A differential algebra based importance sampling method for impact probability computation on Earth resonant returns of Near Earth Objects

A differential algebra based importance sampling method for uncertainty propagation and impact probability computation on the first resonant returns of Near Earth Objects is presented in this paper. Starting from the results of an orbit determination process, we use a differential algebra based automatic domain pruning to estimate resonances and automatically propagate in time the regions of the initial uncertainty set that include the resonant return of interest. The result is a list of polynomial state vectors, each mapping specific regions of the uncertainty set from the observation epoch to the resonant return. Then, we employ a Monte Carlo importance sampling technique on the generated subsets for impact probability computation. We assess the performance of the proposed approach on the case of asteroid (99942) Apophis. A sensitivity analysis on the main parameters of the technique is carried out, providing guidelines for their selection. We finally compare the results of the proposed method to standard and advanced orbital sampling techniques

Brownian Motion in a Weyl Chamber, Non-Colliding Particles, and Random Matrices

Let $n$ particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is $A_{n-1}$, the symmetric group. For any starting positions, we compute a determinant formula for the density function for the particles to be at specified positions at time $t$ without having collided by time $t$. We show that the probability that there will be no collision up to time $t$ is asymptotic to a constant multiple of $t^{-n(n-1)/4}$ as $t$ goes to infinity, and compute the constant as a polynomial of the starting positions. We have analogous results for the other classical Weyl groups; for example, the hyperoctahedral group $B_n$ gives a model of $n$ independent particles with a wall at $x=0$. We can define Brownian motion on a Lie algebra, viewing it as a vector space; the eigenvalues of a point in the Lie algebra correspond to a point in the Weyl chamber, giving a Brownian motion conditioned never to exit the chamber. If there are $m$ roots in $n$ dimensions, this shows that the radial part of the conditioned process is the same as the $n+2m$-dimensional Bessel process. The conditioned process also gives physical models, generalizing Dyson's model for $A_{n-1}$ corresponding to ${\mathfrak s}{\mathfrak u}_n$ of $n$ particles moving in a diffusion with a repelling force between two particles proportional to the inverse of the distance between them

Energy correlations for a random matrix model of disordered bosons

Linearizing the Heisenberg equations of motion around the ground state of an interacting quantum many-body system, one gets a time-evolution generator in the positive cone of a real symplectic Lie algebra. The presence of disorder in the physical system determines a probability measure with support on this cone. The present paper analyzes a discrete family of such measures of exponential type, and does so in an attempt to capture, by a simple random matrix model, some generic statistical features of the characteristic frequencies of disordered bosonic quasi-particle systems. The level correlation functions of the said measures are shown to be those of a determinantal process, and the kernel of the process is expressed as a sum of bi-orthogonal polynomials. While the correlations in the bulk scaling limit are in accord with sine-kernel or GUE universality, at the low-frequency end of the spectrum an unusual type of scaling behavior is found.Comment: 20 pages, 3 figures, references adde

Random Lie group actions on compact manifolds: A perturbative analysis

A random Lie group action on a compact manifold generates a discrete time Markov process. The main object of this paper is the evaluation of associated Birkhoff sums in a regime of weak, but sufficiently effective coupling of the randomness. This effectiveness is expressed in terms of random Lie algebra elements and replaces the transience or Furstenberg's irreducibility hypothesis in related problems. The Birkhoff sum of any given smooth function then turns out to be equal to its integral w.r.t. a unique smooth measure on the manifold up to errors of the order of the coupling constant. Applications to the theory of products of random matrices and a model of a disordered quantum wire are presented.Comment: Published in at http://dx.doi.org/10.1214/10-AOP544 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Information Physics: The New Frontier

At this point in time, two major areas of physics, statistical mechanics and quantum mechanics, rest on the foundations of probability and entropy. The last century saw several significant fundamental advances in our understanding of the process of inference, which make it clear that these are inferential theories. That is, rather than being a description of the behavior of the universe, these theories describe how observers can make optimal predictions about the universe. In such a picture, information plays a critical role. What is more is that little clues, such as the fact that black holes have entropy, continue to suggest that information is fundamental to physics in general. In the last decade, our fundamental understanding of probability theory has led to a Bayesian revolution. In addition, we have come to recognize that the foundations go far deeper and that Cox's approach of generalizing a Boolean algebra to a probability calculus is the first specific example of the more fundamental idea of assigning valuations to partially-ordered sets. By considering this as a natural way to introduce quantification to the more fundamental notion of ordering, one obtains an entirely new way of deriving physical laws. I will introduce this new way of thinking by demonstrating how one can quantify partially-ordered sets and, in the process, derive physical laws. The implication is that physical law does not reflect the order in the universe, instead it is derived from the order imposed by our description of the universe. Information physics, which is based on understanding the ways in which we both quantify and process information about the world around us, is a fundamentally new approach to science.Comment: 17 pages, 6 figures. Knuth K.H. 2010. Information physics: The new frontier. J.-F. Bercher, P. Bessi\`ere, and A. Mohammad-Djafari (eds.) Bayesian Inference and Maximum Entropy Methods in Science and Engineering (MaxEnt 2010), Chamonix, France, July 201