Searcharxiv⌕ Search

arXiv subjects

Thomas Machon

Publications and source records attributed to Thomas Machon.

At least 19 recordsLinked to original sources
arXiv2024

Universal limiting behaviour of reaction-diffusion systems with conservation laws

Making sense of complex inhomogeneous systems composed of many interacting species is a grand challenge that pervades basically all natural sciences. Phase separation and pattern formation in reaction-diffusion systems have been largely studied as two separate paradigms. Here we show that in reaction-diffusion systems composed of many species, the presence of a conservation law constrains the evolution of the conserved quantity to be governed by a Cahn-Hilliard-like equation. This establishes a direct link with the paradigm of coexistence and recent "active" field theories. Hence, even for complex many-species systems a dramatically simplified but accurate description emerges over coarse spatio-temporal scales. Using the nullcline (the line of homogeneous steady states) as the central motif, we develop a geometrical framework which endows chemical space with a basis and suitable coordinates. This framework allows us to capture and understand the effect of eliminating fast non-conserved degrees of freedom, and to explicitly construct coefficients of the coarse field theory. We expect that the theory we develop here will be particularly relevant to advance our understanding of biomolecular condensates.

cond-mat.soft↗
arXiv2023

Biases in Inverse Ising Estimates of Near-Critical Behaviour

Inverse Ising inference allows pairwise interactions of complex binary systems to be reconstructed from empirical correlations. Typical estimators used for this inference, such as Pseudo-likelihood maximization (PLM), are biased. Using the Sherrington-Kirkpatrick (SK) model as a benchmark, we show that these biases are large in critical regimes close to phase boundaries, and may alter the qualitative interpretation of the inferred model. In particular, we show that the small-sample bias causes models inferred through PLM to appear closer-to-criticality than one would expect from the data. Data-driven methods to correct this bias are explored and applied to a functional magnetic resonance imaging (fMRI) dataset from neuroscience. Our results indicate that additional care should be taken when attributing criticality to real-world datasets.

cond-mat.dis-nn↗
arXiv2022

Poisson structures on sets of Maurer-Cartan elements

Given a differential graded Lie algebra (dgla) L satisfying certain conditions, we construct Poisson structures on the gauge orbits of its set of Maurer-Cartan (MC) elements, termed Maurer-Cartan-Poisson (MCP) structures. They associate a compatible Batalin-Vilkovisky algebra to each MC element of L. An MCP structure is shown to exist for a number of dglas associated to commutative Frobenius algebras, deformations of Poisson and symplectic structures, as well as the Chevally-Eilenberg complex. MCP structures yield a notion of hamiltonian flow of MC elements, and also define Lie algebroids on gauge orbits, whose isotropy algebras give invariants of MC elements. As an example, this gives a finite-dimensional two-step nilpotent graded Lie algebra associated to any closed symplectic manifold.

math.DG↗
arXiv2022

Anti-Holomorphic Modes in Vortex Lattices

A continuum theory of linearized Helmholtz-Kirchoff point vortex dynamics about a steadily rotating lattice state is developed by two separate methods: firstly by a direct procedure, secondly by taking the long-wavelength limit of Tkachenko's exact solution for a triangular vortex lattice. Solutions to the continuum theory are found, described by arbitrary anti-holomorphic functions, and give power-law localized edge modes. Numerical results for finite lattices show excellent agreement to the theory.

physics.flu-dyn↗
arXiv2021

Uniaxial versus Biaxial Pathways in One-Dimensional Cholesteric Liquid Crystals

Cholesteric liquid crystals exhibit great morphological richness of static metastable states. Understanding the transitions between such states is key for the development of switchable devices. We show, using a quasi-one-dimensional model, that cholesterics exhibit distinct uniaxial and biaxial pathways between distinct minima. We study transitions between different layer numbers and prove, and show, that transition states are distinguished either through splay-mediated untwisting, understood through contact topology, or the presence of biaxiality. Furthermore we characterise a menagerie of additional saddle points that dictate the connectivity of the solution landscape.

cond-mat.soft↗
arXiv2021

The Topological Origin of the Peierls-Nabarro Barrier

Crystals and other condensed matter systems described by density waves often exhibit dislocations. Here we show, by considering the topology of the ground state manifolds (GSMs) of such systems, that dislocations in the density phase field always split into disclinations, and that the disclinations themselves are constrained to sit at particular points in the GSM. Consequently, the topology of the GSM forbids zero-energy dislocation glide, giving rise to a Peirels-Nabarro barrier.

cond-mat.soft↗
arXiv2021

Stability of highly-twisted Skyrmions from contact topology

We describe a topological protection mechanism for highly-twisted two-dimensional Skyrmions in systems with Dyloshinskii-Moriya (DM) coupling, where non-zero DM energy density (dubbed "twisting energy density") acts as a kind of band gap in real space, yielding an N invariant for highly twisted Skyrmions. We prove our result through the application of contact topology by extending our system along a fictitious third dimension, and further establish the structural stability of highly-twisted Skyrmions under arbitrary chirality-preserving distortions. Our results apply for all two-dimensional systems hosting Skyrmion excitations including spin-orbit coupled systems exhibiting quantum Hall ferromagnetism.

cond-mat.mes-hall↗
arXiv2020

A Poisson bracket on the space of Poisson structures

Let $M$ be a smooth closed orientable manifold and $\mathcal{P}(M)$ the space of Poisson structures on $M$. We construct a Poisson bracket on $\mathcal{P}(M)$ depending on a choice of volume form. The Hamiltonian flow of the bracket acts on $\mathcal{P}(M)$ by volume-preserving diffeomorphism of $M$. We then define an invariant of a Poisson structure that describes fixed points of the flow equation and compute it for regular Poisson 3-manifolds, where it detects unimodularity. For unimodular Poisson structures we define a further, related Poisson bracket and show that for symplectic structures the associated invariant counting fixed points of the flow equation is given in terms of the $d d^\Lambda$ and $d+ d^\Lambda$ symplectic cohomology groups defined by Tseng and Yau.

