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Michael Wilkinson

Publications and source records attributed to Michael Wilkinson.

At least 37 records · Page 2Linked to original sources
arXiv2019

Numerical Coalescence of Chaotic Trajectories

Pairs of numerically computed trajectories of a chaotic system may coalesce because of finite arithmetic precision. We analyse an example of this phenomenon, showing that it occurs surprisingly frequently. We argue that our model belongs to a universality class of chaotic systems where this numerical coincidence effect can be described by mapping it to a first-passage process. Our results are applicable to aggregation of small particles in random flows, as well as to numerical investigation of chaotic systems.

nlin.CD↗
arXiv2019

Nests and Chains of Hofstadter Butterflies

The \lq Hofstadter butterfly', a plot of the spectrum of an electron in a two-dimensional periodic potential with a uniform magnetic field, contains subsets which resemble small, distorted images of the entire plot. We show how the sizes of these sub-images are determined, and calculate scaling factors describing their self-similar nesting, revealing an un-expected simplicity in the fractal structure of the spectrum. We also characterise semi-infinite chains of sub-images, showing one end of the chain is a result of gap closure, and the other end is at an accumulation point.

nlin.CD↗
arXiv2018

Lacunarity Exponents

Many physical processes result in very uneven, apparently random, distributions of matter, characterized by fluctuations of the local density over orders of magnitude. The density of matter in the sparsest regions can have a power-law distribution, with an exponent that we term the lacunarity exponent. We discuss a mechanism which explains the wide occurrence of these power laws, and give analytical expressions for the exponent in some simple models.

nlin.CD↗
arXiv2017

A Matrix Contraction Process

We consider a stochastic process in which independent identically distributed random matrices are multiplied and where the Lyapunov exponent of the product is positive. We continue multiplying the random matrices as long as the norm, $ε$, of the product is \emph{less} than unity. If the norm is greater than unity we reset the matrix to a multiple of the identity and then continue the multiplication. We address the problem of determining the probability density function of the norm, $P_ε$. We argue that, in the limit as $ε\to 0$, $P_ε\sim (\ln (1/ε))^με^γ$, where $μ$ and $γ$ are two real parameters. Our motivation for analysing this \emph{matrix contraction process} is that it serves as a model for describing the fine-structure of strange attractors, where a dense concentration of trajectories results from the differential of the flow being contracting in some region. We exhibit a matrix-product model for the differential of the flow in a random velocity field, and show that there is a phase transition, with the parameter $μ$ changing abruptly from $μ=0$ to $μ=-\frac{3}{2}$ as a parameter of the flow field model is varied.

cond-mat.stat-mech↗
arXiv2017

Uniformity Transition for Ray Intensities in Random Media

This paper analyses a model for the intensity of distribution for rays propagating without absorption in a random medium. The random medium is modelled as a dynamical map. After $N$ iterations, the intensity is modelled as a sum $S$ of ${\cal N}$ contributions from different trajectories, each of which is a product of $N$ independent identically distributed random variables $x_k$, representing successive focussing or de-focussing events. The number of ray trajectories reaching a given point is assumed to proliferate exponentially: ${\cal N}=Λ^N$, for some $Λ>1$. We investigate the probability distribution of $S$. We find a phase transition as parameters of the model are varied. There is a phase where the fluctuations of $S$ are suppressed as $N\to \infty$, and a phase where the $S$ has large fluctuations, for which we provide a large deviation analysis.

physics.optics↗
arXiv2017

Stability in Chaos

Intrinsic instability of trajectories characterizes chaotic dynamical systems. We report here that trajectories can exhibit a surprisingly high degree of stability, over a very long time, in a chaotic dynamical system. We provide a detailed quantitative description of this effect for a one-dimensional model of inertial particles in a turbulent flow using large-deviation theory. Specifically, the determination of the entropy function for the distribution of finite-time Lyapunov exponents reduces to the analysis of a Schrödinger equation, which is tackled by semi-classical methods.

