From deterministic dynamics to kinetic phenomena

We investigate a one-dimenisonal Hamiltonian system that describes a system of particles interacting through short-range repulsive potentials. Depending on the particle mean energy, $\epsilon$, the system demonstrates a spectrum of kinetic regimes, characterized by their transport properties ranging from ballistic motion to localized oscillations through anomalous diffusion regimes. We etsablish relationships between the observed kinetic regimes and the "thermodynamic" states of the system. The nature of heat conduction in the proposed model is discussed.Comment: 4 pages, 4 figure

Molecular motor with a build-in escapement device

We study dynamics of a classical particle in a one-dimensional potential, which is composed of two periodic components, that are time-independent, have equal amplitudes and periodicities. One of them is externally driven by a random force and thus performs a diffusive-type motion with respect to the other. We demonstrate that here, under certain conditions, the particle may move unidirectionally with a constant velocity, despite the fact that the random force averages out to zero. We show that the physical mechanism underlying such a phenomenon resembles the work of an escapement-type device in watches; upon reaching certain level, random fluctuations exercise a locking function creating the points of irreversibility in particle's trajectories such that the particle gets uncompensated displacements. Repeated (randomly) in each cycle, this process ultimately results in a random ballistic-type motion. In the overdamped limit, we work out simple analytical estimates for the particle's terminal velocity. Our analytical results are in a very good agreement with the Monte Carlo data.Comment: 7 pages, 4 figure

Saltatory drift in a randomly driven two-wave potential

Dynamics of a classical particle in a one-dimensional, randomly driven potential is analysed both analytically and numerically. The potential considered here is composed of two identical spatially-periodic saw-tooth-like components, one of which is externally driven by a random force. We show that under certain conditions the particle may travel against the averaged external force performing a saltatory unidirectional drift with a constant velocity. Such a behavior persists also in situations when the external force averages out to zero. We demonstrate that the physics behind this phenomenon stems from a particular behavior of fluctuations in random force: upon reaching a certain level, random fluctuations exercise a locking function creating points of irreversibility which the particle can not overpass. Repeated (randomly) in each cycle, this results in a saltatory unidirectional drift. This mechanism resembles the work of an escapement-type device in watches. Considering the overdamped limit, we propose simple analytical estimates for the particle's terminal velocity.Comment: 14 pages, 6 figures; appearing in Journal of Physics: Condensed Matter, special issue on Molecular Motors and Frictio

Anomalous escape governed by thermal 1/f noise

We present an analytic study for subdiffusive escape of overdamped particles out of a cusp-shaped parabolic potential well which are driven by thermal, fractional Gaussian noise with a $1/\omega^{1-\alpha}$ power spectrum. This long-standing challenge becomes mathematically tractable by use of a generalized Langevin dynamics via its corresponding non-Markovian, time-convolutionless master equation: We find that the escape is governed asymptotically by a power law whose exponent depends exponentially on the ratio of barrier height and temperature. This result is in distinct contrast to a description with a corresponding subdiffusive fractional Fokker-Planck approach; thus providing experimentalists an amenable testbed to differentiate between the two escape scenarios

First passage times and asymmetry of DNA translocation

Motivated by experiments in which single-stranded DNA with a short hairpin loop at one end undergoes unforced diffusion through a narrow pore, we study the first passage times for a particle, executing one-dimensional brownian motion in an asymmetric sawtooth potential, to exit one of the boundaries. We consider the first passage times for the case of classical diffusion, characterized by a mean-square displacement of the form $\sim t$, and for the case of anomalous diffusion or subdiffusion, characterized by a mean-square displacement of the form $\sim t^{\gamma}$ with $0<\gamma<1$. In the context of classical diffusion, we obtain an expression for the mean first passage time and show that this quantity changes when the direction of the sawtooth is reversed or, equivalently, when the reflecting and absorbing boundaries are exchanged. We discuss at which numbers of `teeth' $N$ (or number of DNA nucleotides) and at which heights of the sawtooth potential this difference becomes significant. For large $N$, it is well known that the mean first passage time scales as $N^2$. In the context of subdiffusion, the mean first passage time does not exist. Therefore we obtain instead the distribution of first passage times in the limit of long times. We show that the prefactor in the power relation for this distribution is simply the expression for the mean first passage time in classical diffusion. We also describe a hypothetical experiment to calculate the average of the first passage times for a fraction of passage events that each end within some time $t^*$. We show that this average first passage time scales as $N^{2/\gamma}$ in subdiffusion.Comment: 10 pages, 4 figures We incorporated reviewers' suggestions from Physical Review E. We reformulated a few paragraphs in the introduction and further clarified the issue of the (a)symmetry of passage times. In the results section, we re-expressed the results in a form that manifest the important features. We also added a few references concerning anomalous diffusion. The look (but not the content) of figure 1 was also change