Computability of entropy and information in classical Hamiltonian systems

We consider the computability of entropy and information in classical Hamiltonian systems. We define the information part and total information capacity part of entropy in classical Hamiltonian systems using relative information under a computable discrete partition. Using a recursively enumerable nonrecursive set it is shown that even though the initial probability distribution, entropy, Hamiltonian and its partial derivatives are computable under a computable partition, the time evolution of its information capacity under the original partition can grow faster than any recursive function. This implies that even though the probability measure and information are conserved in classical Hamiltonian time evolution we might not actually compute the information with respect to the original computable partition

Two Body Relaxation in Simulated Cosmological Haloes

This paper aims at quantifying discreetness effects, born of finite particle number, on the dynamics of dark matter haloes forming in the context of cosmological simulations. By generalising the standard calculation of two body relaxation to the case when the size and mass distribution are variable, and parametrising the time evolution using established empirical relations, we find that the dynamics of a million particle halo is noise-dominated within the inner percent of the final virial radius. Far larger particle numbers (~ 10^8) are required for the RMS perturbations to the velocity to drop to the 10 % level there. The radial scaling of the relaxation time is simple and strong: t_relax ~ r^2, implying that numbers >> 10^8 are required to faithfully model the very inner regions; artificial relaxation may thus constitute an important factor, contributing to the contradictory claims concerning the persistence of a power law density cusp to the very centre. The cores of substructure haloes can be many relaxation times old. Since relaxation first causes their expansion before recontraction occurs, it may render them either more difficult or easier to disrupt, depending on their orbital parameters. It may thus modify the characteristics of the subhalo distribution and effects of interactions with the parent. We derive simple closed form formulas for the characteristic relaxation times, as well as for the weak N-scaling reported by Diemand et al. when the main contribution comes from relaxing subhaloes (abridged).Comment: 11 Pages, 7 figs, Monthly Notices styl

Does the Fornax dwarf spheroidal have a central cusp or core?

The dark matter dominated Fornax dwarf spheroidal has five globular clusters orbiting at ~1 kpc from its centre. In a cuspy CDM halo the globulars would sink to the centre from their current positions within a few Gyrs, presenting a puzzle as to why they survive undigested at the present epoch. We show that a solution to this timing problem is to adopt a cored dark matter halo. We use numerical simulations and analytic calculations to show that, under these conditions, the sinking time becomes many Hubble times; the globulars effectively stall at the dark matter core radius. We conclude that the Fornax dwarf spheroidal has a shallow inner density profile with a core radius constrained by the observed positions of its globular clusters. If the phase space density of the core is primordial then it implies a warm dark matter particle and gives an upper limit to its mass of ~0.5 keV, consistent with that required to significantly alleviate the substructure problem.Comment: 6 pages, 5 figures, accepted for publication in MNRAS, high resolution simulations include

Quantification of the performance of iterative and non-iterative computational methods of locating partial discharges using RF measurement techniques

Partial discharge (PD) is an electrical discharge phenomenon that occurs when the insulation materialof high voltage equipment is subjected to high electric field stress. Its occurrence can be an indication ofincipient failure within power equipment such as power transformers, underground transmission cableor switchgear. Radio frequency measurement methods can be used to detect and locate discharge sourcesby measuring the propagated electromagnetic wave arising as a result of ionic charge acceleration. Anarray of at least four receiving antennas may be employed to detect any radiated discharge signals, thenthe three dimensional position of the discharge source can be calculated using different algorithms. These algorithms fall into two categories; iterative or non-iterative. This paper evaluates, through simulation, the location performance of an iterative method (the standardleast squares method) and a non-iterative method (the Bancroft algorithm). Simulations were carried outusing (i) a "Y" shaped antenna array and (ii) a square shaped antenna array, each consisting of a four-antennas. The results show that PD location accuracy is influenced by the algorithm's error bound, thenumber of iterations and the initial values for the iterative algorithms, as well as the antenna arrangement for both the non-iterative and iterative algorithms. Furthermore, this research proposes a novel approachfor selecting adequate error bounds and number of iterations using results of the non-iterative method, thus solving some of the iterative method dependencies