Searcharxiv⌕ Search

arXiv subjects

Colm Connaughton

Publications and source records attributed to Colm Connaughton.

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

Empirical analysis of the variability in the flow-density relationship for smart motorways

The fundamental diagram is an assumed functional relationship between traffic flow and traffic density. In practice, this relationship is noisy and exhibits significant statistical variability. On smart motorways, this variability is increased by variable speed limits that are not captured by the fundamental diagram. To study this variability, it is appropriate to consider the joint probability distribution function (pdf) of density and flow. We perform an empirical study of the variability in the relationship between flow and density using 74 days of data from 64 sections of London's M25. The objectives are to determine how much of the variability in the flow-density relationship results from variable speed limits and to assess whether particular functional forms of the fundamental diagram are systematically preferred. Empirically, the joint pdf of flow and density is strongly bimodal, illustrating that traffic flows are often found in high-density or low-density regimes but rarely in between. We find that the high-density regime is strongly affected by variable speed limits whereas the low-density regime is not. The Daganzo-Newell (triangular) model of the fundamental diagram systematically fits best to the data. However, the optimal parameters vary with location. Clustering analysis of these parameters suggests three qualitatively different types of flow-density relationships applying to different sections of the M25. These clusters have natural interpretations in terms of the frequency and severity of flow breakdown. Accident rates also depend on cluster type suggesting possible links to other properties of traffic flows beyond the flow-density relationship.

cs.CY↗
arXiv2018

Stationary mass distribution and nonlocality in models of coalescence and shattering

We study the asymptotic properties of the steady state mass distribution for a class of collision kernels in an aggregation-shattering model in the limit of small shattering probabilities. It is shown that the exponents characterizing the large and small mass asymptotic behavior of the mass distribution depend on whether the collision kernel is local (the aggregation mass flux is essentially generated by collisions between particles of similar masses), or non-local (collision between particles of widely different masses give the main contribution to the mass flux). We show that the non-local regime is further divided into two sub-regimes corresponding to weak and strong non-locality. We also observe that at the boundaries between the local and non-local regimes, the mass distribution acquires logarithmic corrections to scaling and calculate these corrections. Exact solutions for special kernels and numerical simulations are used to validate some non-rigorous steps used in the analysis. Our results show that for local kernels, the scaling solutions carry a constant flux of mass due to aggregation, whereas for the non-local case there is a correction to the constant flux exponent. Our results suggest that for general scale-invariant kernels, the universality classes of mass distributions are labeled by two parameters: the homogeneity degree of the kernel and one further number measuring the degree of the non-locality of the kernel.

cond-mat.stat-mech↗
arXiv2017

Fitness voter model: damped oscillations and anomalous consensus

We study the dynamics of opinion formation in a heterogeneous voter model on a complete graph, in which each agent is endowed with an integer fitness parameter $k \ge 0$, in addition to its $+$ or $-$ opinion state. The evolution of the distribution of $k$--values and the opinion dynamics are coupled together, so as to allow the system to dynamically develop heterogeneity and memory in a simple way. When two agents with different opinions interact, their $k$--values are compared and, with probability $p$ the agent with the lower value adopts the opinion of the one with the higher value, while with probability $1-p$ the opposite happens. The winning agent then increments its $k$--value by one. We study the dynamics of the system in the entire $0 \le p \le 1$ range and compare with the case $p=1/2$, which corresponds to the standard voter model. When $0 \le p < 1/2$, the system approaches exponentially fast to the consensus state of the initial majority opinion. The mean consensus time $τ$ grows logarithmically with the number of agents $N$, and it is greatly decreased relative to the linear behaviour $τ\sim N$ found in the voter model. When $1/2 < p \le 1$, the system initially relaxes to a state with an even coexistence of opinions, but eventually reaches consensus by finite-size fluctuations. The approach to coexistence is monotonic for $1/2 < p < 0.8$, while for $0.8 \le p \le 1$ there are damped oscillations around the coexistence value. The final approach to coexistence is approximately a power law $t^{-b(p)}$ in both regimes, where $b$ increases with $p$. Also, $τ$ increases respect to the voter model, although it still scales linearly with $N$. The $p=1$ case is special, with a relaxation to coexistence that scales as $t^{-2.73}$ and a consensus time that scales as $τ\sim N^β$, with $β\simeq 1.45$.

