arXiv subjects
Publications and source records attributed to Martin Grant.
We prove that, in an arbitrary ordered field, L'H\^{o}pital's Rule is true if and only if the Least Upper Bound Property is true. We do the same for Taylor's Theorem with Peano Remainder, and for one other property sometimes given as a corollary of L'H\^{o}pital's Rule.
We look at buildings' competition over space in cities through the lens of ecology. Adopting the convex hull of the building's footprint perimeter as a definition of species yields parallels to forest trees' competition, which we expound on. Their perimeter distribution $p(r)$ follows a power-law behavior beyond a critical threshold of the density of the built environment. In this regime, the species coexistence likelihood $p(d)$, where $d$ is the distance to the nearest competitor, which we define to be a building with a larger $r$, bifurcates with the buildings' number $n$. This reveals two different predation laws: a vicious predatory one which is linked spatial homogeneity and segregation, as opposed to another favoring spatial diversity and intermixing between species.
In this paper, we explore the analogy between the refugees' drownings in the sea and the earthquakes' occurrences and focus on the aspect that characterizes the statistics of their spatial and temporal successions. The former is shown to parallel the spatial distribution of consecutive drowning events with the difference that the latter exhibits short-range behavior below $κ= 4km$ and it is characterized by scale-free statistics, with a critical exponent $δ\approx 0.5$, falling within the range of the earthquakes' $δ= 0.65 \pm 0.20$, as well as finite size scaling beyond $κ= 4km$, while the distribution of events' rates exhibits no similarity with that of the earthquakes. Finally, the events' velocity distribution is also recovered. $κ$ is suspected to be related to the radar and mobile network's coverage ranges and thus effectively represents a cut-off in the ability of picking up signals on drownings in the sea.
\begin{abstract} In this Letter we follow the asymptotic spatial correlation of buildings' heights $G_{\infty}(r)$ in the whole of the Netherlands \cite{bag3d}, which comprises $\approx 10,000,000$ buildings, for the purpose of recovering its scaling with respect to space and time given respectively by the exponents $r^{2 α}$ and $t^{2β}$. This allows us to identify the universality class of the evolution of the urban skyline seen as a dynamically evolving interface. Two major classes of cities were identified based on the recovered value of $α=0.4$ and $α= 0$, which correspond respectively to the KPZ and the EW universality classes. Picking a discrete model from each of these classes and mapping it to physical rules for constructions in cities we conclude that imposed restrictions on buildings' heights are reflected in the exponent $α$ and thus have implications on how the skyline evolves.
We use a sharp interface model for active cells to study the jamming transition point and behavior near it by varying cell concentration, active velocity and elasticity, including a binary mixture of soft and stiff cells. We determine the jamming transition point, as well as behavior near the transition,including the effective diffusion, and sixfold bond correlations. Finally, we expand on previous studies by showing the Voronoi dimensionless cell shape can be treated as an order parameter at any concentration.
In order to study the effect of cell elastic properties on the behavior of assemblies of motile cells, this paper describes an alternative to the cell phase field (CPF) \cite{Palmieri2015} we have previously proposed. The CPF is a multi-scale approach to simulating many cells which tracked individual cells and allowed for large deformations. Though results were largely in agreement with experiment that focus on the migration of a soft cancer cell in a confluent layer of normal cells \cite{Lee2012}, simulations required large computing resources, making more detailed study unfeasible. In this work we derive a sharp interface limit of CPF, including all interactions and parameters. This new model offers over $200$ fold speedup when compared to our original CPF implementation. We demonstrate that this model captures similar behavior and allows us to obtain new results that were previously intractable. We obtain the full velocity distribution for a large range of degrees of confluence, $ρ$, and show regimes where its tail is heavier and lighter than a normal distribution. Furthermore, we fully characterize the velocity distribution with a single parameter, and its dependence on $ρ$ is fully determined. Finally, cell motility is shown to linearly decrease with increasing $ρ$, consistent with previous theoretical results.
We present a model to generate power spectrum noise with intensity proportional to 1/f as a function of frequency f. The model arises from a broken-symmetry variable which corresponds to absolute pitch, where fluctuations occur in an attempt to restore that symmetry, influenced by interactions in the creation of musical melodies.
A new isotropic magneto-elastic phase field crystal (PFC) model to study the relation between morphological structure and magnetic properties of pure ferromagnetic solids is introduced. Analytic calculations were used to determine the phase diagram and obtain the relationship between elastic strains and magnetization. Time dependent numerical simulations were used to demonstrate the effect of grain boundaries on the formation of magnetic domains. It was shown that the grain boundaries act as nucleating sites for domains of reverse magnetization. Finally, we derive a relation for coercivity versus grain mis-orientation in the isotropic limit.
Plastic deformation in solids induced by external shear stress is of huge practical interest. Presence of local crystalline order in polycrystals, consisting of many grains, distinguishes its deformation pattern from that of amorphous materials. Despite strong anisotropy, induced by external stress, the plastic flow and the consequent deformation field show strong dynamical heterogeneity. The distribution $P(u)$ of particle displacements ($u$) shows three distinct regimes including a power law scaling regime at moderate displacements. Using a phase field simulation we show how polycrystals generate saddle and vortex like flow patterns, which hitherto have been termed as elementary plastic events in the context of amorphous materials. Interestingly, such events here find natural explanation in terms of the underlying dislocation dynamics. We also characterize the spatial distribution of the flow field using Okubo-Weiss measure.