math.DG↗
arXiv2020

Good Fibrations: Packing Rules for Diabolic Domains

We describe a theory of packing hyperboloid 'diabolic' domains in bend-free textures of liquid crystals. The domains sew together continuously, providing a menagerie of bend-free textures akin to the packing of focal conic domains in smectic liquid crystals. We show how distinct domains may be related to each other by Lorentz transformations, and that this process may lower the elastic energy of the system. We discuss a number of phases that may be formed as a result, including splay-twist analogues of blue phases. We also discuss how these diabolic domains may be subject to "superluminal boosts", yielding defects analogous to shocks waves. We explore the geometry of these textures, demonstrating their relation to Milnor fibrations of the Hopf link. Finally, we show how the theory of these domains is unified in four-dimensional space.

cond-mat.soft↗
arXiv2020

The Godbillon-Vey Invariant as Topological Vorticity Compression and Obstruction to Steady Flow in Ideal Fluids

If the vorticity field of an ideal fluid is tangent to a foliation, additional conservation laws arise. For a class of zero-helicity vorticity fields the Godbillon-Vey (GV) invariant of foliations is defined and is shown to be an invariant purely of the vorticity, becoming a higher-order helicity-type invariant of the flow. GV non-zero gives both a global topological obstruction to steady flow and, in a particular form, a local obstruction. GV is interpreted as helical compression and stretching of vortex lines. Examples are given where the value of GV is determined by a set of distinguished closed vortex lines.

math-ph↗
arXiv2019

A Bj\"orling Representation for Jacobi Fields on Minimal Surfaces and Soap Film Instabilities

We develop a general framework for the description of instabilities on soap films using the Bj\"orling representation of minimal surfaces. The construction is naturally geometric and the instability has the interpretation as being specified by its amplitude and transverse gradient along any curve lying in the minimal surface. When the amplitude vanishes, the curve forms part of the boundary to a critically stable domain, while when the gradient vanishes the Jacobi field is maximal along the curve. In the latter case, we show that the Jacobi field is maximally localised if its amplitude is taken to be the lowest eigenfunction of a one-dimensional Schr\"odinger operator. We present examples for the helicoid, catenoid, circular helicoids and planar Enneper minimal surfaces, and emphasise that the geometric nature of the Bj\"orling representation allows direct connection with instabilities observed in soap films.

cond-mat.soft↗
arXiv2018

Inverse Ising inference, hyperuniformity and absorbing states in the Manna model

Using inverse Ising inference we show that the absorbing states of the one-dimensional Manna model can be described by an equilibrium model with an emergent interaction displaying short-ranged repulsion and long-ranged attraction. As the model approaches the critical point the interaction becomes purely repulsive, decaying as $r^{-\alpha}$ and we conjecture the exact value $\alpha=1/2$, suggesting density fluctuations decay as $r^{-3/2}$. We present a simple Gaussian field theory for the long distance behaviour of critical absorbing states and discuss implications for the Manna universality class.

cond-mat.stat-mech↗
arXiv2018

Aspects of Defect Topology in Smectic Liquid Crystals

We study the topology of smectic defects in two and three dimensions. We give a topological classification of smectic point defects and disclination lines in three dimensions. In addition we describe the combination rules for smectic point defects in two and three dimensions, showing how the broken translational symmetry of the smectic confers a path dependence on the result of defect addition.

cond-mat.soft↗
arXiv2018

Woven Nematic Defects, Skyrmions and the Abelian Sandpile Model

We show that a fixed set of woven defect lines in a nematic liquid crystal supports a set of non-singular topological states which can be mapped on to recurrent stable configurations in the Abelian sandpile model or chip-firing game. The physical correspondence between local Skyrmion flux and sandpile height is made between the two models. Using a toy model of the elastic energy, we examine the structure of energy minima as a function of topological class and show that the system admits domain wall Skyrmion solitons.

cond-mat.soft↗
arXiv2017

Contact Topology and the Structure and Dynamics of Cholesterics

Using tools and concepts from contact topology we show that non-vanishing twist implies conservation of the layer structure in cholesteric liquid crystals. This leads to a number of additional topological invariants for cholesteric textures, such as layer numbers, that are not captured by traditional descriptions, characterises the nature and size of the chiral energy barriers between metastable configurations, and gives a geometric characterisation of cholesteric dynamics in any context, including active systems, those in confined geometries or under the influence of an external field.

cond-mat.soft↗
arXiv2017

Composite Dislocations in Smectic Liquid Crystals

Smectic liquid crystals are charcterized by layers that have a preferred uniform spacing and vanishing curvature in their ground state. Dislocations in the smectics play an important role in phase nucleation, layer reorientation, and dynamics. Typically modeled as possessing one line singularity, the layer structure of a dislocation leads to a diverging compression strain as one approaches the defect center, suggesting a large, elastically determined melted core. However, it has been observed that for large charge dislocations, the defect breaks up into two disclinations [C. E. Williams, Philos. Mag. 32, 313 (1975)]. Here we investigate the topology of the composite core. Because the smectic cannot twist, transformations between different disclination geometries are highly constrained. We demonstrate the geometric route between them and show that despite enjoying precisely the topological rules of the three-dimensional nematic, the additional structure of line disclinations in three-dimensional smectics localizes transitions to higher-order point singularities.

cond-mat.soft↗