709 research outputs found
Achievable Qubit Rates for Quantum Information Wires
Suppose Alice and Bob have access to two separated regions, respectively, of
a system of electrons moving in the presence of a regular one-dimensional
lattice of binding atoms. We consider the problem of communicating as much
quantum information, as measured by the qubit rate, through this quantum
information wire as possible. We describe a protocol whereby Alice and Bob can
achieve a qubit rate for these systems which is proportional to N^(-1/3) qubits
per unit time, where N is the number of lattice sites. Our protocol also
functions equally in the presence of interactions modelled via the t-J and
Hubbard models
Higher-order splitting algorithms for solving the nonlinear Schr\"odinger equation and their instabilities
Since the kinetic and the potential energy term of the real time nonlinear
Schr\"odinger equation can each be solved exactly, the entire equation can be
solved to any order via splitting algorithms. We verified the fourth-order
convergence of some well known algorithms by solving the Gross-Pitaevskii
equation numerically. All such splitting algorithms suffer from a latent
numerical instability even when the total energy is very well conserved. A
detail error analysis reveals that the noise, or elementary excitations of the
nonlinear Schr\"odinger, obeys the Bogoliubov spectrum and the instability is
due to the exponential growth of high wave number noises caused by the
splitting process. For a continuum wave function, this instability is
unavoidable no matter how small the time step. For a discrete wave function,
the instability can be avoided only for \dt k_{max}^2{<\atop\sim}2 \pi, where
.Comment: 10 pages, 8 figures, submitted to Phys. Rev.
Averaging approximation to singularly perturbed nonlinear stochastic wave equations
An averaging method is applied to derive effective approximation to the
following singularly perturbed nonlinear stochastic damped wave equation \nu
u_{tt}+u_t=\D u+f(u)+\nu^\alpha\dot{W} on an open bounded domain
\,, \,. Here is a small parameter
characterising the singular perturbation, and \,, \,, parametrises the strength of the noise. Some scaling transformations
and the martingale representation theorem yield the following effective
approximation for small , u_t=\D u+f(u)+\nu^\alpha\dot{W} to an error of
\ord{\nu^\alpha}\,.Comment: 16 pages. Submitte
Linear superposition in nonlinear wave dynamics
We study nonlinear dispersive wave systems described by hyperbolic PDE's in
R^{d} and difference equations on the lattice Z^{d}. The systems involve two
small parameters: one is the ratio of the slow and the fast time scales, and
another one is the ratio of the small and the large space scales. We show that
a wide class of such systems, including nonlinear Schrodinger and Maxwell
equations, Fermi-Pasta-Ulam model and many other not completely integrable
systems, satisfy a superposition principle. The principle essentially states
that if a nonlinear evolution of a wave starts initially as a sum of generic
wavepackets (defined as almost monochromatic waves), then this wave with a high
accuracy remains a sum of separate wavepacket waves undergoing independent
nonlinear evolution. The time intervals for which the evolution is considered
are long enough to observe fully developed nonlinear phenomena for involved
wavepackets. In particular, our approach provides a simple justification for
numerically observed effect of almost non-interaction of solitons passing
through each other without any recourse to the complete integrability. Our
analysis does not rely on any ansatz or common asymptotic expansions with
respect to the two small parameters but it uses rather explicit and
constructive representation for solutions as functions of the initial data in
the form of functional analytic series.Comment: New introduction written, style changed, references added and typos
correcte
Analytical Investigation of Innovation Dynamics Considering Stochasticity in the Evaluation of Fitness
We investigate a selection-mutation model for the dynamics of technological
innovation,a special case of reaction-diffusion equations. Although mutations
are assumed to increase the variety of technologies, not their average success
("fitness"), they are an essential prerequisite for innovation. Together with a
selection of above-average technologies due to imitation behavior, they are the
"driving force" for the continuous increase in fitness. We will give analytical
solutions for the probability distribution of technologies for special cases
and in the limit of large times.