physics.soc-ph↗
arXiv2017

Universality properties of steady driven coagulation with collisional evaporation

Irreversible aggregation is an archetypal example of a system driven far from equilibrium by sources and sinks of a conserved quantity (mass). The source is a steady input of monomers and the evaporation of colliding particles with a small probability is the sink. Using exact and heuristic analyses, we find a universal regime and two distinct non-universal regimes distinguished by the relative importance of mergers between small and large particles. At the boundary between the regimes we find an analogue of the logarithmic correction conjectured by Kraichnan for two-dimensional turbulence.

cond-mat.stat-mech↗
arXiv2016

Non-equilibrium phase diagram for a model with coalescence, evaporation and deposition

We study a d-dimensional lattice model of diffusing coalescing massive particles, with two parameters controlling deposition and evaporation of monomers. The unique stationary distribution for the system exhibits a phase transition in all dimensions d greater or equal to one between a growing phase, in which the expected mass is infinite at each site, and an exponential phase in which the expected mass is finite. We establish rigorous upper and lower bounds on the critical curve describing the phase transition for this system, and some asymptotics for large or small deposition rates.

math.PR↗
arXiv2015

Importance Sampling Variance Reduction for the Fokker-Planck Rarefied Gas Particle Method

Models and methods that are able to accurately and efficiently predict the flows of low-speed rarefied gases are in high demand, due to the increasing ability to manufacture devices at micro and nano scales. One such model and method is a Fokker-Planck approximation to the Boltzmann equation, which can be solved numerically by a stochastic particle method. The stochastic nature of this method leads to noisy estimates of the thermodynamic quantities one wishes to sample when the signal is small in comparison to the thermal velocity of the gas. Recently, Gorji et al have proposed a method which is able to greatly reduce the variance of the estimators, by creating a correlated stochastic process which acts as a control variate for the noisy estimates. However, there are potential difficulties involved when the geometry of the problem is complex, as the method requires the density to be solved for independently. Importance sampling is a variance reduction technique that has already been shown to successfully reduce the noise in direct simulation Monte Carlo calculations. In this paper we propose an importance sampling method for the Fokker-Planck stochastic particle scheme. The method requires minimal change to the original algorithm, and dramatically reduces the variance of the estimates. We test the importance sampling scheme on a homogeneous relaxation, planar Couette flow and a lid-driven-cavity flow, and find that our method is able to greatly reduce the noise of estimated quantities. Significantly, we find that as the characteristic speed of the flow decreases, the variance of the noisy estimators becomes independent of the characteristic speed.

physics.comp-ph↗
arXiv2015

Explosive condensation in symmetric mass transport models

We study the dynamics of condensation in a misanthrope process with nonlinear jump rates and factorized stationary states. For large enough density, it is known that such models have a phase separated state, with a non-zero fraction of the total mass concentrating in a single lattice site. It has been established in [B Waclaw and M R Evans, Phys. Rev. Lett., 108(7):070601, 2012] for asymmetric dynamics that such processes exhibit explosive condensation, where the time to reach the stationary state vanishes with increasing system size. This constitutes a spatially extended version of instantaneous gelation which has previously been studied only in mean-field coagulation models. We show that this phenomenon also occurs for symmetric dynamics in one dimension if the non-linearity is strong enough, and we find a coarsening regime where the time to stationarity diverges with the system size for weak non-linearity. In higher space dimensions explosive condensation is expected to be generic for all parameter values. Our results are based on heuristic mean field arguments which are confirmed by simulation data.

