10,177 research outputs found
Diffusion coefficients for multi-step persistent random walks on lattices
We calculate the diffusion coefficients of persistent random walks on
lattices, where the direction of a walker at a given step depends on the memory
of a certain number of previous steps. In particular, we describe a simple
method which enables us to obtain explicit expressions for the diffusion
coefficients of walks with two-step memory on different classes of one-, two-
and higher-dimensional lattices.Comment: 27 pages, 2 figure
Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards
We perform numerical measurements of the moments of the position of a tracer
particle in a two-dimensional periodic billiard model (Lorentz gas) with
infinite corridors. This model is known to exhibit a weak form of
super-diffusion, in the sense that there is a logarithmic correction to the
linear growth in time of the mean-squared displacement. We show numerically
that this expected asymptotic behavior is easily overwhelmed by the subleading
linear growth throughout the time-range accessible to numerical simulations. We
compare our simulations to the known analytical results for the variance of the
anomalously-rescaled limiting normal distributions.Comment: 10 pages, 4 figure
Transport properties of L\'evy walks: an analysis in terms of multistate processes
Continuous time random walks combining diffusive and ballistic regimes are
introduced to describe a class of L\'evy walks on lattices. By including
exponentially-distributed waiting times separating the successive jump events
of a walker, we are led to a description of such L\'evy walks in terms of
multistate processes whose time-evolution is shown to obey a set of coupled
delay differential equations. Using simple arguments, we obtain asymptotic
solutions to these equations and rederive the scaling laws for the mean squared
displacement of such processes. Our calculation includes the computation of all
relevant transport coefficients in terms of the parameters of the models.Comment: 5 pages, 2 figures. New references adde
L\'evy walks on lattices as multi-state processes
Continuous-time random walks combining diffusive scattering and ballistic
propagation on lattices model a class of L\'evy walks. The assumption that
transitions in the scattering phase occur with exponentially-distributed
waiting times leads to a description of the process in terms of multiple
states, whose distributions evolve according to a set of delay differential
equations, amenable to analytic treatment. We obtain an exact expression of the
mean squared displacement associated with such processes and discuss the
emergence of asymptotic scaling laws in regimes of diffusive and superdiffusive
(subballistic) transport, emphasizing, in the latter case, the effect of
initial conditions on the transport coefficients. Of particular interest is the
case of rare ballistic propagation, in which case a regime of superdiffusion
may lurk underneath one of normal diffusion.Comment: 27 pages, 4 figure
Machta-Zwanzig regime of anomalous diffusion in infinite-horizon billiards
We study diffusion on a periodic billiard table with infinite horizon in the
limit of narrow corridors. An effective trapping mechanism emerges according to
which the process can be modeled by a L\'evy walk combining
exponentially-distributed trapping times with free propagation along paths
whose precise probabilities we compute. This description yields an
approximation of the mean squared displacement of infinite-horizon billiards in
terms of two transport coefficients which generalizes to this anomalous regime
the Machta-Zwanzig approximation of normal diffusion in finite-horizon
billiards [Phys. Rev. Lett. 50, 1959 (1983)].Comment: 5 pages, 3 figure
Chaos in cylindrical stadium billiards via a generic nonlinear mechanism
We describe conditions under which higher-dimensional billiard models in
bounded, convex regions are fully chaotic, generalizing the Bunimovich stadium
to dimensions above two. An example is a three-dimensional stadium bounded by a
cylinder and several planes; the combination of these elements may give rise to
defocusing, allowing large chaotic regions in phase space. By studying families
of marginally-stable periodic orbits that populate the residual part of phase
space, we identify conditions under which a nonlinear instability mechanism
arises in their vicinity. For particular geometries, this mechanism rather
induces stable nonlinear oscillations, including in the form of
whispering-gallery modes.Comment: 4 pages, 4 figure
Steady-state conduction in self-similar billiards
The self-similar Lorentz billiard channel is a spatially extended
deterministic dynamical system which consists of an infinite one-dimensional
sequence of cells whose sizes increase monotonically according to their
indices. This special geometry induces a nonequilibrium stationary state with
particles flowing steadily from the small to the large scales. The
corresponding invariant measure has fractal properties reflected by the
phase-space contraction rate of the dynamics restricted to a single cell with
appropriate boundary conditions. In the near-equilibrium limit, we find
numerical agreement between this quantity and the entropy production rate as
specified by thermodynamics
A simplified DNA extraction protocol for unsorted bulk arthropod samples that maintains exoskeletal integrity
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