arXiv subjects

M. Wilkinson

Publications and source records attributed to M. Wilkinson.

34 records · Page 2Linked to original sources

arXiv2006

Generalised Ornstein-Uhlenbeck processes

We solve a physically significant extension of a classic problem in the theory of diffusion, namely the Ornstein-Uhlenbeck process [G. E. Ornstein and L. S. Uhlenbeck, Phys. Rev. 36, 823, (1930)]. Our generalised Ornstein-Uhlenbeck systems include a force which depends upon the position of the particle, as well as upon time. They exhibit anomalous diffusion at short times, and non-Maxwellian velocity distributions in equilibrium. Two approaches are used. Some statistics are obtained from a closed-form expression for the propagator of the Fokker-Planck equation for the case where the particle is initially at rest. In the general case we use spectral decomposition of a Fokker-Planck equation, employing nonlinear creation and annihilation operators to generate the spectrum which consists of two staggered ladders.

cond-mat.stat-mech↗

arXiv2005

Staggered Ladder Spectra

We exactly solve a Fokker-Planck equation by determining its eigenvalues and eigenfunctions: we construct nonlinear second-order differential operators which act as raising and lowering operators, generating ladder spectra for the odd and even parity states. These are staggered: the odd-even separation differs from even-odd. The Fokker-Planck equation describes, in the limit of weak damping, a generalised Ornstein-Uhlenbeck process where the random force depends upon position as well as time. Our exact solution exhibits anomalous diffusion at short times and a stationary non-Maxwellian momentum distribution.

cond-mat.soft↗

arXiv2005

Clustering in mixing flows

We calculate the Lyapunov exponents for particles suspended in a random three-dimensional flow, concentrating on the limit where the viscous damping rate is small compared to the inverse correlation time. In this limit Lyapunov exponents are obtained as a power series in epsilon, a dimensionless measure of the particle inertia. Although the perturbation generates an asymptotic series, we obtain accurate results from a Pade-Borel summation. Our results prove that particles suspended in an incompressible random mixing flow can show pronounced clustering when the Stokes number is large and we characterise two distinct clustering effects which occur in that limit.

cond-mat.dis-nn↗

arXiv2004

Caustics in turbulent aerosols

Networks of caustics can occur in the distribution of particles suspended in a randomly moving gas. These can facilitate coagulation of particles by bringing them into close proximity, even in cases where the trajectories do not coalesce. We show that the long-time morphology of these caustic patterns is determined by the Lyapunov exponents lambda_1, lambda_2 of the suspended particles, as well as the rate J at which particles encounter caustics. We develop a theory determining the quantities J, lambda_1, lambda_2 from the statistical properties of the gas flow, in the limit of short correlation times.

cond-mat.stat-mech↗

arXiv2004

On the Aggregation of Inertial Particles in Random Flows

We describe a criterion for particles suspended in a randomly moving fluid to aggregate. Aggregation occurs when the expectation value of a random variable is negative. This random variable evolves under a stochastic differential equation. We analyse this equation in detail in the limit where the correlation time of the velocity field of the fluid is very short, such that the stochastic differential equation is a Langevin equation.

cond-mat.stat-mech↗

arXiv2004

Spectroscopic survey of the Galaxy with Gaia I. Design and performance of the Radial Velocity Spectrometer

The definition and optimisation studies for the Gaia satellite spectrograph, the Radial Velocity Spectrometer (RVS), converged in late 2002 with the adoption of the instrument baseline. This paper reviews the characteristics of the selected configuration and presents its expected performance. The RVS is a 2.0 by 1.6 degree integral field spectrograph, dispersing the light of all sources entering its field of view with a resolving power R=11 500 over the wavelength range [848, 874] nm. The RVS will continuously and repeatedly scan the sky during the 5 years of the Gaia mission. On average, each source will be observed 102 times over this period. The RVS will collect the spectra of about 100-150 million stars up to magnitude V~17-18. At the end of the mission, the RVS will provide radial velocities with precisions of ~2 km/s at V=15 and \~15-20 km/s at V=17, for a solar metallicity G5 dwarf. The RVS will also provide rotational velocities, with precisions (at the end of the mission) for late type stars of sigma_vsini ~5 km/s at V~15 as well as atmospheric parameters up to V~14-15. The individual abundances of elements such as Silicon and Magnesium, vital for the understanding of Galactic evolution, will be obtained up to V~12-13. Finally, the presence of the 862.0 nm Diffuse Interstellar Band (DIB) in the RVS wavelength range will make it possible to derive the three dimensional structure of the interstellar reddening.

astro-ph↗

arXiv2004

Mapping the Stellar Dynamics of M31

Using the Planetary Nebula Spectrograph, we have observed and measured the velocities for some 2764 PNe in the disk and halo of the Andromeda galaxy. Preliminary analysis using a basic ring model shows a rotation curve in good agreement with that obtained from HI data out to ~20kpc. Some substructure has also been detected within the velocity field, which can be modeled as the continuation of the tidal-remnant known as the Southern Stream, as it passes through Andromeda's disk.