cond-mat.stat-mech↗
arXiv2014

Rossby and Drift Wave Turbulence and Zonal Flows: the Charney-Hasegawa-Mima model and its extensions

A detailed study of the Charney-Hasegawa-Mima model and its extensions is presented. These simple nonlinear partial differential equations suggested for both Rossby waves in the atmosphere and also drift waves in a magnetically-confined plasma exhibit some remarkable and nontrivial properties, which in their qualitative form survive in more realistic and complicated models, and as such form a conceptual basis for understanding the turbulence and zonal flow dynamics in real plasma and geophysical systems. Two idealised scenarios of generation of zonal flows by small-scale turbulence are explored: a modulational instability and turbulent cascades. A detailed study of the generation of zonal flows by the modulational instability reveals that the dynamics of this zonal flow generation mechanism differ widely depending on the initial degree of nonlinearity. A numerical proof is provided for the extra invariant in Rossby and drift wave turbulence -zonostrophy and the invariant cascades are shown to be characterised by the zonostrophy pushing the energy to the zonal scales. A small scale instability forcing applied to the model demonstrates the well-known drift wave - zonal flow feedback loop in which the turbulence which initially leads to the zonal flow creation, is completely suppressed and the zonal flows saturate. The turbulence spectrum is shown to diffuse in a manner which has been mathematically predicted. The insights gained from this simple model could provide a basis for equivalent studies in more sophisticated plasma and geophysical fluid dynamics models in an effort to fully understand the zonal flow generation, the turbulent transport suppression and the zonal flow saturation processes in both the plasma and geophysical contexts as well as other wave and turbulence systems where order evolves from chaos.

physics.flu-dyn↗
arXiv2013

Wave turbulence in the two-layer ocean model

This paper looks at the two-layer ocean model from a wave turbulence perspective. A symmetric form of the two-layer kinetic equation for Rossby waves is derived using canonical variables, allowing the turbulent cascade of energy between the barotropic and baroclinic modes to be studied. It turns out that energy is transferred via local triad interactions from the large-scale baroclinic modes to the baroclinic and barotropic modes at the Rossby deformation scale. From there it is then transferred to the large-scale barotropic modes via a nonlocal inverse transfer. Using scale separation a sys- tem of coupled equations were obtained for the small-scale baroclinic component and the large-scale barotropic component. Since the total energy of the small-scale component is not conserved, but the total barotropic plus baroclinic energy is conserved, the baroclinic energy loss at small scales will be compensated by the growth of the barotropic energy at large scales. It is found that this transfer is mostly anisotropic and mostly to the zonal component.

physics.flu-dyn↗
arXiv2013

Modulational Instability in Basic Plasma and Geophysical Models

This is a review of the theory of the modulational instability in idealised fluid models of strongly magnetised plasmas and reduced models of geophysical fluid dynamics, particularly the role it plays in the formation of zonal flows. The discussion focusses on the Charney-Hasegawa-Mima and Hasegawa-Wakatani models. Particular attention is paid to the wave turbulence - zonal flow feedback loop whereby large scale zonal flows which are initially generated by modulational instability of small-scale drift/Rossby waves tend to subsequently suppress these small scale waves by their shearing action. This negative feedback can result in a dynamic equilibrium in which large scale zonal flows grow by drawing energy from small scale turbulence but suppress the small scale turbulence in the process until a balance is reached. In this regime, the level of small scale turbulence is greatly reduced compared to the level one would observe in the absence of the zonal flows.

