977 research outputs found
On the Aggregation of Inertial Particles in Random Flows
We describe a criterion for particles suspended in a randomly moving fluid to
aggregate. Aggregation occurs when the expectation value of a random variable
is negative. This random variable evolves under a stochastic differential
equation. We analyse this equation in detail in the limit where the correlation
time of the velocity field of the fluid is very short, such that the stochastic
differential equation is a Langevin equation.Comment: 16 pages, 2 figure
The discretised harmonic oscillator: Mathieu functions and a new class of generalised Hermite polynomials
We present a general, asymptotical solution for the discretised harmonic
oscillator. The corresponding Schr\"odinger equation is canonically conjugate
to the Mathieu differential equation, the Schr\"odinger equation of the quantum
pendulum. Thus, in addition to giving an explicit solution for the Hamiltonian
of an isolated Josephon junction or a superconducting single-electron
transistor (SSET), we obtain an asymptotical representation of Mathieu
functions. We solve the discretised harmonic oscillator by transforming the
infinite-dimensional matrix-eigenvalue problem into an infinite set of
algebraic equations which are later shown to be satisfied by the obtained
solution. The proposed ansatz defines a new class of generalised Hermite
polynomials which are explicit functions of the coupling parameter and tend to
ordinary Hermite polynomials in the limit of vanishing coupling constant. The
polynomials become orthogonal as parts of the eigenvectors of a Hermitian
matrix and, consequently, the exponential part of the solution can not be
excluded. We have conjectured the general structure of the solution, both with
respect to the quantum number and the order of the expansion. An explicit proof
is given for the three leading orders of the asymptotical solution and we
sketch a proof for the asymptotical convergence of eigenvectors with respect to
norm. From a more practical point of view, we can estimate the required effort
for improving the known solution and the accuracy of the eigenvectors. The
applied method can be generalised in order to accommodate several variables.Comment: 18 pages, ReVTeX, the final version with rather general expression
Classical and Quantum Chaos in a quantum dot in time-periodic magnetic fields
We investigate the classical and quantum dynamics of an electron confined to
a circular quantum dot in the presence of homogeneous magnetic
fields. The classical motion shows a transition to chaotic behavior depending
on the ratio of field magnitudes and the cyclotron
frequency in units of the drive frequency. We determine a
phase boundary between regular and chaotic classical behavior in the
vs plane. In the quantum regime we evaluate the quasi-energy
spectrum of the time-evolution operator. We show that the nearest neighbor
quasi-energy eigenvalues show a transition from level clustering to level
repulsion as one moves from the regular to chaotic regime in the
plane. The statistic confirms this
transition. In the chaotic regime, the eigenfunction statistics coincides with
the Porter-Thomas prediction. Finally, we explicitly establish the phase space
correspondence between the classical and quantum solutions via the Husimi phase
space distributions of the model. Possible experimentally feasible conditions
to see these effects are discussed.Comment: 26 pages and 17 PstScript figures, two large ones can be obtained
from the Author
On the de Haas-van Alphen effect in inhomogeneous alloys
We show that Landau level broadening in alloys occurs naturally as a
consequence of random variations in the local quasiparticle density, without
the need to consider a relaxation time. This approach predicts
Lorentzian-broadened Landau levels similar to those derived by Dingle using the
relaxation-time approximation. However, rather than being determined by a
finite relaxation time , the Landau-level widths instead depend directly
on the rate at which the de Haas-van Alphen frequency changes with alloy
composition. The results are in good agreement with recent data from three very
different alloy systems.Comment: 5 pages, no figure
Surface effects on nanowire transport: numerical investigation using the Boltzmann equation
A direct numerical solution of the steady-state Boltzmann equation in a
cylindrical geometry is reported. Finite-size effects are investigated in large
semiconducting nanowires using the relaxation-time approximation. A nanowire is
modelled as a combination of an interior with local transport parameters
identical to those in the bulk, and a finite surface region across whose width
the carrier density decays radially to zero. The roughness of the surface is
incorporated by using lower relaxation-times there than in the interior.
