Entropic particle transport in periodic channels.

The dynamics of Brownian motion has widespread applications extending from transport in designed micro-channels up to its prominent role for inducing transport in molecular motors and Brownian motors. Here, Brownian transport is studied in micro-sized, two-dimensional periodic channels, exhibiting periodically varying cross-sections. The particles in addition are subjected to an external force acting alongside the direction of the longitudinal channel axis. For a fixed channel geometry, the dynamics of the two-dimensional problem is characterized by a single dimensionless parameter which is proportional to the ratio of the applied force and the temperature of the particle environment. In such structures entropic effects may play a dominant role. Under certain conditions the two-dimensional dynamics can be approximated by an effective one-dimensional motion of the particle in the longitudinal direction. The Langevin equation describing this reduced, one-dimensional process is of the type of the Fick-Jacobs equation. It contains an entropic potential determined by the varying extension of the eliminated channel direction, and a correction to the diffusion constant that introduces a space dependent diffusion. Different forms of this correction term have been suggested before, which we here compare for a particular class of models. We analyze the regime of validity of the Fick-Jacobs equation, both by means of analytical estimates and the comparisons with numerical results for the full two-dimensional stochastic dynamics. For the nonlinear mobility we find a temperature dependence which is opposite to that known for particle transport in periodic potentials. The influence of entropic effects is discussed for both, the nonlinear mobility and the effective diffusion constant.

Absolute negative mobility induced by thermal equilibrium fluctuations.

A novel transport phenomenon is identified that is induced by inertial Brownian particles which move in simple one-dimensional, symmetric periodic potentials under the influence of both a time periodic and a constant, biasing driving force. Within tailored parameter regimes, thermal equilibrium fluctuations induce the phenomenon of absolute negative mobility (ANM), which means that the particle noisily moves backwards against a small constant bias. When no thermal fluctuations act, the transport vanishes identically in these tailored regimes. ANM can also occur in the absence of fluctuations on grounds which are rooted solely in the complex, inertial deterministic dynamics. The experimental verification of this new transport scheme is elucidated for the archetype symmetric physical system: a convenient setup consisting of a resistively and capacitively shunted Josephson junction device.

Dynamical bimodality in equilibrium monostable systems.

General features of the stochastic dynamics of classical systems approaching a thermodynamic equilibrium Gibbs state are studied via the numerical analysis of time-dependent solutions of the Fokker-Planck equation for an overdamped particle in various monostable potentials. A large class of initial states can dynamically bifurcate during its time evolution into bimodal transient states, which in turn wear off when approaching the long-time regime. Suitable quantifiers characterizing this transient dynamical bimodality, such as its lifetime, the positions of maxima, and the time-dependent well depth of the probability distribution, are analyzed. Some potential applications are pointed out that make use of this interesting principle which is based on an appropriately chosen initial preparation procedure.

Quantum diffusion in biased washboard potentials: strong friction limit.

Diffusive transport properties of a quantum Brownian particle moving in a tilted spatially periodic potential and strongly interacting with a thermostat are explored. Apart from the average stationary velocity, we foremost investigate the diffusive behavior by evaluating the effective diffusion coefficient together with the corresponding Peclet number. Corrections due to quantum effects, such as quantum tunneling and quantum fluctuations, are shown to substantially enhance the effectiveness of diffusive transport if only the thermostat temperature resides within an appropriate interval of intermediate values.

Flow profiling of a surface-acoustic-wave nanopump.

The flow profile in a capillary gap and the pumping efficiency of an acoustic micropump employing surface acoustic waves is investigated both experimentally and theoretically. Ultrasonic surface waves on a piezoelectric substrate strongly couple to a thin liquid layer and generate a quadrupolar streaming pattern within the fluid. We use fluorescence correlation spectroscopy and fluorescence microscopy as complementary tools to investigate the resulting flow profile. The velocity was found to depend on the applied power approximately linearly and to decrease with the inverse third power of the distance from the ultrasound generator on the chip. The found properties reveal acoustic streaming as a promising tool for the controlled agitation during microarray hybridization.

Consistent description of quantum Brownian motors operating at strong friction.