nlin.CD↗
arXiv2017

Aggregation-fragmentation-diffusion model for trail dynamics

We investigate statistical properties of trails formed by a random process incorporating aggregation, fragmentation, and diffusion. In this stochastic process, which takes place in one spatial dimension, two neighboring trails may combine to form a larger one and also, one trail may split into two. In addition, trails move diffusively. The model is defined by two parameters which quantify the fragmentation rate and the fragment size. In the long-time limit, the system reaches a steady state, and our focus is the limiting distribution of trail weights. We find that the density of trail weight has power-law tail $P(w) \sim w^{-γ}$ for small weight $w$. We obtain the exponent $γ$ analytically, and find that it varies continuously with the two model parameters. The exponent $γ$ can be positive or negative, so that in one range of parameters small-weight tails are abundant, and in the complementary range, they are rare.

cond-mat.stat-mech↗
arXiv2017

Convergent Chaos

Chaos is widely understood as being a consequence of sensitive dependence upon initial conditions. This is the result of an instability in phase space, which separates trajectories exponentially. Here, we demonstrate that this criterion should be refined. Despite their overall intrinsic instability, trajectories may be very strongly convergent in phase space over extremely long periods, as revealed by our investigation of a simple chaotic system (a realistic model for small bodies in a turbulent flow). We establish that this strong convergence is a multi-facetted phenomenon, in which the clustering is intense, widespread and balanced by lacunarity of other regions. Power laws, indicative of scale-free features, characterise the distribution of particles in the system. We use large-deviation and extreme-value statistics to explain the effect. Our results show that the interpretation of the 'butterfly effect' needs to be carefully qualified. We argue that the combination of mixing and clustering processes makes our specific model relevant to understanding the evolution of simple organisms. Lastly, this notion of 'convergent chaos', which implies the existence of conditions for which uncertainties are unexpectedly small, may also be relevant to the valuation of insurance and futures contracts.

nlin.CD↗
arXiv2016

Non-Thermal Einstein Relations

We consider a particle moving with equation of motion $\dot x=f(t)$, where $f(t)$ is a random function with statistics which are independent of $x$ and $t$, with a finite drift velocity $v=\langle f\rangle$ and in the presence of a reflecting wall. Far away from the wall, translational invariance implies that the stationary probability distribution is $P(x)\sim \exp(αx)$. A classical example of a problem of this type is sedimentation equilibrium, where $α$ is determined by temperature. In this work we do not introduce a thermal reservoir and $α$ is determined from the equation of motion. We consider a general approach to determining $α$ which is not always in agreement with Einstein's relation between the mean velocity and the diffusion coefficient. We illustrate our results with a model inspired by the Boltzmann equation.

nlin.CD↗
arXiv2015

Analysis of the Correlation Dimension for Inertial Particles

We obtain an implicit equation for the correlation dimension which describes clustering of inertial particles in a complex flow onto a fractal measure. Our general equation involves a propagator of a nonlinear stochastic process in which the velocity gradient of the fluid appears as additive noise. When the long-time limit of the propagator is considered our equation reduces to an existing large-deviation formalism, from which it is difficult to extract concrete results. In the short-time limit, however, our equation reduces to a solvability condition on a partial differential equation. We show how this approach leads to a perturbative expansion of the correlation dimension, for which the coefficients can be obtained exactly and in principle to any order. We derive the perturbation series for the correlation dimension of inertial particles suspended in three-dimensional spatially smooth random flows with white-noise time correlations, obtaining the first 33 non-zero coefficients exactly.

physics.flu-dyn↗
arXiv2015

Collisional Aggregation due to Turbulence

Collisions between particles suspended in a fluid play an important role in many physical processes. As an example, collisions of microscopic water droplets in clouds are a necessary step in the production of macroscopic raindrops. Collisions of dust grains are also conjectured to be important for planet formation in the gas surrounding young stars, and also to play a role in the dynamics of sand storms. In these processes, collisions are favoured by fast turbulent motions. Here we review recent advances in the understanding of collisional aggregation due to turbulence. We discuss the role of fractal clustering of particles, and caustic singularities of their velocities. We also discuss limitations of the Smoluchowski equation for modelling these processes. These advances lead to a semi-quantitative understanding on the influence of turbulence on collision rates, and point to deficiencies in the current understanding of rainfall and planet formation.

