Heavy Tails, Importance Sampling and Cross-Entropy

We consider the problem of estimating P (Y1+ ... +Yn > x) by importance sampling when the Yi are i.i.d. and heavy-tailed. The idea is to exploit the cross-entropy method as a tool for choosing good parameters in the importance sampling distribution; in doing so, we use the asymptotic description that given P(Y1+ ... +Yn > x,) n-1 of the Yi have distribution F and one the conditional distribution of Y given Y > x. We show in some parametric examples (Pareto and Weibull) how this leads to precise answers, which as demonstrated numerically, are close to being variance minimal within the parametric class under consideration. Related problems for M/G/1 and GI/G/1 queues are also discussed

The transform likelihood ratio method for rare event simulation with heavy tails

We present a novel method, called the transform likelihood ratio (TLR) method, for estimation of rare event probabilities with heavy-tailed distributions. Via a simple transformation ( change of variables) technique the TLR method reduces the original rare event probability estimation with heavy tail distributions to an equivalent one with light tail distributions. Once this transformation has been established we estimate the rare event probability via importance sampling, using the classical exponential change of measure or the standard likelihood ratio change of measure. In the latter case the importance sampling distribution is chosen from the same parametric family as the transformed distribution. We estimate the optimal parameter vector of the importance sampling distribution using the cross-entropy method. We prove the polynomial complexity of the TLR method for certain heavy-tailed models and demonstrate numerically its high efficiency for various heavy-tailed models previously thought to be intractable. We also show that the TLR method can be viewed as a universal tool in the sense that not only it provides a unified view for heavy-tailed simulation but also can be efficiently used in simulation with light-tailed distributions. We present extensive simulation results which support the efficiency of the TLR method

Instabilities of wave function monopoles in Bose-Einstein condensates

We present analytic and numerical results for a class of monopole solutions to the two-component Gross-Pitaevski equation for a two-species Bose condensate in an effectively two-dimensional trap. We exhibit dynamical instabilities involving vortex production as one species pours through another, from which we conclude that the sub-optical sharpness of potentials exerted by matter waves makes condensates ideal tools for manipulating condensates. We also show that there are two equally valid but drastically different hydrodynamic descriptions of a two-component condensate, and illustrate how different phenomena may appear simpler in each.Comment: 4 pages, 9 figures (compressed figures become legible when zoomed or when paper is actually printed

The cross-entropy method for continuous multi-extremal optimization

In recent years, the cross-entropy method has been successfully applied to a wide range of discrete optimization tasks. In this paper we consider the cross-entropy method in the context of continuous optimization. We demonstrate the effectiveness of the cross-entropy method for solving difficult continuous multi-extremal optimization problems, including those with non-linear constraints

Static and dynamic properties of large polymer melts in equilibrium

We present a detailed study of the static and dynamic behavior of long semiflexible polymer chains in a melt. Starting from previously obtained fully equilibrated high molecular weight polymer melts [{\it Zhang et al.} ACS Macro Lett. 3, 198 (2014)] we investigate their static and dynamic scaling behavior as predicted by theory. We find that for semiflexible chains in a melt, results of the mean square internal distance, the probability distributions of the end-to-end distance, and the chain structure factor are well described by theoretical predictions for ideal chains. We examine the motion of monomers and chains by molecular dynamics simulations using the ESPResSo++ package. The scaling predictions of the mean squared displacement of inner monomers, center of mass, and relations between them based on the Rouse and the reptation theory are verified, and related characteristic relaxation times are determined. Finally we give evidence that the entanglement length $N_{e,PPA}$ as determined by a primitive path analysis (PPA) predicts a plateau modulus, $G_N^0=\frac{4}{5}(\rho k_BT/N_e)$, consistent with stresses obtained from the Green-Kubo relation. These comprehensively characterized equilibrium structures, which offer a good compromise between flexibility, small $N_e$, computational efficiency, and small deviations from ideality provide ideal starting states for future non-equilibrium studies.Comment: 13 pages, 10 figures, to be published in J. Chem. Phys. (2016

Vortex states of rapidly rotating dilute Bose-Einstein condensates

We show that, in the Thomas-Fermi regime, the cores of vortices in rotating dilute Bose-Einstein condensates adjust in radius as the rotation velocity, $\Omega$, grows, thus precluding a phase transition associated with core overlap at high vortex density. In both a harmonic trap and a rotating hard-walled bucket, the core size approaches a limiting fraction of the intervortex spacing. At large rotation speeds, a system confined in a bucket develops, within Thomas-Fermi, a hole along the rotation axis, and eventually makes a transition to a giant vortex state with all the vorticity contained in the hole.Comment: 4 pages, 2 figures, RevTex4. Version as published; discussion extended, some references added and update

Periodic One-Dimensional Hopping Model with one Mobile Directional Impurity

Analytic solution is given in the steady state limit for the system of Master equations describing a random walk on one-dimensional periodic lattices with arbitrary hopping rates containing one mobile, directional impurity (defect bond). Due to the defect, translational invariance is broken, even if all other rates are identical. The structure of Master equations lead naturally to the introduction of a new entity, associated with the walker-impurity pair which we call the quasi-walker. The velocities and diffusion constants for both the random walker and impurity are given, being simply related to that of the quasi-particle through physically meaningful equations. Applications in driven diffusive systems are shown, and connections with the Duke-Rubinstein reptation models for gel electrophoresis are discussed.Comment: 31 LaTex pages, 5 Postscript figures included, to appear in Journal of Statistical Physic

Mechanical Properties of End-crosslinked Entangled Polymer Networks using Sliplink Brownian Dynamics Simulations

The mechanical properties of a polymeric network containing both crosslinks and sliplinks (entanglements) are studied using a multi-chain Brownian dynamics simulation. We coarse-grain at the level of chain segments connecting consecutive nodes (cross- or sliplinks), with particular attention to the Gaussian statistics of the network. Affine displacement of nodes is not imposed: their displacement as well as sliding of monomers through sliplinks is governed by force balances. The simulation results of stress in uniaxial extension and the full stress tensor in simple shear including the (non-zero) second normal stress difference are presented for monodisperse chains with up to 18 entanglements between two crosslinks. The cases of two different force laws of the subchains (Gaussian chains and chains with finite extensibility) for two different numbers of monomers in a subchain (no = 50 and no = 100) are examined. It is shown that the additivity assumption of slip- and crosslink contribution holds for sufficiently long chains with two or more entanglements, and that it can be used to construct the strain response of a network of infinitely long chains. An important consequence is that the contribution of sliplinks to the small-strain shear modulus is about ⅔ of the contribution of a crosslink