3,911 research outputs found
Towards deterministic equations for Levy walks: the fractional material derivative
Levy walks are random processes with an underlying spatiotemporal coupling.
This coupling penalizes long jumps, and therefore Levy walks give a proper
stochastic description for a particle's motion with broad jump length
distribution. We derive a generalized dynamical formulation for Levy walks in
which the fractional equivalent of the material derivative occurs. Our approach
will be useful for the dynamical formulation of Levy walks in an external force
field or in phase space for which the description in terms of the continuous
time random walk or its corresponding generalized master equation are less well
suited
Optimal target search on a fast folding polymer chain with volume exchange
We study the search process of a target on a rapidly folding polymer (`DNA')
by an ensemble of particles (`proteins'), whose search combines 1D diffusion
along the chain, Levy type diffusion mediated by chain looping, and volume
exchange. A rich behavior of the search process is obtained with respect to the
physical parameters, in particular, for the optimal search.Comment: 4 pages, 3 figures, REVTe
Aging renewal theory and application to random walks
The versatility of renewal theory is owed to its abstract formulation.
Renewals can be interpreted as steps of a random walk, switching events in
two-state models, domain crossings of a random motion, etc. We here discuss a
renewal process in which successive events are separated by scale-free waiting
time periods. Among other ubiquitous long time properties, this process
exhibits aging: events counted initially in a time interval [0,t] statistically
strongly differ from those observed at later times [t_a,t_a+t]. In complex,
disordered media, processes with scale-free waiting times play a particularly
prominent role. We set up a unified analytical foundation for such anomalous
dynamics by discussing in detail the distribution of the aging renewal process.
We analyze its half-discrete, half-continuous nature and study its aging time
evolution. These results are readily used to discuss a scale-free anomalous
diffusion process, the continuous time random walk. By this we not only shed
light on the profound origins of its characteristic features, such as weak
ergodicity breaking. Along the way, we also add an extended discussion on aging
effects. In particular, we find that the aging behavior of time and ensemble
averages is conceptually very distinct, but their time scaling is identical at
high ages. Finally, we show how more complex motion models are readily
constructed on the basis of aging renewal dynamics.Comment: 21 pages, 7 figures, RevTe
Fine structure of distributions and central limit theorem in diffusive billiards
We investigate deterministic diffusion in periodic billiard models, in terms
of the convergence of rescaled distributions to the limiting normal
distribution required by the central limit theorem; this is stronger than the
usual requirement that the mean square displacement grow asymptotically
linearly in time. The main model studied is a chaotic Lorentz gas where the
central limit theorem has been rigorously proved. We study one-dimensional
position and displacement densities describing the time evolution of
statistical ensembles in a channel geometry, using a more refined method than
histograms. We find a pronounced oscillatory fine structure, and show that this
has its origin in the geometry of the billiard domain. This fine structure
prevents the rescaled densities from converging pointwise to gaussian
densities; however, demodulating them by the fine structure gives new densities
which seem to converge uniformly. We give an analytical estimate of the rate of
convergence of the original distributions to the limiting normal distribution,
based on the analysis of the fine structure, which agrees well with simulation
results. We show that using a Maxwellian (gaussian) distribution of velocities
in place of unit speed velocities does not affect the growth of the mean square
displacement, but changes the limiting shape of the distributions to a
non-gaussian one. Using the same methods, we give numerical evidence that a
non-chaotic polygonal channel model also obeys the central limit theorem, but
with a slower convergence rate.Comment: 16 pages, 19 figures. Accepted for publication in Physical Review E.
Some higher quality figures at http://www.maths.warwick.ac.uk/~dsander
Harmonic operation of a free-electron laser
Harmonic operation of a free-electron-laser amplifier is studied. The key issue investigated here is suppression of the fundamental. For a tapered amplifier with the right choice of parameters, it is found that the presence of the harmonic mode greatly reduces the growth rate of the fundamental. A limit on the reflection coefficient of the fundamental mode that will ensure stable operation is derived. The relative merits of tripling the frequency by operating at the third harmonic versus decreasing the wiggler period by a factor of 3 are discussed
Thermodynamics and Fractional Fokker-Planck Equations
The relaxation to equilibrium in many systems which show strange kinetics is
described by fractional Fokker-Planck equations (FFPEs). These can be
considered as phenomenological equations of linear nonequilibrium theory. We
show that the FFPEs describe the system whose noise in equilibrium funfills the
Nyquist theorem. Moreover, we show that for subdiffusive dynamics the solutions
of the corresponding FFPEs are probability densities for all cases where the
solutions of normal Fokker-Planck equation (with the same Fokker-Planck
operator and with the same initial and boundary conditions) exist. The
solutions of the FFPEs for superdiffusive dynamics are not always probability
densities. This fact means only that the corresponding kinetic coefficients are
incompatible with each other and with the initial conditions
Helix untwisting and bubble formation in circular DNA
The base pair fluctuations and helix untwisting are examined for a circular
molecule. A realistic mesoscopic model including twisting degrees of freedom
and bending of the molecular axis is proposed. The computational method, based
on path integral techniques, simulates a distribution of topoisomers with
various twist numbers and finds the energetically most favorable molecular
conformation as a function of temperature. The method can predict helical
repeat, openings loci and bubble sizes for specific sequences in a broad
temperature range. Some results are presented for a short DNA circle recently
identified in mammalian cells.Comment: The Journal of Chemical Physics, vol. 138 (2013), in pres
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