The dynamics of glass formation in monatomic and binary liquids are studied numerically using a microscopic field theory for the evolution of the time-averaged atomic number density. A stochastic framework combining phase field crystal free energies and dynamic density functional theory is shown to successfully describe several aspects of glass formation over multiple time scales. Agreement with mode coupling theory is demonstrated for underdamped liquids at moderate supercoolings, and a rapidly growing dynamic correlation length is found to be associated with fragile behavior.
The significant role of space in maintaining species coexistence and determining community structure and function is well established. However, community ecology studies have mainly focused on simple competition and predation systems, and the relative impact of positive interspecific interactions in shaping communities in a spatial context is not well understood. Here we employ a spatially explicit metacommunity model to investigate the effect of local dispersal on the structure and function of communities in which species are linked through an interaction web comprising mutualism, competition and exploitation. Our results show that function, diversity and interspecific interactions of locally linked communities undergo a phase transition with changes in the rate of species dispersal. We find that low spatial interconnectedness favors the spontaneous emergence of strongly mutualistic communities which are more stable but less productive and diverse. On the other hand, high spatial interconnectedness promotes local biodiversity at the expense of local stability and supports communities with a wide range of interspecific interactions. We argue that investigations of the relationship between spatial processes and the self-organization of complex interaction webs are critical to understanding the geographic structure of interactions in real landscapes.
We use molecular dynamics to study the nucleation of cracks in a two dimensional material without pre-existing cracks. We study models with zero and non-zero shear modulus. In both situations the time required for crack formation obeys an Arrhenius law, from which the energy barrier and pre-factor are extracted for different system sizes. For large systems, the characteristic time of rupture is found to decrease with system size, in agreement with classical Weibull theory. In the case of zero shear modulus, the energy opposing rupture is identified with the breakage of a single atomic layer. In the case of non-zero shear modulus, thermally activated fracture can only be studied within a reasonable time at very high strains. In this case the energy barrier involves the stretching of bonds within several layers, accounting for a much higher barrier compared to the zero shear modulus case. This barrier is understood within adiabatic simulations.
In this paper we introduce a three-dimensional version of the Mercedes-Benz model to describe water molecules. In this model van der Waals interactions and hydrogen bonds are given explicitly through a Lennard-Jones potential and a Gaussian orientation-dependent terms, respectively. At low temperature the model freezes forming Ice-I and it reproduces the main peaks of the experimental radial distribution function of water. In addition to these structural properties, the model also captures the thermodynamical anomalies of water: the anomalous density profile, the negative thermal expansivity, the large heat capacity and the minimum in the isothermal compressibility.
Liquid-crystal networks consist of weakly crosslinked polymers that are coupled to liquid-crystal molecules. The resultant hybrid system has rich elastic properties. We develop a phase field model to describe mechanical properties of a hexagonal liquid-crystal network. The hexagonal liquid-crystal network is found to have soft shear deformations. The elastic properties are predicted analytically and confirmed with numerical simulations. In addition the model incorporates non-linear elasticity and dislocations or disclinations.
We present a spatial, individual-based predator-prey model in which dispersal is dependent on the local community. We determine species suitability to the biotic conditions of their local environment through a time and space varying fitness measure. Dispersal of individuals to nearby communities occurs whenever their fitness falls below a predefined tolerance threshold. The spatiotemporal dynamics of the model is described in terms of this threshold. We compare this dynamics with the one obtained through density-independent dispersal and find marked differences. In the community-driven scenario, the spatial correlations in the population density do not vary in a linear fashion as we increase the tolerance threshold. Instead we find the system to cross different dynamical regimes as the threshold is raised. Spatial patterns evolve from disordered, to scale-free complex patterns, to finally becoming well-organized domains. This model therefore predicts that natural populations, the dispersal strategies of which are likely to be influenced by their local environment, might be subject to complex spatiotemporal dynamics.
Dislocation and grain boundary melting are studied in three dimensions using the Phase Field Crystal method. Isolated dislocations are found to melt radially outward from their core, as the localized excess elastic energy drives a power law divergence in the melt radius. Dislocations within low-to-mid angle grain boundaries melt similarly until an angle-dependent first order wetting transition occurs when neighboring melted regions coalesce. High angle boundaries are treated within a screening approximation, and issues related to ensembles, metastability, and grain size are discussed.
Phase field crystals (PFC) are a tool for simulating materials at the atomic level. They combine the small length-scale resolution of molecular dynamics (MD) with the ability to simulate dynamics on mesoscopic time scales. We show how PFC can be interpreted as the result of applying coarse-graining in time to the microscopic density field of molecular dynamics simulations. We take the form of the free energy for the phase field from the classical density functional theory of inhomogeneous liquids and then choose coefficients to match the structure factor of the time coarse-grained microscopic density field. As an example, we show how to construct a PFC free energy for Weber and Stillinger's two-dimensional square crystal potential which models a system of proteins suspended in a membrane.
We elucidate the mechanism of cold denaturation through constant-pressure simulations for a model of hydrophobic molecules in an explicit solvent. We find that the temperature dependence of the hydrophobic effect is the driving force/induces/facilitates cold denaturation. The physical mechanism underlying this phenomenon is identified as the destabilization of hydrophobic contact in favor of solvent separated configurations, the same mechanism seen in pressure induced denaturation. A phenomenological explanation proposed for the mechanism is suggested as being responsible for cold denaturation in real proteins.