astro-ph↗

arXiv2003

Path Coalescence in Spatially Correlated Random Walks

A particle subject to successive, random displacements is said to execute a random walk (in position or some other coordinate). The mathematical properties of random walks have been very thoroughly investigated, and the model is used in many areas of science and engineering as well as other fields such as finance and the life sciences. This letter describes a phenomenon occurring in a natural extension of this model: we consider the motion of a large number of particles subject to successive random displacements which are correlated in space, but not in time. If these random displacements are smaller than their correlation length, the trajectories coalesce onto a decreasing number of trails. This surprising effect is explained and quantitative results are obtained. Various possible realisations are discussed, ranging from coalescence of the tracks of water droplets blown off a windshield to migration patterns of animals.

cond-mat.stat-mech↗

arXiv2003

The path-coalescence transition and its applications

We analyse the motion of a system of particles subjected a random force fluctuating in both space and time, and experiencing viscous damping. When the damping exceeds a certain threshold, the system undergoes a phase transition: the particle trajectories coalesce. We analyse this transition by mapping it to a Kramers problem which we solve exactly. In the limit of weak random force we characterise the dynamics by computing the rate at which caustics are crossed, and the statistics of the particle density in the coalescing phase. Last but not least we describe possible realisations of the effect, ranging from trajectories of raindrops on glass surfaces to animal migration patterns.

cond-mat.dis-nn↗

arXiv2003

Coagulation by Random Velocity Fields as a Kramers Problem

We analyse the motion of a system of particles suspended in a fluid which has a random velocity field. There are coagulating and non-coagulating phases. We show that the phase transition is related to a Kramers problem, and use this to determine the phase diagram, as a function of the dimensionless inertia of the particles, epsilon, and a measure of the relative intensities of potential and solenoidal components of the velocity field, Gamma. We find that the phase line is described by a function which is non-analytic at epsilon=0, and which is related to escape over a barrier in the Kramers problem. We discuss the physical realisations of this phase transition.

cond-mat.dis-nn↗

arXiv2003

Mapping the Remote Milky Way Halo using BHB stars at 70 < r < 130 kpc

We increase the number of remote halo tracers by using blue horizontal branch (BHB) stars out to Galactocentric distances of 130 kpc. We use SDSS EDR photometry and the VLT to detect 16 BHB stars at Galactocentric distances 70 < r < 130, and to measure their radial velocities. We find the mass of the Milky Way is M =1.7^{+3.0}_{-0.6} x 10^{12} solar masses. When completed this survey will: (i) substantially reduce the errors in the total mass and extent of the Milky Way halo, (ii) map the velocity space in a hitherto unexplored region of the halo.

astro-ph↗

arXiv2000

Semiclassical trace formulae using coherent states

We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate-space formulation. Using a coherent-state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum-mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.

cond-mat.mes-hall↗

arXiv1999

Non-universality of chaotic classical dynamics: implications for quantum chaos

It might be anticipated that there is statistical universality in the long-time classical dynamics of chaotic systems, corresponding to the universal correspondence of their quantum spectral statistics with random matrix models. We argue that no such universality exists. We discuss various statistical properties of long period orbits: the distribution of the phase-space density of periodic orbits of fixed length and a correlation function of periodic-orbit actions, corresponding to the universal quantum spectral two-point correlation function. We show that bifurcations are a mechanism for correlations of periodic-orbit actions. They lead to a result which is non-universal, and which in general may not be an analytic function of the action difference.

chao-dyn↗

arXiv1999

Absorption of radiation by small metallic particles: a general self-consistent approach

We introduce a theory for the absorption of electromagnetic radiation by small metal particles, which generalises the random phase approximation by incorporating both electric and magnetic dipole absorption within a unified self-consistent scheme. We demonstrate the equivalence of the new approach to a superficially dissimilar perturbative approach. We show how to obtain solutions to the self-consistent equations using a classical approximation, taking into account the non-locality of the polarisability and the conductivity tensor. We discuss the nature of the self-consistent solutions for diffusive and ballistic electron dynamics.

physics.class-ph↗

arXiv1999

Spectral correlations: understanding oscillatory contributions

We give a transparent derivation of a relation obtained using a supersymmetric non-linear sigma model by Andreev and Altshuler [Phys. Rev. Lett. 72, 902, (1995)], which connects smooth and oscillatory components of spectral correlation functions. We show that their result is not specific to random matrix theory. Also, we show that despite an apparent contradiction, the results obtained using their formula are consistent with earlier perspectives on random matrix models. In particular, the concept of resurgence is not required.

cond-mat.dis-nn↗

arXiv1997

Magnetic Dipole Absorption of Radiation in Small Conducting Particles

We give a theoretical treatment of magnetic dipole absorption of electromagnetic radiation in small conducting particles, at photon energies which are large compared to the single particle level spacing, and small compared to the plasma frequency. We discuss both diffusive and ballistic electron dynamics for particles of arbitrary shape. The conductivity becomes non-local when the frequency is smaller than the frequency ω_c characterising the transit of electrons from one side of the particle to the other, but in the diffusive case ω_c plays no role in determining the absorption coefficient. In the ballistic case, the absorption coefficient is proportional to ω^2 for ω<< ω_c, but is a decreasing function of ωfor ω>> ω_c.

cond-mat↗