234 research outputs found
The fast sampling algorithm for Lie-Trotter products
A fast algorithm for path sampling in path integral Monte Carlo simulations
is proposed. The algorithm utilizes the Levy-Ciesielski implementation of
Lie-Trotter products to achieve a mathematically proven computational cost of
n*log_2(n) with the number of time slices n, despite the fact that each path
variable is updated separately, for reasons of optimality. In this respect, we
demonstrate that updating a group of random variables simultaneously results in
loss of efficiency.Comment: 4 pages, 1 figure; fast rejection from Phys. Rev. Letts; transfered
to PRE as a Rapid Communication. Eq. 6 to 10 contained some inconsistencies
that have been repaired in the present version; A sample code implementing
the algorithm for LJ clusters is available from the author upon reques
On the efficient Monte Carlo implementation of path integrals
We demonstrate that the Levy-Ciesielski implementation of Lie-Trotter
products enjoys several properties that make it extremely suitable for
path-integral Monte Carlo simulations: fast computation of paths, fast Monte
Carlo sampling, and the ability to use different numbers of time slices for the
different degrees of freedom, commensurate with the quantum effects. It is
demonstrated that a Monte Carlo simulation for which particles or small groups
of variables are updated in a sequential fashion has a statistical efficiency
that is always comparable to or better than that of an all-particle or
all-variable update sampler. The sequential sampler results in significant
computational savings if updating a variable costs only a fraction of the cost
for updating all variables simultaneously or if the variables are independent.
In the Levy-Ciesielski representation, the path variables are grouped in a
small number of layers, with the variables from the same layer being
statistically independent. The superior performance of the fast sampling
algorithm is shown to be a consequence of these observations. Both mathematical
arguments and numerical simulations are employed in order to quantify the
computational advantages of the sequential sampler, the Levy-Ciesielski
implementation of path integrals, and the fast sampling algorithm.Comment: 14 pages, 3 figures; submitted to Phys. Rev.
Upon the existence of short-time approximations of any polynomial order for the computation of density matrices by path integral methods
In this article, I provide significant mathematical evidence in support of
the existence of short-time approximations of any polynomial order for the
computation of density matrices of physical systems described by arbitrarily
smooth and bounded from below potentials. While for Theorem 2, which is
``experimental'', I only provide a ``physicist's'' proof, I believe the present
development is mathematically sound. As a verification, I explicitly construct
two short-time approximations to the density matrix having convergence orders 3
and 4, respectively. Furthermore, in the Appendix, I derive the convergence
constant for the trapezoidal Trotter path integral technique. The convergence
orders and constants are then verified by numerical simulations. While the two
short-time approximations constructed are of sure interest to physicists and
chemists involved in Monte Carlo path integral simulations, the present article
is also aimed at the mathematical community, who might find the results
interesting and worth exploring. I conclude the paper by discussing the
implications of the present findings with respect to the solvability of the
dynamical sign problem appearing in real-time Feynman path integral
simulations.Comment: 19 pages, 4 figures; the discrete short-time approximations are now
treated as independent from their continuous version; new examples of
discrete short-time approximations of order three and four are given; a new
appendix containing a short review on Brownian motion has been added; also,
some additional explanations are provided here and there; this is the last
version; to appear in Phys. Rev.
Hard Discs on the Hyperbolic Plane
We examine a simple hard disc fluid with no long range interactions on the
two dimensional space of constant negative Gaussian curvature, the hyperbolic
plane. This geometry provides a natural mechanism by which global crystalline
order is frustrated, allowing us to construct a tractable model of disordered
monodisperse hard discs. We extend free area theory and the virial expansion to
this regime, deriving the equation of state for the system, and compare its
predictions with simulation near an isostatic packing in the curved space.Comment: 4 pages, 3 figures, included, final versio
Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model
This paper deals with a one--dimensional model for granular materials, which
boils down to an inelastic version of the Kac kinetic equation, with
inelasticity parameter . In particular, the paper provides bounds for
certain distances -- such as specific weighted --distances and the
Kolmogorov distance -- between the solution of that equation and the limit. It
is assumed that the even part of the initial datum (which determines the
asymptotic properties of the solution) belongs to the domain of normal
attraction of a symmetric stable distribution with characteristic exponent
\a=2/(1+p). With such initial data, it turns out that the limit exists and is
just the aforementioned stable distribution. A necessary condition for the
relaxation to equilibrium is also proved. Some bounds are obtained without
introducing any extra--condition. Sharper bounds, of an exponential type, are
exhibited in the presence of additional assumptions concerning either the
behaviour, near to the origin, of the initial characteristic function, or the
behaviour, at infinity, of the initial probability distribution function
Brownian Motion and Polymer Statistics on Certain Curved Manifolds
We have calculated the probability distribution function G(R,L|R',0) of the
end-to-end vector R-R' and the mean-square end-to-end distance (R-R')^2 of a
Gaussian polymer chain embedded on a sphere S^(D-1) in D dimensions and on a
cylinder, a cone and a curved torus in 3-D.
