252 research outputs found
Quantum Diffusion and Delocalization for Band Matrices with General Distribution
We consider Hermitian and symmetric random band matrices in
dimensions. The matrix elements , indexed by , are independent and their variances satisfy \sigma_{xy}^2:=\E
\abs{H_{xy}}^2 = W^{-d} f((x - y)/W) for some probability density . We
assume that the law of each matrix element is symmetric and exhibits
subexponential decay. We prove that the time evolution of a quantum particle
subject to the Hamiltonian is diffusive on time scales . We
also show that the localization length of the eigenvectors of is larger
than a factor times the band width . All results are uniform in
the size \abs{\Lambda} of the matrix. This extends our recent result
\cite{erdosknowles} to general band matrices. As another consequence of our
proof we show that, for a larger class of random matrices satisfying
for all , the largest eigenvalue of is bounded
with high probability by for any ,
where M \deq 1 / (\max_{x,y} \sigma_{xy}^2).Comment: Corrected typos and some inaccuracies in appendix
Alternative sampling for variational quantum Monte Carlo
Expectation values of physical quantities may accurately be obtained by the
evaluation of integrals within Many-Body Quantum mechanics, and these
multi-dimensional integrals may be estimated using Monte Carlo methods. In a
previous publication it has been shown that for the simplest, most commonly
applied strategy in continuum Quantum Monte Carlo, the random error in the
resulting estimates is not well controlled. At best the Central Limit theorem
is valid in its weakest form, and at worst it is invalid and replaced by an
alternative Generalised Central Limit theorem and non-Normal random error. In
both cases the random error is not controlled. Here we consider a new `residual
sampling strategy' that reintroduces the Central Limit Theorem in its strongest
form, and provides full control of the random error in estimates. Estimates of
the total energy and the variance of the local energy within Variational Monte
Carlo are considered in detail, and the approach presented may be generalised
to expectation values of other operators, and to other variants of the Quantum
Monte Carlo method.Comment: 14 pages, 9 figure
Mixing by polymers: experimental test of decay regime of mixing
By using high molecular weight fluorescent passive tracers with different
diffusion coefficients and by changing the fluid velocity we study dependence
of a characteristic mixing length on the Peclet number, , which controls
the mixing efficiency. The mixing length is found to be related to by a
power law, , and increases faster than
expected for an unbounded chaotic flow. Role of the boundaries in the mixing
length abnormal growth is clarified. The experimental findings are in a good
quantitative agreement with the recent theoretical predictions.Comment: 4 pages,5 figures. accepted for publication in PR
Quantification of the performance of chaotic micromixers on the basis of finite time Lyapunov exponents
Chaotic micromixers such as the staggered herringbone mixer developed by
Stroock et al. allow efficient mixing of fluids even at low Reynolds number by
repeated stretching and folding of the fluid interfaces. The ability of the
fluid to mix well depends on the rate at which "chaotic advection" occurs in
the mixer. An optimization of mixer geometries is a non trivial task which is
often performed by time consuming and expensive trial and error experiments. In
this paper an algorithm is presented that applies the concept of finite-time
Lyapunov exponents to obtain a quantitative measure of the chaotic advection of
the flow and hence the performance of micromixers. By performing lattice
Boltzmann simulations of the flow inside a mixer geometry, introducing massless
and non-interacting tracer particles and following their trajectories the
finite time Lyapunov exponents can be calculated. The applicability of the
method is demonstrated by a comparison of the improved geometrical structure of
the staggered herringbone mixer with available literature data.Comment: 9 pages, 8 figure
Dissipation time and decay of correlations
We consider the effect of noise on the dynamics generated by
volume-preserving maps on a d-dimensional torus. The quantity we use to measure
the irreversibility of the dynamics is the dissipation time. We focus on the
asymptotic behaviour of this time in the limit of small noise. We derive
universal lower and upper bounds for the dissipation time in terms of various
properties of the map and its associated propagators: spectral properties,
local expansivity, and global mixing properties. We show that the dissipation
is slow for a general class of non-weakly-mixing maps; on the opposite, it is
fast for a large class of exponentially mixing systems which include uniformly
expanding maps and Anosov diffeomorphisms.Comment: 26 Pages, LaTex. Submitted to Nonlinearit
Convergence of the stochastic Euler scheme for locally Lipschitz coefficients
Stochastic differential equations are often simulated with the Monte Carlo
Euler method. Convergence of this method is well understood in the case of
globally Lipschitz continuous coefficients of the stochastic differential
equation. The important case of superlinearly growing coefficients, however,
has remained an open question. The main difficulty is that numerically weak
convergence fails to hold in many cases of superlinearly growing coefficients.
In this paper we overcome this difficulty and establish convergence of the
Monte Carlo Euler method for a large class of one-dimensional stochastic
differential equations whose drift functions have at most polynomial growth.Comment: Published at http://www.springerlink.com/content/g076w80730811vv3 in
the Foundations of Computational Mathematics 201
Growth of uniform infinite causal triangulations
We introduce a growth process which samples sections of uniform infinite
causal triangulations by elementary moves in which a single triangle is added.
A relation to a random walk on the integer half line is shown. This relation is
used to estimate the geodesic distance of a given triangle to the rooted
boundary in terms of the time of the growth process and to determine from this
the fractal dimension. Furthermore, convergence of the boundary process to a
diffusion process is shown leading to an interesting duality relation between
the growth process and a corresponding branching process.Comment: 27 pages, 6 figures, small changes, as publishe
Linear response theory and transient fluctuation theorems for diffusion processes: a backward point of view
On the basis of perturbed Kolmogorov backward equations and path integral
representation, we unify the derivations of the linear response theory and
transient fluctuation theorems for continuous diffusion processes from a
backward point of view. We find that a variety of transient fluctuation
theorems could be interpreted as a consequence of a generalized
Chapman-Kolmogorov equation, which intrinsically arises from the Markovian
characteristic of diffusion processes
Dependent coordinates in path integral measure factorization
The transformation of the path integral measure under the reduction procedure
in the dynamical systems with a symmetry is considered. The investigation is
carried out in the case of the Wiener--type path integrals that are used for
description of the diffusion on a smooth compact Riemannian manifold with the
given free isometric action of the compact semisimple unimodular Lie group. The
transformation of the path integral, which factorizes the path integral
measure, is based on the application of the optimal nonlinear filtering
equation from the stochastic theory. The integral relation between the kernels
of the original and reduced semigroup are obtained.Comment: LaTeX2e, 28 page
- …