arXiv subjects

Thomas Hudson

Publications and source records attributed to Thomas Hudson.

At least 19 recordsLinked to original sources

arXiv2026

Curve-Supported Concentrations of Vortices and Screw Dislocations Subject to Confinement

We study the concentration of logarithmically interacting defects, such as Ginzburg--Landau vortices and screw dislocations, on closed curves in two dimensions, subject to a confining field. We prove that a measure supported on a smooth closed curve solves the mean-field transport equation exactly when, in Lagrangian labels, the curve moves with the negative induced velocity and each label keeps its mass. This contour law is a gradient flow of the mean-field energy. We prove that the gradient of the energy on a curve has a leading term proportional to the operator $|D|$ with Fourier symbol $|k|$, which prevents every smooth equilibrium curve from being a local minimizer, in a gradient-flow analogue of the Kelvin--Helmholtz instability of vortex sheets. In a radially symmetric case with quadratic confinement, we find an explicit circular equilibrium whose linearization is diagonal in Fourier modes, and we show that, as $N\rightarrow \infty $, the linear spectrum of the regular $N$-gon converges, mode by mode, to that of the circle. Finally, we compute a critical point on two concentric circles with lower energy than the single circle, indicating a rich family of multi-circle equilibria.

math.AP↗

arXiv2026

On the linearised force balance condition in the analysis of atomistic dislocation models

A technical condition arising in the analysis of atomistic models for dislocations is studied. Key estimates for the far-field strain behaviour proved in Ehrlacher, Ortner, Shapeev (2016) rely on a summation-by-parts argument for linearised forces, under conditions of sufficient decay and of vanishing net force in an infinite system. In particular, the latter condition is required for an application of this theory to the standard far-field dislocation predictor, but the vanishing of the associated force sum has not been fully justified. Here, the missing verification of this condition is provided and its role in determining the far-field decay of the corrector is clarified. It is moreover shown that for more general physically compatible predictors, the linearised force sum need not vanish, leading to slower decay and potentially divergent finite-domain approximations.

math.NA↗

arXiv2026

On first-order thermodynamic equilibrium conditions for fluid-fluid and solid-phase interfaces

Starting from the constrained variational formulation of Larché and Cahn for solids in contact with fluids, a unified thermodynamic framework for such systems is developed. Both bulk and interfacial equilibrium conditions arise as stationarity conditions of a single thermodynamic functional. While earlier theory neglects a full treatment of interfacial contributions and therefore cannot describe systems in which surface effects are significant, the present formulation incorporates interfacial thermodynamics directly into the variational principle. The framework is introduced first for fluid-fluid systems to establish the underlying mathematical structure, and is then extended to solid-fluid interfaces. Within this setting, classical equilibrium relations for heterogeneous systems emerge naturally from different classes of admissible variations, providing a common theoretical basis for bulk, interfacial, and configurational thermodynamics.

cond-mat.stat-mech↗

arXiv2026

Numerical stability revisited: A family of benchmark problems for the analysis of explicit stochastic differential equation integrators

We revisit the numerical stability of four well-established explicit stochastic integration schemes through a new generic benchmark stochastic differential equation designed to assess asymptotic statistical accuracy and stability properties. This one-parameter benchmark equation is derived from a general one-dimensional first-order SDE using spatio-temporal nondimensionalization and is employed to evaluate the performance of the (1) Euler-Maruyama, (2) Milstein, (3) Stochastic Heun, and (4) three-stage Runge-Kutta schemes. Our findings reveal that lower-order schemes can outperform higher-order ones over a range of time step sizes, depending on the benchmark parameters and application context. The theoretical results are validated through a series of numerical experiments, and we discuss their implications for more general applications, including a nonlinear example. Our results suggest that the insights obtained from the linear benchmark problem provide reliable guidance for time-stepping strategies when simulating nonlinear SDEs.

math.NA↗

arXiv2026

Accurate and Efficient Interatomic Potentials for Dislocations in InP

We present Atomic Cluster Expansion (ACE) and MACE models trained on a new dataset of Density Functional Theory (DFT) calculations, constructed for the task of studying the mobility of dislocations in Indium Phosphide (InP). The models are validated in a suite of tests against RSCAN DFT, and compared with previously published potentials from literature. Our new models act as much better surrogates for DFT than the literature models: errors on partial dislocation formation energies are at most 4% for both ACE and MACE, compared with 18% for the MACE-MPA foundation model and 42-50% for earlier bespoke potentials. The bespoke MACE model achieves this accuracy while being around five times faster to evaluate than the MP0 and MPA foundation models.

cond-mat.mtrl-sci↗

arXiv2026

Bridging Atomistic and Continuum Descriptions of Nanoscale Dislocation Loops in Tungsten