We showed that: surface curvature induces a geometrical localization area; at
short length the polymer is locally "flat" and (R-R')^2 = L l in all cases; at
large scales, (R-R')^2 is constant for the sphere, it is linear in L for the
cylinder and reaches different constant values for the torus. The cone vertex
induces (function of opening angle and R') contraction of the chain for all
lengths. Explicit crossover formulas are derived.Comment: 9 pages, 4 figures, RevTex, uses amssymb.sty and multicol.sty, to
appear in Phys. Rev
Persistent spins in the linear diffusion approximation of phase ordering and zeros of stationary gaussian processes
The fraction r(t) of spins which have never flipped up to time t is studied
within a linear diffusion approximation to phase ordering. Numerical
simulations show that, even in this simple context, r(t) decays with time like
a power-law with a non-trival exponent which depends on the space
dimension. The local dynamics at a given point is a special case of a
stationary gaussian process of known correlation function and the exponent
is shown to be determined by the asymptotic behavior of the
probability distribution of intervals between consecutive zero-crossings of
this process. An approximate way of computing this distribution is proposed, by
taking the lengths of the intervals between successive zero-crossings as
independent random variables. The approximation gives values of the exponent
in close agreement with the results of simulations.Comment: 10 pages, 2 postscript files. Submitted to PRL. Reference screwup
correcte
Curvature in Noncommutative Geometry
Our understanding of the notion of curvature in a noncommutative setting has
progressed substantially in the past ten years. This new episode in
noncommutative geometry started when a Gauss-Bonnet theorem was proved by
Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral
geometry and heat kernel asymptotic expansions suggest a general way of
defining local curvature invariants for noncommutative Riemannian type spaces
where the metric structure is encoded by a Dirac type operator. To carry
explicit computations however one needs quite intriguing new ideas. We give an
account of the most recent developments on the notion of curvature in
noncommutative geometry in this paper.Comment: 76 pages, 8 figures, final version, one section on open problems
added, and references expanded. Appears in "Advances in Noncommutative
Geometry - on the occasion of Alain Connes' 70th birthday
Self-similarity and power-like tails in nonconservative kinetic models
In this paper, we discuss the large--time behavior of solution of a simple
kinetic model of Boltzmann--Maxwell type, such that the temperature is time
decreasing and/or time increasing. We show that, under the combined effects of
the nonlinearity and of the time--monotonicity of the temperature, the kinetic
model has non trivial quasi-stationary states with power law tails. In order to
do this we consider a suitable asymptotic limit of the model yielding a
Fokker-Planck equation for the distribution. The same idea is applied to
investigate the large-time behavior of an elementary kinetic model of economy
involving both exchanges between agents and increasing and/or decreasing of the
mean wealth. In this last case, the large-time behavior of the solution shows a
Pareto power law tail. Numerical results confirm the previous analysis
The direction of research into visual disability and quality of life in glaucoma
<p>Abstract</p> <p>Background</p> <p>Glaucoma will undoubtedly impact on a person's ability to function as they go about their day-to-day life. The purpose of this study is to investigate the amount of published knowledge in quality of life (QoL) and visual disability studies for glaucoma, and make comparisons with similar research in other chronic conditions.</p> <p>Methods</p> <p>A systematic literature search of the Global Health, EMBASE Psychiatry and MEDLINE databases. Title searches for glaucoma and six other example chronic diseases were entered alongside a selection of keywords chosen to capture studies focusing on QoL and everyday task ability. These results were further filtered during a manual search of resulting abstracts. Outcomes were the number of publications per year for each disease, number relating to QoL and type of glaucoma QoL research.</p> <p>Results</p> <p>Fifteen years ago there were no published studies relating to the impact of glaucoma on QoL but by 2009 this had risen to 1.2% of all glaucoma articles. The number of papers relating to QoL as a proportion of all papers in glaucoma in the past 10 years (0.6%) is smaller than for AMD and some other disabling chronic diseases. Most QoL studies in glaucoma (82%) involve questionnaires.</p> <p>Conclusion</p> <p>QoL studies in glaucoma are increasing in number but represent a tiny minority of the total publications in glaucoma research. There are fewer QoL articles in glaucoma compared to some other disabling chronic conditions. The majority of QoL articles in glaucoma research use questionnaires; performance-based measures of visual disability may offer an additional method of determining how the disease impacts on QoL.</p
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