-dependent contribution to the mobility at zero force and to the related effective diffusion coefficient are proportional to T(-3/2) and T(-1/2), respectively, and have a logarithmic singularity at l→0.