physics.flu-dyn↗
arXiv2015

Large deviations and rain showers

Rainfall from ice-free cumulus clouds requires collisions of large numbers of microscopic droplets to create every raindrop. The onset of rain showers can be surprisingly rapid, much faster than the mean time required for a single collision. Large-deviation theory is used to explain this observation.

physics.ao-ph↗
arXiv2015

Stochastic Cascade Amplification of Fluctuations

We consider a dynamical system which has a stable attractor and which is perturbed by an additive noise. Under some quite typical conditions, the fluctuations from the attractor are intermittent and have a probability distribution with power-law tails. We show that this results from a stochastic cascade of amplification of fluctuations due to transient periods of instability. The exponent of the power-law is interpreted as a negative fractal dimension.

nlin.CD↗
arXiv2014

Advection diffusion equation with absorbing boundary

We consider a spatially homogeneous advection-diffusion equation in which the diffusion tensor and drift velocity are time-independent, but otherwise general. We derive asymptotic expressions, valid at large distances from a steady point source, for the flux onto a completely permeable boundary and onto an absorbing boundary. The absorbing case is treated by making a source of antiparticles at the boundary. In both cases there is an exponential decay as the distance from the source increases; we find that the exponent is the same for both boundary conditions.

nlin.CD↗
arXiv2014

Convective Ripening and Initiation of Rainfall

This paper discusses the evolution of the droplet size distribution for a liquid-in-gas aerosol contained in a Rayleigh-Bénard cell. It introduces a non-collisional model for broadening the droplet size distribution, termed 'convective ripening'. The paper also considers the initiation of rainfall from ice-free cumulus clouds. It is argued that while collisional mechanisms cannot explain the production of rain from clouds with water droplet diameters of $20\ μ{\rm m}$, the non-collisional convective ripening mechanism gives a much faster route to increasing the size of the small fraction of droplets that grow into raindrops.

physics.ao-ph↗
arXiv2014

Triangular Constellations in Fractal Measures

The local structure of a fractal set is described by its dimension $D$, which is the exponent of a power-law relating the mass ${\cal N}$ in a ball to its radius $ε$: ${\cal N}\sim ε^D$. It is desirable to characterise the {\em shapes} of constellations of points sampling a fractal measure, as well as their masses. The simplest example is the distribution of shapes of triangles formed by triplets of points, which we investigate for fractals generated by chaotic dynamical systems. The most significant parameter describing the triangle shape is the ratio $z$ of its area to the radius of gyration squared. We show that the probability density of $z$ has a phase transition: $P(z)$ is independent of $ε$ and approximately uniform below a critical flow compressibility $β_{\rm c}$, but for $β>β_{\rm c}$ it is described by two power laws: $P(z)\sim z^{α_1}$ when $1\gg z\gg z_{\rm c}(ε)$, and $P(z)\sim z^{α_2}$ when $z\ll z_{\rm c}(ε)$.

physics.flu-dyn↗
arXiv2014

Collision rate for suspensions at large Stokes numbers - comparing Navier-Stokes and synthetic turbulence

The use of simplified models of turbulent flows provides an appealing possibility to study the collision rate of turbulent suspensions, especially in conditions relevant to astrophysics, which require large time scale separations. To check the validity of such approaches, we used a direct numerical simulation (DNS) velocity field, which satisfies the Navier-Stokes equations (although it neglects the effect of the suspended particles on the flow field), and a kinematic simulation (KS) velocity field, which is a random field designed so that its statistics are in accord with the Kolmogorov theory for fully-developed turbulence. In the limit where the effects of particle inertia (characterised by the Stokes number) are negligible, the collision rates from the two approaches agree. As the Stokes number St increases, however, we show that the DNS collision rate exceeds the KS collision rate by orders of magnitude. We propose an explanation for this phenomenon and explore its consequences. We discuss the collision rate $R$ for particles in high Reynolds number flows at large Stokes number, and present evidence that $R\propto \sqrt{\rm St}$.

physics.flu-dyn↗