5,009 research outputs found
Some generic properties of level spacing distributions of 2D real random matrices
We study the level spacing distribution of 2D real random matrices
both symmetric as well as general, non-symmetric. In the general case we
restrict ourselves to Gaussian distributed matrix elements, but different
widths of the various matrix elements are admitted. The following results are
obtained: An explicit exact formula for is derived and its behaviour
close to S=0 is studied analytically, showing that there is linear level
repulsion, unless there are additional constraints for the probability
distribution of the matrix elements. The constraint of having only positive or
only negative but otherwise arbitrary non-diagonal elements leads to quadratic
level repulsion with logarithmic corrections. These findings detail and extend
our previous results already published in a preceding paper. For the {\em
symmetric} real 2D matrices also other, non-Gaussian statistical distributions
are considered. In this case we show for arbitrary statistical distribution of
the diagonal and non-diagonal elements that the level repulsion exponent
is always , provided the distribution function of the matrix elements
is regular at zero value. If the distribution function of the matrix elements
is a singular (but still integrable) power law near zero value of , the
level spacing distribution is a fractional exponent pawer law at small
. The tail of depends on further details of the matrix element
statistics. We explicitly work out four cases: the constant (box) distribution,
the Cauchy-Lorentz distribution, the exponential distribution and, as an
example for a singular distribution, the power law distribution for near
zero value times an exponential tail.Comment: 21 pages, no figures, submitted to Zeitschrift fuer Naturforschung
Extended phase diagram of the Lorenz model
The parameter dependence of the various attractive solutions of the three
variable nonlinear Lorenz model equations for thermal convection in
Rayleigh-B\'enard flow is studied. Its bifurcation structure has commonly been
investigated as a function of r, the normalized Rayleigh number, at fixed
Prandtl number \sigma. The present work extends the analysis to the entire
(r,\sigma) parameter plane. An onion like periodic pattern is found which is
due to the alternating stability of symmetric and non-symmetric periodic
orbits. This periodic pattern is explained by considering non-trivial limits of
large r and \sigma. In addition to the limit which was previously analyzed by
Sparrow, we identify two more distinct asymptotic regimes in which either
\sigma/r or \sigma^2/r is constant. In both limits the dynamics is
approximately described by Airy functions whence the periodicity in parameter
space can be calculated analytically. Furthermore, some observations about
sequences of bifurcations and coexistence of attractors, periodic as well as
chaotic, are reported.Comment: 36 pages, 20 figure
Kohn-Sham equations for nanowires with direct current
The paper describes the derivation of the Kohn-Sham equations for a nanowire
with direct current. A value of the electron current enters the problem as an
input via a subsidiary condition imposed by pointwise Lagrange multiplier.
Using the constrained minimization of the Hohenberg-Kohn energy functional, we
derive a set of self-consistent equations for current carrying orbitals of the
molecular wire
Spectra of Harmonium in a magnetic field using an initial value representation of the semiclassical propagator
For two Coulombically interacting electrons in a quantum dot with harmonic
confinement and a constant magnetic field, we show that time-dependent
semiclassical calculations using the Herman-Kluk initial value representation
of the propagator lead to eigenvalues of the same accuracy as WKB calculations
with Langer correction. The latter are restricted to integrable systems,
however, whereas the time-dependent initial value approach allows for
applications to high-dimensional, possibly chaotic dynamics and is extendable
to arbitrary shapes of the potential.Comment: 11 pages, 1 figur
Finite size corrections to scaling in high Reynolds number turbulence
We study analytically and numerically the corrections to scaling in
turbulence which arise due to the finite ratio of the outer scale of
turbulence to the viscous scale , i.e., they are due to finite size
effects as anisotropic forcing or boundary conditions at large scales. We find
that the deviations \dzm from the classical Kolmogorov scaling of the velocity moments \langle |\u(\k)|^m\rangle \propto k^{-\zeta_m}
decrease like . Our numerics employ a
reduced wave vector set approximation for which the small scale structures are
not fully resolved. Within this approximation we do not find independent
anomalous scaling within the inertial subrange. If anomalous scaling in the
inertial subrange can be verified in the large limit, this supports the