In order to predict the long-term effects of irradiation on the material properties of tungsten, a continuum approach to simulating the interactions of dislocation loops, which arise from radiation damage, is proposed. Continuum models of the displacement, strain and stress fields produced by dislocation loops exhibit unphysical singularities near the defect core, but are thought to accurately capture atomistic displacements in the far-field. A linear elastic model of nanoscale dislocation loops in tungsten is developed, and the model is verified using atomistic simulations to ensure that the model is informed by lower-length scale phenomena such that the physics of the problem is correctly captured. We discuss the model and its advantages, and show that predictions produced by atomistic simulations do indeed agree well with the far-field behaviour of the continuum model when dislocation loops are far from material boundaries. In particular, we robustly demonstrate that the decay rate of atomistic results and continuum results coincide with one another, and show that the results converge as the size of the atomistic simulations approach the far-field limit.

cond-mat.mtrl-sci↗

arXiv2025

The $C_2$-equivariant ordinary cohomology of complex quadrics II: The symmetric case

In this, the second of three papers about $C_2$-equivariant complex quadrics, we calculate the equivariant ordinary cohomology of smooth symmetric quadrics graded on the representation ring of $ΠBU(1)$ and with coefficients in the Burnside Mackey functor. These calculations exhibit various interesting properties, including the first naturally occurring example we are aware of where the cohomology is not just the sum of shifted copies of the cohomology of a point, but also has summands that are shifted copies of the cohomology of the free orbit $C_2/e$.

math.AT↗

arXiv2025

The $C_2$-equivariant ordinary cohomology of complex quadrics III: Exceptional cases

In this, the last of three papers about $C_2$-equivariant complex quadrics, we complete the calculation of the equivariant ordinary cohomology of smooth symmetric quadrics in the cases where the fixed sets have more than two components. These calculations imply one for a $C_2$-equivariant Grassmannian, which we use to prove an equivariant refinement of the result that there are 27 lines on a cubic surface in $\mathbb{P}^3$.

math.AT↗

arXiv2025

The $C_2$-equivariant ordinary cohomology of complex quadrics I: The antisymmetric case

In this, the first of three papers about $C_2$-equivariant complex quadrics, we calculate the equivariant ordinary cohomology of smooth antisymmetric quadrics. One of these quadrics coincides with a $C_2$-equivariant Grassmannian, and we use this calculation to prove an equivariant refinement of the result that there are 27 lines on a cubic surface in $\mathbb{P}^3$.

math.AT↗

arXiv2025

A quantitative model for the Frank-Read dislocation source based on pinned mean curvature flow

This work introduces a simple quantitative model for the Frank--Read source, considered to be one of the most important micro-mechanical mechanisms of dislocation creation in crystalline materials. It has long been known that these sources create dislocations in a repetitive, oscillatory process, which is driven by an external shear force. Unlike the existing explanations in the literature, the model introduced in the present article is based on just a few simple physical principles, namely line tension and dislocation motion due to a single slip plane flow rule, together with a pinning constraint on the ends of the central dislocation line. A complete discretisation, including suitable re-meshing and ``topological cutting'' algorithms, is described and simulation results are discussed. Despite its conceptual simplicity, the model and discretisation described in the present work yield remarkably accurate predictions about the shape and properties of the Frank--Read source. In particular, it is shown that only one dimensionless parameter controls the dynamics of the Frank--Read source if one neglects crystal anisotropy. This allows to derive an emergent law about the length of dislocation line generated per shear energy.

cond-mat.mtrl-sci↗

arXiv2025

Existence and uniqueness for a class of fractional drift-diffusion equations

This work establishes the existence and uniqueness of solutions to the fractional diffusion equation $$\frac{\partial^αu}{\partial t^α} + K(-Δ)^β u - \nabla \cdot (\nabla V u) = f$$ on a $d$-dimensional torus, subject to sufficient conditions on the input parameters. The focus is on fractional orders $α$ and $β$ less than 1. The strategy uses a Galerkin method and focuses on the additional complexity that comes from the fractional-order derivatives. Additional Sobolev regularity of the solution is shown. The spectral approach to the existence proof suggests an algorithm to compute explicit solutions numerically, and the regularity results are used to support a rigorous convergence analysis of the proposed numerical scheme.