An argument supported by our numerical results challenges a commonly used
zero-width parametrization of the surface layer. In the non-degenerate limit,
appropriate for moderately doped semiconductors, a finite surface width model
does produce a positive longitudinal magneto-conductance, in agreement with
existing theory. However, the effect is seen to be quite small (a few per cent)
for realistic values of the wire parameters even at the highest practical
magnetic fields. Physical insights emerging from the results are discussed.Comment: 15 pages, 7 figure
Complex Periodic Orbits and Tunnelling in Chaotic Potentials
We derive a trace formula for the splitting-weighted density of states
suitable for chaotic potentials with isolated symmetric wells. This formula is
based on complex orbits which tunnel through classically forbidden barriers.
The theory is applicable whenever the tunnelling is dominated by isolated
orbits, a situation which applies to chaotic systems but also to certain
near-integrable ones. It is used to analyse a specific two-dimensional
potential with chaotic dynamics. Mean behaviour of the splittings is predicted
by an orbit with imaginary action. Oscillations around this mean are obtained
from a collection of related orbits whose actions have nonzero real part
The Coulomb phase shift revisited
We investigate the Coulomb phase shift, and derive and analyze new and more
precise analytical formulae. We consider next to leading order terms to the
Stirling approximation, and show that they are important at small values of the
angular momentum and other regimes. We employ the uniform approximation.
The use of our expressions in low energy scattering of charged particles is
discussed and some comparisons are made with other approximation methods.Comment: 13 pages, 5 figures, 1 tabl
Beyond the Heisenberg time: Semiclassical treatment of spectral correlations in chaotic systems with spin 1/2
The two-point correlation function of chaotic systems with spin 1/2 is
evaluated using periodic orbits. The spectral form factor for all times thus
becomes accessible. Equivalence with the predictions of random matrix theory
for the Gaussian symplectic ensemble is demonstrated. A duality between the
underlying generating functions of the orthogonal and symplectic symmetry
classes is semiclassically established
Statistics of selectively neutral genetic variation
Random models of evolution are instrumental in extracting rates of
microscopic evolutionary mechanisms from empirical observations on genetic
variation in genome sequences. In this context it is necessary to know the
statistical properties of empirical observables (such as the local homozygosity
for instance). Previous work relies on numerical results or assumes Gaussian
approximations for the corresponding distributions. In this paper we give an
analytical derivation of the statistical properties of the local homozygosity
and other empirical observables assuming selective neutrality. We find that
such distributions can be very non-Gaussian.Comment: 4 pages, 4 figure
Pauli principle and chaos in a magnetized disk
We present results of a detailed quantum mechanical study of a gas of
noninteracting electrons confined to a circular boundary and subject to
homogeneous dc plus ac magnetic fields , with
). We earlier found a one-particle {\it classical}
phase diagram of the (scaled) Larmor frequency
{\rm vs} that
separates regular from chaotic regimes. We also showed that the quantum
spectrum statistics changed from Poisson to Gaussian orthogonal ensembles in
the transition from classically integrable to chaotic dynamics. Here we find
that, as a function of and , there are clear
quantum signatures in the magnetic response, when going from the
single-particle classically regular to chaotic regimes. In the quasi-integrable
regime the magnetization non-monotonically oscillates between diamagnetic and
paramagnetic as a function of . We quantitatively understand this behavior
from a perturbation theory analysis. In the chaotic regime, however, we find
that the magnetization oscillates as a function of but it is {\it always}
diamagnetic. Equivalent results are also presented for the orbital currents. We
also find that the time-averaged energy grows like in the
quasi-integrable regime but changes to a linear dependence in the chaotic
regime. In contrast, the results with Bose statistics are akin to the
single-particle case and thus different from the fermionic case. We also give
an estimate of possible experimental parameters were our results may be seen in
semiconductor quantum dot billiards.Comment: 22 pages, 7 GIF figures, Phys. Rev. E. (1999
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