nlin.CD↗
arXiv2013

A nonlinear least squares method for the inverse droplet coagulation problem

If the rates, $K(x,y)$, at which particles of size $x$ coalesce with particles of size $y$ is known, then the mean-field evolution of the particle-size distribution of an ensemble of irreversibly coalescing particles is described by the Smoluchowski equation. We study the corresponding inverse problem which aims to determine the coalescence rates, $K(x,y)$ from measurements of the particle size distribution. We assume that $K(x,y)$ is a homogeneous function of its arguments, a case which occurs commonly in practice. The problem of determining, $K(x,y)$, a function to two variables, then reduces to a simpler problem of determining a function of a single variable plus two exponents, $μ$ and $ν$, which characterise the scaling properties of $K(x,y)$. The price of this simplification is that the resulting least squares problem is nonlinear in the exponents $μ$ and $ν$. We demonstrate the effectiveness of the method on a selection of coalescence problems arising in polymer physics, cloud science and astrophysics. The applications include examples in which the particle size distribution is stationary owing to the presence of sources and sinks of particles and examples in which the particle size distribution is undergoing self-similar relaxation in time.

cond-mat.stat-mech↗
arXiv2012

Percolation transition in the kinematics of nonlinear resonance broadening in Charney-Hasegawa-Mima model of Rossby wave turbulence

We study the kinematics of nonlinear resonance broadening of interacting Rossby waves as modelled by the Charney-Hasegawa-Mima equation on a biperiodic domain. We focus on the set of wave modes which can interact quasi-resonantly at a particular level of resonance broadening and aim to characterise how the structure of this set changes as the level of resonance broadening is varied. The commonly held view that resonance broadening can be thought of as a thickening of the resonant manifold is misleading. We show that in fact the set of modes corresponding to a single quasi-resonant triad has a nontrivial structure and that its area in fact diverges for a finite degree of broadening. We also study the connectivity of the network of modes which is generated when quasi-resonant triads share common modes. This network has been argued to form the backbone for energy transfer in Rossby wave turbulence. We show that this network undergoes a percolation transition when the level of resonance broadening exceeds a critical value. Below this critical value, the largest connected component of the quasi-resonant network contains a negligible fraction of the total number of modes in the system whereas above this critical value a finite fraction of the total number of modes in the system are contained in the largest connected component. We argue that this percolation transition should correspond to the transition to turbulence in the system.

nlin.CD↗
arXiv2012

Collective oscillations in driven coagulation

We present a novel form of collective oscillatory behavior in the kinetics of irreversible coagulation with a constant input of monomers and removal of large clusters. For a broad class of collision rates, this system reaches a non-equilibrium stationary state at large times and the cluster size distribution tends to a universal form characterised by a constant flux of mass through the space of cluster sizes. Universality, in this context, means that the stationary state becomes independent of the cut-off as the cut-off grows. This universality is lost, however, if the aggregation rate between large and small clusters increases sufficiently steeply as a function of cluster sizes. We identify a transition to a regime in which the stationary state vanishes as the cut-off grows. This non-universal stationary state becomes unstable, however, as the cut-off is increased and undergoes a Hopf bifurcation. After this bifurcation, the stationary kinetics are replaced by persistent and periodic collective oscillations. These oscillations carry pulses of mass through the space of cluster sizes. As a result, the average mass flux remains constant. Furthermore, universality is partially restored in the sense that the scaling of the period and amplitude of oscillation is inherited from the dynamical scaling exponents of the universal regime. The implications of this new type of long-time asymptotic behaviour for other driven non-equilibrium systems are discussed.

cond-mat.stat-mech↗
arXiv2012

Driven Brownian coagulation of polymers

We present an analysis of the mean-field kinetics of Brownian coagulation of droplets and polymers driven by input of monomers which aims to characterize the long time behavior of the cluster size distribution as a function of the inverse fractal dimension, $a$, of the aggregates. We find that two types of long time behavior are possible. For $0\leq a < 1/2$ the size distribution reaches a stationary state with a power law distribution of cluster sizes having exponent 3/2. The amplitude of this stationary state is determined exactly as a function of $a$. For $1/2 < a \leq 1$, the cluster size distribution never reaches a stationary state. Instead a bimodal distribution is formed in which a narrow population of small clusters near the monomer scale is separated by a gap (where the cluster size distribution is effectively zero) from a population of large clusters which continue to grow for all time by absorbing small clusters. The marginal case, $a=1/2$, is difficult to analyze definitively, but we argue that the cluster size distribution becomes stationary and there is a logarithmic correction to the algebraic tail.

