arXiv subjects
Publications and source records attributed to Michael Wilkinson.
There is a dimensionless parameter which enters into the equation for the evolution of supersaturation in Ostwald ripening processes. This parameter is typically a large number. Here it is argued that the consequent stiffness of the equation results in the evolution of the supersaturation being unstable. The instability is evident in numerical simulations of Ostwald ripening.
This paper considers an Ostwald ripening process in which new droplets are injected at a constant rate, with a fixed distribution of radii, and in which droplets are removed when they grow to a specified maximum radius. This process exhibits a transition from a steady state to a limit cycle as a parameter is varied. The instability is shown to be related to the roots of the Laplace transform of a response kernel. A model is described which gives a good approximation of the period of the limit cycle. The model may also exhibit chaotic behaviour. The relevance of the model to atmospheric precipitation is discussed.
If a body of inviscid fluid is disturbed, it will typically eject a jet of fluid. If the effects of gravity and surface tension are negligible, these jets travel in straight lines, with the tips approaching a constant velocity. It has been observed that these jets can have a broad base, tapering progressively toward the tip, but the mathematical form of their profile has not been successfully analysed in earlier works. In this paper, we describe the simplest case, in two dimensions: an infinitely deep body of inviscid fluid, with no surface tension or gravitational forces acting, responds to an impulsive disturbance. We find that, contrary to some earlier suggestions, the jet has a hyperbolic profile (away from its tip and its base).
We obtain the full equations of motion for a wide, pivoted, slider bearing. These are used to review the choice of the optimal position for the pivot point, to discuss its response to time-dependent sliding velocity, and to determine the stability of the motion. The case of an abrupt acceleration of the slider, which results in a large transient increase of the resistive force, is surprisingly complicated. We also discuss a general \emph{transversion} formula for changing the dependent variables in Stokes flow problems.
We consider a random field $\phi(\mathbf{r})$ in $d$ dimensions which is largely concentrated around small `hotspots', with `weights', $w_i$. These weights may have a very broad distribution, such that their mean does not exist, or else is not a useful estimate. In such cases, the median $\overline W$ of the total weight $W$ in a region of size $R$ is an informative characterisation of the weights. We define the function $F$ by $\ln \overline W=F(\ln R)$. If $F'(x)>d$, the distribution of hotspots is dominated by the largest weights. In the case where $F'(x)-d$ approaches a constant positive value when $R\to \infty$, the hotspots distribution has a type of scale-invariance which is different from that of fractal sets, and which we term \emph{ultradimensional}. The form of the function $F(x)$ is determined for a model of diffusion in a random potential.
We discuss a model for directed percolation in which the flux of material along each bond is a dynamical variable. The model includes a physically significant limiting case where the total flux of material is conserved. We show that the distribution of fluxes is asymptotic to a power law at small fluxes. We give an implicit equation for the exponent, in terms of probabilities characterising site occupations. In one dimension the site occupations are exactly independent, and the model is exactly solvable. In two dimensions, the independent-occupation assumption gives a good approximation. We explore the relationship between this model and traditional models for directed percolation.
If a flat, horizontal, plate settles onto a flat surface, it is known that the gap $h$ decreases with time $t$ as a power-law: $h\sim t^{-1/2}$. We consider what happens if the plate is not initially horizontal, and/or the centre of mass is not symmetrically positioned: does one edge contact the surface in finite time, or does the plate approach the horizontal without making contact? The dynamics of this system is analysed and shown to be remarkably complex.
In the international standard for system and software engineering ISO/IEC/IEEE 15288: 2015, the output of the stakeholder needs and the business or mission analysis technical processes are transformed into a technical view of the system by the system requirements definition process. In model-based systems engineering, functional needs can be modeled by use case diagrams. Intended outcomes of system requirements definition include resolution of disagreement about requirements, explicit agreement between stakeholders, and traceability. However, stakeholder needs are often elicited in a siloed manner and may be inconsistent. The lack of mathematically based systematic approaches for requirements definition poses a challenge to model-based transformation of needs into a technical view of the system that achieves agreement between stakeholders. This paper specifies and demonstrates mathematical frameworks for rationalizing and synthesizing functional needs that have been captured through an elicitation process. Benefits of this approach include but are not limited to supporting rigorous identification and resolution of disagreements and facilitating systematic analysis of change impact to achieve stakeholder agreement all with minimal intervention by the system engineers.
The aviation literature gives relatively little guidance to practitioners about the specifics of architecting systems for safety, particularly the impact of architecture on allocating safety requirements, or the relative ease of system assurance resulting from system or subsystem level architectural choices. As an exemplar, this paper considers common architectural patterns used within traditional aviation systems and explores their safety and safety assurance implications when applied in the context of integrating artificial intelligence (AI) and machine learning (ML) based functionality. Considering safety as an architectural property, we discuss both the allocation of safety requirements and the architectural trade-offs involved early in the design lifecycle. This approach could be extended to other assured properties, similar to safety, such as security. We conclude with a discussion of the safety considerations that emerge in the context of candidate architectural patterns that have been proposed in the recent literature for enabling autonomy capabilities by integrating AI and ML. A recommendation is made for the generation of a property-driven architectural pattern catalogue.