A quantum Smoluchowski equation is put forward that consistently describes thermal quantum states. In particular, it notably does not induce a violation of the second law of thermodynamics. This so modified kinetic equation is applied to study analytically directed quantum transport at strong friction in arbitrarily shaped ratchet potentials that are driven by nonthermal two-state noise. Depending on the mutual interplay of quantum tunneling and quantum reflection these quantum corrections can induce both, a sizable enhancement or a suppression of transport. Moreover, the threshold for current reversals becomes markedly shifted due to such quantum fluctuations.

Brownian motors: current fluctuations and rectification efficiency.

With this work, we investigate an often neglected aspect of Brownian motor transport, namely, the role of fluctuations of the noise-induced current and its consequences for the efficiency of rectifying noise. In doing so, we consider a Brownian inertial motor that is driven by an unbiased monochromatic, time-periodic force and thermal noise. Typically, we find that the asymptotic, time-, and noise-averaged transport velocities are small, possessing rather broad velocity fluctuations. This implies a corresponding poor performance for the rectification power. However, for tailored profiles of the ratchet potential and appropriate drive parameters, we can identify a drastic enhancement of the rectification efficiency. This regime is marked by persistent, unidirectional motion of the Brownian motor with few back-turns only. The corresponding asymmetric velocity distribution is then rather narrow, with a support that predominantly favors only one sign for the velocity.

Anomalous diffusion and phase relaxation.

The diffusion and relaxation of a phase are investigated on the basis of several stochastic models. A simple relation between the diffusional behavior of the extended phase and the relaxation of periodic phase observables is found in the case of Gaussian and Lèvy distributed increments. In these cases, an anomalous diffusion gives rise to a stretched exponential relaxation of phase observables. Continuous time random walks may lead, even in the case of normal diffusion, to a slow algebraic relaxation.

Refined scaling hypothesis for breakdown coefficients in turbulence.

We study the statistics of Novikov's breakdown coefficients, which represent ratios between energy dissipation rates at different length scales in turbulence. The distribution of their logarithms is shown to be very closely reproduced by an analytic function that we obtain from a hierarchical stochastic process for the turbulent cascade. Correlations and deviation from Gaussianity in the model are accounted for by two parameters, one of which can be interpreted as a generalized dimension. Finally, we illustrate the lack of power-law scaling in the moments of the breakdown coefficients and propose an analytical approximation scheme for them.

Power spectrum and detrended fluctuation analysis: application to daily temperatures

The variability measures of fluctuation analysis (FA) and detrended fluctuation analysis (DFA) are expressed in terms of the power spectral density and of the autocovariance of a given process. The diagnostic potential of these methods is tested on several model power spectral densities. In particular we find that both FA and DFA reveal an algebraic singularity of the power spectral density at small frequencies corresponding to an algebraic decay of the autocovariance. A scaling behavior of the power spectral density in an intermediate frequency regime is better reflected by DFA than by FA. We apply FA and DFA to ambient temperature data from the 20th century with the primary goal to resolve the controversy in literature whether the low frequency behavior of the corresponding power spectral densities are better described by a power law or a stretched exponential. As a third possible model we suggest a Weibull distribution. However, it turns out that neither FA nor DFA can reliably distinguish between the proposed models.

Analysis of energy cascade models of turbulence.

The exponent tau(p) that describes the scaling of the order-p moment of the energy dissipation field in fully developed turbulence is studied in a range which extends to negative values of p. The curve tau(p) vs p and its derivative tau(')(p) are estimated in various ways, including a two-scale method. Predictions of recent cascade models agree with our findings, within the statistical errors, for moderately large, positive p. Evident discrepancies already appear, however, in the interval pin[0,1], and sometimes become dramatic for p<0. In the discussion of a class of cascade models, we present a scaling law which relates different moments to one another.

Biasymptotic formula for the turbulent energy cascade.

We present a family of differential models for the scaling exponents tau(p) which characterize the moments of the energy dissipation rate in turbulence. This scheme interpolates between the asymptotic values of the derivative tau'p of tau(p) versus p in the limits p-->+/-infinity and reproduces the negative-p part of the exponents spectrum tau(p) as well, in contrast with other recent conjectures. Each member of the family is defined by a sigmoidal function, the form of which remains open to theoretical investigations.