cond-mat.stat-mech↗
arXiv2012

Externally forced triads of resonantly interacting waves: boundedness and integrability properties

We revisit the problem of a triad of resonantly interacting nonlinear waves driven by an external force applied to the unstable mode of the triad. The equations are Hamiltonian, and can be reduced to a dynamical system for 5 real variables with 2 conservation laws. If the Hamiltonian, $H$, is zero we reduce this dynamical system to the motion of a particle in a one-dimensional time-independent potential and prove that the system is integrable. Explicit solutions are obtained for some particular initial conditions. When explicit solution is not possible we present a novel numerical/analytical method for approximating the dynamics. Furthermore we show analytically that when H=0 the motion is generically bounded. That is to say the waves in the forced triad are bounded in amplitude for all times for any initial condition with the single exception of one special choice of initial condition for which the forcing is in phase with the nonlinear oscillation of the triad. This means that the energy in the forced triad generically remains finite for all time despite the fact that there is no dissipation in the system. We provide a detailed characterisation of the dependence of the period and maximum energy of the system on the conserved quantities and forcing intensity. When $H \neq 0$ we reduce the problem to the motion of a particle in a one-dimensional time-periodic potential. Poincaré sections of this system provide strong evidence that the motion remains bounded when $H \neq 0$ and is typically quasi-periodic although periodic orbits can certainly be found. Throughout our analyses, the phases of the modes in the triad play a crucial role in understanding the dynamics.

nlin.CD↗
arXiv2011

Instantaneous Gelation in Smoluchowski's Coagulation Equation Revisited

We study the solutions of the Smoluchowski coagulation equation with a regularisation term which removes clusters from the system when their mass exceeds a specified cut-off size, M. We focus primarily on collision kernels which would exhibit an instantaneous gelation transition in the absence of any regularisation. Numerical simulations demonstrate that for such kernels with monodisperse initial data, the regularised gelation time decreases as M increases, consistent with the expectation that the gelation time is zero in the unregularised system. This decrease appears to be a logarithmically slow function of M, indicating that instantaneously gelling kernels may still be justifiable as physical models despite the fact that they are highly singular in the absence of a cut-off. We also study the case when a source of monomers is introduced in the regularised system. In this case a stationary state is reached. We present a complete analytic description of this regularised stationary state for the model kernel, K(m_1,m_2)=max{m_1,m_2}^v, which gels instantaneously when M tends to infinity if v>1. The stationary cluster size distribution decays as a stretched exponential for small cluster sizes and crosses over to a power law decay with exponent v for large cluster sizes. The total particle density in the stationary state slowly vanishes as (Log M^(v-1))^-1/2 when M gets large. The approach to the stationary state is non-trivial : oscillations about the stationary state emerge from the interplay between the monomer injection and the cut-off, M, which decay very slowly when M is large. A quantitative analysis of these oscillations is provided for the addition model which describes the situation in which clusters can only grow by absorbing monomers.

cond-mat.stat-mech↗
arXiv2011

Some remarks on the inverse Smoluchowski problem for cluster-cluster aggregation

It is proposed to revisit the inverse problem associated with Smoluchowski's coagulation equation. The objective is to reconstruct the functional form of the collision kernel from observations of the time evolution of the cluster size distribution. A regularised least squares method originally proposed by Wright and Ramkrishna (1992) based on the assumption of self-similarity is implemented and tested on numerical data generated for a range of different collision kernels. This method expands the collision kernel as a sum of orthogonal polynomials and works best when the kernel can be expressed exactly in terms of these polynomials. It is shown that plotting an "L-curve" can provide an a-priori understanding of the optimal value of the regularisation parameter and the reliability of the inversion procedure. For kernels which are not exactly expressible in terms of the orthogonal polynomials it is found empirically that the performance of the method can be enhanced by choosing a more complex regularisation function.

cond-mat.stat-mech↗