19,744 research outputs found
Three-Body Recombination in One Dimension
We study the three-body problem in one dimension for both zero and finite
range interactions using the adiabatic hyperspherical approach. Particular
emphasis is placed on the threshold laws for recombination, which are derived
for all combinations of the parity and exchange symmetries. For bosons, we
provide a numerical demonstration of several universal features that appear in
the three-body system, and discuss how certain universal features in three
dimensions are different in one dimension. We show that the probability for
inelastic processes vanishes as the range of the pair-wise interaction is taken
to zero and demonstrate numerically that the recombination threshold law
manifests itself for large scattering length.Comment: 15 pages 7 figures Submitted to Physical Review
Constant Rank Bimatrix Games are PPAD-hard
The rank of a bimatrix game (A,B) is defined as rank(A+B). Computing a Nash
equilibrium (NE) of a rank-, i.e., zero-sum game is equivalent to linear
programming (von Neumann'28, Dantzig'51). In 2005, Kannan and Theobald gave an
FPTAS for constant rank games, and asked if there exists a polynomial time
algorithm to compute an exact NE. Adsul et al. (2011) answered this question
affirmatively for rank- games, leaving rank-2 and beyond unresolved.
In this paper we show that NE computation in games with rank , is
PPAD-hard, settling a decade long open problem. Interestingly, this is the
first instance that a problem with an FPTAS turns out to be PPAD-hard. Our
reduction bypasses graphical games and game gadgets, and provides a simpler
proof of PPAD-hardness for NE computation in bimatrix games. In addition, we
get:
* An equivalence between 2D-Linear-FIXP and PPAD, improving a result by
Etessami and Yannakakis (2007) on equivalence between Linear-FIXP and PPAD.
* NE computation in a bimatrix game with convex set of Nash equilibria is as
hard as solving a simple stochastic game.
* Computing a symmetric NE of a symmetric bimatrix game with rank is
PPAD-hard.
* Computing a (1/poly(n))-approximate fixed-point of a (Linear-FIXP)
piecewise-linear function is PPAD-hard.
The status of rank- games remains unresolved
A two-species continuum model for aeolian sand ripples
We formulate a continuum model for aeolian sand ripples consisting of two
species of grains: a lower layer of relatively immobile clusters, with an upper
layer of highly mobile grains moving on top. We predict analytically the ripple
wavelength, initial ripple growth rate and threshold saltation flux for ripple
formation. Numerical simulations show the evolution of realistic ripple
profiles from initial surface roughness via ripple growth and merger.Comment: 9 pages, 3 figure
Level statistics for quantum -core percolation
Quantum -core percolation is the study of quantum transport on -core
percolation clusters where each occupied bond must have at least occupied
neighboring bonds. As the bond occupation probability, , is increased from
zero to unity, the system undergoes a transition from an insulating phase to a
metallic phase. When the lengthscale for the disorder, , is much greater
than the coherence length, , earlier analytical calculations of quantum
conduction on the Bethe lattice demonstrate that for the metal-insulator
transition (MIT) is discontinuous, suggesting a new universality class of
disorder-driven quantum MITs. Here, we numerically compute the level spacing
distribution as a function of bond occupation probability and system size
on a Bethe-like lattice. The level spacing analysis suggests that for ,
, the quantum percolation critical probability, is greater than , the
geometrical percolation critical probability, and the transition is continuous.
In contrast, for , and the transition is discontinuous such that
these numerical findings are consistent with our previous work to reiterate a
new universality class of disorder-driven quantum MITs.Comment: 8 pages, 11 figure
Asymptotics of skew orthogonal polynomials
Exact integral expressions of the skew orthogonal polynomials involved in
Orthogonal (beta=1) and Symplectic (beta=4) random matrix ensembles are
obtained: the (even rank) skew orthogonal polynomials are average
characteristic polynomials of random matrices. From there, asymptotics of the
skew orthogonal polynomials are derived.Comment: 17 pages, Late
Integrability of the critical point of the Kagom\'e three-state Potts mode
The vicinity of the critical point of the three-state Potts model on a
Kagom\'e lattice is studied by mean of Random Matrix Theory. Strong evidence
that the critical point is integrable is given.Comment: 1 LaTex file + 3 eps files 7 page
Intruder States and their Local Effect on Spectral Statistics
The effect on spectral statistics and on the revival probability of intruder
states in a random background is analysed numerically and with perturbative
methods. For random coupling the intruder does not affect the GOE spectral
statistics of the background significantly, while a constant coupling causes
very strong correlations at short range with a fourth power dependence of the
spectral two-point function at the origin.The revival probability is
significantly depressed for constant coupling as compared to random coupling.Comment: 18 pages, 10 Postscript figure
Spectral Properties of Statistical Mechanics Models
The full spectrum of transfer matrices of the general eight-vertex model on a
square lattice is obtained by numerical diagonalization. The eigenvalue spacing
distribution and the spectral rigidity are analyzed. In non-integrable regimes
we have found eigenvalue repulsion as for the Gaussian orthogonal ensemble in
random matrix theory. By contrast, in integrable regimes we have found
eigenvalue independence leading to a Poissonian behavior, and, for some points,
level clustering. These first examples from classical statistical mechanics
suggest that the conjecture of integrability successfully applied to quantum
spin systems also holds for classical systems.Comment: 4 pages, 1 Revtex file and 4 postscript figures tarred, gzipped and
uuencode
Poisson-to-Wigner crossover transition in the nearest-neighbor spacing statistics of random points on fractals
We show that the nearest-neighbor spacing distribution for a model that
consists of random points uniformly distributed on a self-similar fractal is
the Brody distribution of random matrix theory. In the usual context of
Hamiltonian systems, the Brody parameter does not have a definite physical
meaning, but in the model considered here, the Brody parameter is actually the
fractal dimension. Exploiting this result, we introduce a new model for a
crossover transition between Poisson and Wigner statistics: random points on a
continuous family of self-similar curves with fractal dimension between 1 and
2. The implications to quantum chaos are discussed, and a connection to
conservative classical chaos is introduced.Comment: Low-resolution figure is included here. Full resolution image
available (upon request) from the author
Splitting of Andreev levels in a Josephson junction by spin-orbit coupling
We consider the effect of spin-orbit coupling on the energy levels of a
single-channel Josephson junction below the superconducting gap. We investigate
quantitatively the level splitting arising from the combined effect of
spin-orbit coupling and the time-reversal symmetry breaking by the phase
difference between the superconductors. Using the scattering matrix approach we
establish a simple connection between the quantum mechanical time delay matrix
and the effective Hamiltonian for the level splitting. As an application we
calculate the distribution of level splittings for an ensemble of chaotic
Josephson junctions. The distribution falls off as a power law for large
splittings, unlike the exponentially decaying splitting distribution given by
the Wigner surmise -- which applies for normal chaotic quantum dots with
spin-orbit coupling in the case that the time-reversal symmetry breaking is due
to a magnetic field.Comment: 6 pages, 3 figure
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