4,098 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
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
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
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
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)
Yang-Lee zeroes for an urn model for the separation of sand
We apply the Yang-Lee theory of phase transitions to an urn model of
separation of sand. The effective partition function of this nonequilibrium
system can be expressed as a polynomial of the size-dependent effective
fugacity . Numerical calculations show that in the thermodynamic limit, the
zeros of the effective partition function are located on the unit circle in the
complex -plane. In the complex plane of the actual control parameter certain
roots converge to the transition point of the model. Thus the Yang-Lee theory
can be applied to a wider class of nonequilibrium systems than those considered
previously.Comment: 4 pages, 3 eps figures include
First Order Phase Transition in a Reaction-Diffusion Model With Open Boundary: The Yang-Lee Theory Approach
A coagulation-decoagulation model is introduced on a chain of length L with
open boundary. The model consists of one species of particles which diffuse,
coagulate and decoagulate preferentially in the leftward direction. They are
also injected and extracted from the left boundary with different rates. We
will show that on a specific plane in the space of parameters, the steady state
weights can be calculated exactly using a matrix product method. The model
exhibits a first-order phase transition between a low-density and a
high-density phase. The density profile of the particles in each phase is
obtained both analytically and using the Monte Carlo Simulation. The two-point
density-density correlation function in each phase has also been calculated. By
applying the Yang-Lee theory we can predict the same phase diagram for the
model. This model is further evidence for the applicability of the Yang-Lee
theory in the non-equilibrium statistical mechanics context.Comment: 10 Pages, 3 Figures, To appear in Journal of Physics A: Mathematical
and Genera
Developed turbulence: From full simulations to full mode reductions
Developed Navier-Stokes turbulence is simulated with varying wavevector mode
reductions. The flatness and the skewness of the velocity derivative depend on
the degree of mode reduction. They show a crossover towards the value of the
full numerical simulation when the viscous subrange starts to be resolved. The
intermittency corrections of the scaling exponents of the pth order velocity
structure functions seem to depend mainly on the proper resolution of the
inertial subrange. Universal scaling properties (i.e., independent of the
degree of mode reduction) are found for the relative scaling exponents rho
which were recently defined by Benzi et al.Comment: 4 pages, 5 eps-figures, replaces version from August 5th, 199
Aging and intermittency in a p-spin model of a glass
We numerically analyze the statistics of the heat flow between an aging
system and its thermal bath, following a method proposed and tested for a
spin-glass model in a recent Letter (P. Sibani and H.J. Jensen, Europhys.
Lett.69, 563 (2005)). The present system, which lacks quenched randomness,
consists of Ising spins located on a cubic lattice, with each plaquette
contributing to the total energy the product of the four spins located at its
corners. Similarly to our previous findings, energy leaves the system in rare
but large, so called intermittent, bursts which are embedded in reversible and
equilibrium-like fluctuations of zero average. The intermittent bursts, or
quakes, dissipate the excess energy trapped in the initial state at a rate
which falls off with the inverse of the age. This strongly heterogeneous
dynamical picture is explained using the idea that quakes are triggered by
energy fluctuations of record size, which occur independently within a number
of thermalized domains. From the temperature dependence of the width of the
reversible heat fluctuations we surmise that these domains have an exponential
density of states. Finally, we show that the heat flow consists of a
temperature independent term and a term with an Arrhenius temperature
dependence. Microscopic dynamical and structural information can thus be
extracted from numerical intermittency data. This type of analysis seems now
within the reach of time resolved micro-calorimetry techniques.Comment: 9 pages, 6 figures, europhysics letter style, to appear in Physical
Review
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