In this third decade of systems engineering in the twenty-first century, it is important to develop and demonstrate practical methods to exploit machine-readable models in the engineering of systems. Substantial investment has been made in languages and modelling tools for developing models. A key problem is that system architects and engineers work in a multidisciplinary environment in which models are not the product of any one individual. This paper provides preliminary results of a formal approach to specify models and structure preserving transformations between them that support model synchronization. This is an important area of research and practice in software engineering. However, it is limited to synchronization at the code level of systems. This paper leverages previous research of the authors to define a core fractal for interpretation of concepts into model specifications and transformation between models. This fractal is used to extend the concept of synchronization of models to the system level and is demonstrated through a practical engineering example for an advanced driver assistance system.
It is difficult to explain rainfall from ice-free clouds, because the timescale for the onset of rain showers is shorter than the mean time for collisions between microscopic water droplets. It has been suggested that raindrops are produced from very rare ' lucky' droplets, which undergo a large number of collisions on a timescale which is short compared to the mean time for a the first collision. This work uses large deviation theory to develop estimates for the timescale for the onset of a rain shower, as a function of the collision rate coefficients. The growth history of the fast-growing droplets which do become raindrops is discussed. It is shown that their first few collisions are always approximately equally spaced in time, regardless of how the mean time for typical droplets varies as a function of the number of collisions.
We investigate the probability distribution of Chern numbers (quantum Hall effect integers) for a parametric version of the GUE random matrix ensemble, which is a model for a chaotic or disordered system. The numerically-calculated single-band Chern number statistics agree well with predictions based on an earlier study [O. Gat and M. Wilkinson, SciPost Phys., 10, 149, (2021)] of the statistics of the quantum adiabatic curvature, when the parametric correlation length is small. However, contrary to an earlier conjecture, we find that the gap Chern numbers are correlated, and that correlation is weak but slowly-decaying. Also, the statistics of weighted sums of Chern numbers for many bands differs markedly from predictions based upon the hypothesis that gap Chern numbers are uncorrelated. All our results are consistent with the universality hypothesis described in the earlier paper, including in the previously unstudied regime of large correlation length, where the Chern statistics is highly non-Gaussian.
Experiments investigating particles floating on a randomly stirred fluid show regions of very low density, which are not well understood. We introduce a simplified model for understanding sparsely occupied regions of the phase space of non-autonomous, chaotic dynamical systems, based upon an extension of the skinny bakers' map. We show how the distribution of the sizes of voids in the phase space can be mapped to the statistics of the running maximum of a Wiener process. We find that the model exhibits a lacunarity transition, which is characterised by regions of the phase space remaining empty as the number of trajectories is increased.
An exact formula is derived, as an integral, for the mean square winding angle of Brownian motion (that is, diffusion) after time t, around an infinitely long impenetrable cylinder of radius a, having started at radius R(>a) from the axis. Strikingly, for the simpler problem with a=0, the mean square winding angle around a straight line, is long known to be instantly infinite however far away the starting point lies. the fractally small, fast, random walk steps of mathematical Brownian motion allow unbounded windings around the zero thickness of the straight line. A remedy if it is required, is to accord the line non-zero thickness, an impenetrable cylinder, as analysed here. The problem straight away reduces to a 2D one of winding around a disc in a plane since the axial component of the 3D Brownian motion is independent of the others. After deriving the exact mean square winding angle, the integral is evaluated in the limit of a narrow cylinder a<<R, highlighting the limits of short and long diffusion times addressed by previous approximate treatments.
It is conceivable that an RNA virus could use a polysome, that is, a string of ribosomes covering the RNA strand, to protect the genetic material from degradation inside a host cell. This paper discusses how such a virus might operate, and how its presence might be detected by ribosome profiling. There are two possible forms for such a polysomally protected virus, depending upon whether just the forward strand or both the forward and complementary strands can be encased by ribosomes (these will be termed type 1 and type 2, respectively). It is argued that in the type 2 case the viral RNA would evolve an ambigrammatic property, whereby the viral genes are free of stop codons in a reverse reading frame (with forward and reverse codons aligned). Recent observations of ribosome profiles of ambigrammatic narnavirus sequences are consistent with our predictions for the type 2 case.
We analyse the correlation function of the quantum curvature in complex quantum systems, using a random matrix model to provide an exemplar of a universal correlation function. We show that the correlation function diverges as the inverse of the distance at small separations. We also define and analyse a correlation function of mixed states, showing that it is finite but singular at small separations. A scaling hypothesis on a universal form for both types of correlations is supported by Monte-Carlo simulations. We relate the correlation function of the curvature to the variance of Chern integers which can describe quantised Hall conductance.
We discuss the combined effects of overdamped motion in a quenched random potential and diffusion, in one dimension, in the limit where the diffusion coefficient is small. Our analysis considers the statistics of the mean first-passage time $T(x)$ to reach position $x$, arising from different realisations of the random potential: specifically, we contrast the median $\bar T(x)$, which is an informative description of the typical course of the dispersion, with the expectation value $\langle T(x)\rangle$, which is dominated by rare events where there is an exceptionally high barrier to diffusion. We show that at relatively short times the median $\bar T(x)$ is explained by a 'flooding' model, where $T(x)$ is predominantly determined by the highest barriers which is encountered before reaching position $x$. These highest barriers are quantified using methods of extreme value statistics.
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.