suggestion that small scale structures should be responsible, originating from
viscosity either in the bulk (vortex tubes or sheets) or from the boundary
layers (plumes or swirls)
Classification of phase transitions of finite Bose-Einstein condensates in power law traps by Fisher zeros
We present a detailed description of a classification scheme for phase
transitions in finite systems based on the distribution of Fisher zeros of the
canonical partition function in the complex temperature plane. We apply this
scheme to finite Bose-systems in power law traps within a semi-analytic
approach with a continuous one-particle density of states for different values of and to a three dimensional harmonically
confined ideal Bose-gas with discrete energy levels. Our results indicate that
the order of the Bose-Einstein condensation phase transition sensitively
depends on the confining potential.Comment: 7 pages, 9 eps-figures, For recent information on physics of small
systems see "http://www.smallsystems.de
Response maxima in modulated turbulence
Isotropic and homogeneous turbulence driven by an energy input modulated in
time is studied within a variable range mean-field theory. The response of the
system, observed in the second order moment of the large-scale velocity
difference D(L,t)=>~Re(t)^2$, is calculated for varying
modulation frequencies w and weak modulation amplitudes. For low frequencies
the system follows the modulation of the driving with almost constant
amplitude, whereas for higher driving frequencies the amplitude of the response
decreases on average 1/w. In addition, at certain frequencies the amplitude of
the response either almost vanishes or is strongly enhanced. These frequencies
are connected with the frequency scale of the energy cascade and multiples
thereof.Comment: 11 pages, 6 figure
Continued Fraction Representation of Temporal Multi Scaling in Turbulence
It was shown recently that the anomalous scaling of simultaneous correlation
functions in turbulence is intimately related to the breaking of temporal scale
invariance, which is equivalent to the appearance of infinitely many times
scales in the time dependence of time-correlation functions. In this paper we
derive a continued fraction representation of turbulent time correlation
functions which is exact and in which the multiplicity of time scales is
explicit. We demonstrate that this form yields precisely the same scaling laws
for time derivatives and time integrals as the "multi-fractal" representation
that was used before. Truncating the continued fraction representation yields
the "best" estimates of time correlation functions if the given information is
limited to the scaling exponents of the simultaneous correlation functions up
to a certain, finite order. It is worth noting that the derivation of a
continued fraction representation obtained here for an operator which is not
Hermitian or anti-Hermitian may be of independent interest.Comment: 7 pages, no figur
Universality in fully developed turbulence
We extend the numerical simulations of She et al. [Phys.\ Rev.\ Lett.\ 70,
3251 (1993)] of highly turbulent flow with Taylor-Reynolds number
up to , employing a reduced wave
vector set method (introduced earlier) to approximately solve the Navier-Stokes
equation. First, also for these extremely high Reynolds numbers ,
the energy spectra as well as the higher moments -- when scaled by the spectral
intensity at the wave number of peak dissipation -- can be described by
{\it one universal} function of for all . Second, the ISR
scaling exponents of this universal function are in agreement with
the 1941 Kolmogorov theory (the better, the large is), as is the
dependence of . Only around viscous damping leads to
slight energy pileup in the spectra, as in the experimental data (bottleneck
phenomenon).Comment: 14 pages, Latex, 5 figures (on request), 3 tables, submitted to Phys.
Rev.
Scaling relations in large-Prandtl-number natural thermal convection
In this study we follow Grossmann and Lohse, Phys. Rev. Lett. 86 (2001), who
derived various scalings regimes for the dependence of the Nusselt number
and the Reynolds number on the Rayleigh number and the Prandtl number
. We focus on theoretical arguments as well as on numerical simulations for
the case of large- natural thermal convection. Based on an analysis of
self-similarity of the boundary layer equations, we derive that in this case
the limiting large- boundary-layer dominated regime is I,
introduced and defined in [1], with the scaling relations and . Our direct numerical
simulations for from to and from 0.1 to 200 show that
the regime I is almost indistinguishable from the regime
III, where the kinetic dissipation is bulk-dominated. With increasing
, the scaling relations undergo a transition to those in IV of
reference [1], where the thermal dissipation is determined by its bulk
contribution
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