math.AP↗

arXiv2024

Chow-Witt rings and topology of flag varieties

The paper computes the Witt-sheaf cohomology rings of partial flag varieties in type A in terms of the Pontryagin classes of the subquotient bundles. The proof is based on a Leray-Hirsch-type theorem for Witt-sheaf cohomology for the maximal rank cases, and a detailed study of cohomology ring presentations and annihilators of characteristic classes for the general case. The computations have consequences for the topology of real flag manifolds: we show that all torsion in the integral cohomology is 2-torsion, which was not known in full generality previously. This allows for example to compute the Poincaré polynomials of complete flag varieties for cohomology with twisted integer coefficients. The computations also allow to describe the Chow-Witt rings of flag varieties, and we sketch an enumerative application to counting flags satisfying multiple incidence conditions to given hypersurfaces.

math.AG↗

arXiv2024

The $C_2$-equivariant ordinary cohomology of $BU(2)$

We calculate the ordinary $C_2$-cohomology, with Burnside ring coefficients, of $BU(2)$, the classifying space for $C_2$-equivariant complex 2-plane bundles, using an extended grading that allows us to capture a more natural set of generators. This allows us to define characteristic classes for such bundles. Combined with earlier calculations, it also allows us to define characteristic numbers for equivariant complex lines and surfaces and we give some sample computations.

math.AT↗

arXiv2024

The $C_2$-equivariant ordinary cohomology of $BT^2$

We calculate the ordinary $C_2$-cohomology of $BT^2$ with Burnside ring coefficients, using an extended grading that allows us to capture a more natural set of generators. We discuss how this cohomology is related to those of $BT^1$ and $BU(2)$, calculated previously, both relationships being more complicated than in the nonequivariant case.

math.AT↗

arXiv2024

Dislocation dynamics in Ni-based superalloys: Parameterising dislocation trajectories from atomistic simulations

Nanoscale precipitates in the microstructure of nickel-based superalloys hinder dislocation motion, which results in an extraordinary strengthening effect at elevated temperatures. We used molecular dynamics (MD) with classical effective potential to observe the movement of an $\frac{a}{2}\langle110\rangle\{111\}$ edge dislocation under shear in pure Ni, which represents the Ni solid solution matrix, and extracted the locations of the dislocations. We show how a Differential Evolution Monte Carlo (DE-MC) analysis is an effective way to find the parameters of an equation of motion for the dislocation lines with quantified uncertainties. The parameters of interest were the effective mass, drag coefficient, and force experienced by the dislocation. The marginal parameter and joint posterior distributions were estimated from the accepted samples produced by the DE-MC algorithm. The equation of motion and parameter distributions were used to predict the dislocation positions and velocities at the simulation timesteps, and the mean fit was found to match the MD trajectories with a root mean square error (RMSE) of \SI{0.2}{\nano\metre}. We also discuss how the selected model can be extended to account for the presence of multiple dislocations as well as dislocation-precipitate interactions. This work serves as the first step towards building a predictive surrogate model that describes the deformation behaviour of Ni-based superalloys.

cond-mat.mtrl-sci↗

arXiv2023

A geometric $C_2$-equivariant Bézout Theorem

Classically, Bézout's theorem says that an intersection of hypersurfaces in a projective space is rationally equivalent to a number of copies of a smaller projective space, the number depending on the degrees of the hypersurfaces. We give a generalization of that result to the context of $C_2$-equivariant hypersurfaces in $C_2$-equivariant linear projective space, expressing the intersection as a linear combination of equivariant Schubert varieties.

math.AT↗

arXiv2023

Dynamical properties of coarse-grained linear SDEs

Coarse-graining or model reduction is a term describing a range of approaches used to extend the time-scale of molecular simulations by reducing the number of degrees of freedom. In the context of molecular simulation, standard coarse-graining approaches approximate the potential of mean force and use this to drive an effective Markovian model. To gain insight into this process, the simple case of a quadratic energy is studied in an overdamped setting. A hierarchy of reduced models is derived and analysed, and the merits of these different coarse-graining approaches are discussed. In particular, while standard recipes for model reduction accurately capture static equilibrium statistics, it is shown that dynamical statistics such as the mean-squared displacement display systematic error, even when a system exhibits large time-scale separation. In the linear setting studied, it is demonstrated both analytically and numerically that such models can be augmented in a simple way to better capture dynamical statistics.

math.DS↗

arXiv2022

An algebraic $C_2$-equivariant Bézout's theorem

Bézout's theorem, nonequivariantly, can be interpreted as a calculation of the Euler class of a sum of line bundles over complex projective space, expressing it in terms of the rank of the bundle and its degree. We give here a generalization to the $C_2$-equivariant context, using the calculation of the cohomology of a $C_2$-complex projective space from an earlier paper. We use ordinary $C_2$-cohomology with Burnside ring coefficients and an extended grading necessary to define the Euler class, which we express in terms of the equivariant rank of the bundle and the degrees of the bundle and its fixed subbundles. We do similar calculations using constant $\mathbb{Z}$ coefficients and Borel cohomology and compare the results.

math.AT↗