The selection dynamics is modelled by a "proportional imitation" of better
technologies. However, the assessment of a technology's fitness may be
imperfect and, therefore, vary stochastically. We will derive conditions, under
which wrong assessment of fitness can accelerate the innovation dynamics, as it
has been found in some surprising numerical investigations.Comment: For related work see http://www.helbing.or
Attempted density blowup in a freely cooling dilute granular gas: hydrodynamics versus molecular dynamics
It has been recently shown (Fouxon et al. 2007) that, in the framework of
ideal granular hydrodynamics (IGHD), an initially smooth hydrodynamic flow of a
granular gas can produce an infinite gas density in a finite time. Exact
solutions that exhibit this property have been derived. Close to the
singularity, the granular gas pressure is finite and almost constant. This work
reports molecular dynamics (MD) simulations of a freely cooling gas of nearly
elastically colliding hard disks, aimed at identifying the "attempted" density
blowup regime. The initial conditions of the simulated flow mimic those of one
particular solution of the IGHD equations that exhibits the density blowup. We
measure the hydrodynamic fields in the MD simulations and compare them with
predictions from the ideal theory. We find a remarkable quantitative agreement
between the two over an extended time interval, proving the existence of the
attempted blowup regime. As the attempted singularity is approached, the
hydrodynamic fields, as observed in the MD simulations, deviate from the
predictions of the ideal solution. To investigate the mechanism of breakdown of
the ideal theory near the singularity, we extend the hydrodynamic theory by
accounting separately for the gradient-dependent transport and for finite
density corrections.Comment: 11 pages, 9 figures, accepted for publication on Physical Review
Towards a continuum theory of clustering in a freely cooling inelastic gas
We performed molecular dynamics simulations to investigate the clustering
instability of a freely cooling dilute gas of inelastically colliding disks in
a quasi-one-dimensional setting. We observe that, as the gas cools, the shear
stress becomes negligibly small, and the gas flows by inertia only. Finite-time
singularities, intrinsic in such a flow, are arrested only when close-packed
clusters are formed. We observe that the late-time dynamics of this system are
describable by the Burgers equation with vanishing viscosity, and predict the
long-time coarsening behavior.Comment: 7 pages, 5 eps figures, to appear in Europhys. Let
Hydrodynamic singularities and clustering in a freely cooling inelastic gas
We employ hydrodynamic equations to follow the clustering instability of a
freely cooling dilute gas of inelastically colliding spheres into a
well-developed nonlinear regime. We simplify the problem by dealing with a
one-dimensional coarse-grained flow. We observe that at a late stage of the
instability the shear stress becomes negligibly small, and the gas flows solely
by inertia. As a result the flow formally develops a finite time singularity,
as the velocity gradient and the gas density diverge at some location. We argue
that flow by inertia represents a generic intermediate asymptotic of unstable
free cooling of dilute inelastic gases.Comment: 4 pages, 4 figure
Generation of linear waves in the flow of Bose-Einstein condensate past an obstacle
The theory of linear wave structures generated in Bose-Einstein condensate
flow past an obstacle is developed. The shape of wave crests and dependence of
amplitude on coordinates far enough from the obstacle are calculated. The
results are in good agreement with the results of numerical simulations
obtained earlier. The theory gives a qualitative description of experiments
with Bose-Einstein condensate flow past an obstacle after condensate's release
from a trap.Comment: 11 pages, 3 figures, to be published in Zh. Eksp. Teor. Fi
Adiabatic nonlinear waves with trapped particles: II. Wave dispersion
A general nonlinear dispersion relation is derived in a nondifferential form
for an adiabatic sinusoidal Langmuir wave in collisionless plasma, allowing for
an arbitrary distribution of trapped electrons. The linear dielectric function
is generalized, and the nonlinear kinetic frequency shift is
found analytically as a function of the wave amplitude . Smooth
distributions yield , as usual. However,
beam-like distributions of trapped electrons result in different power laws, or
even a logarithmic nonlinearity, which are derived as asymptotic limits of the
same dispersion relation. Such beams are formed whenever the phase velocity
changes, because the trapped distribution is in autoresonance and thus evolves
differently from the passing distribution. Hence, even adiabatic is generally nonlocal.Comment: submitted together with Papers I and II
- …