arXiv subjects

Jamie M. Taylor

Publications and source records attributed to Jamie M. Taylor.

26 records · Page 2Linked to original sources

arXiv2018

The excluded volume of two-dimensional convex bodies: Shape reconstruction and non-uniqueness

In the Onsager model of one-component hard-particle systems, the entire phase behaviour is dictated by a function of relative orientation, which represents the amount of space excluded to one particle by another at this relative orientation. We term this function the excluded volume function. Within the context of two-dimensional convex bodies, we investigate this excluded volume function for one-component systems addressing two related questions. Firstly, given a body can we find the excluded volume function?, Secondly, can we reconstruct a body from its excluded volume function? The former is readily answered via an explicit Fourier series representation, in terms of the support function. However we show the latter question is ill-posed in the sense that solutions are not unique for a large class of bodies. This degeneracy is well characterised however, with two bodies admitting the same excluded volume function if and only if the Fourier coefficients of their support functions differ only in phase. Despite the non-uniqueness issue, we then propose and analyse a method for reconstructing a convex body given its excluded volume function, by means of a discretisation procedure where convex bodies are approximated by zonotopes with a fixed number of sides. It is shown that the algorithm will always asymptotically produce a best $L^2$ approximation of the trial function, within the space of excluded volume functions of centrally symmetric bodies. In particular, if a solution exists, it can be found. Results from a numerical implementation are presented, showing that with only desktop computing power, good approximations to solutions can be readily found.

cond-mat.soft↗

arXiv2018

Contributions of Repulsive and Attractive Interactions to Nematic Order

Both repulsive and attractive molecular interactions can be used to explain the onset of nematic order. The object of this paper is to combine these two nematogenic molecular interactions in a unified theory. This attempt is not unprecedented; what is perhaps new is the focus on the understanding of nematics in the high density limit. There, the orientational probability distribution is shown to exhibit a unique feature: it has compact support on configuration space. As attractive interactions are turned on, the behavior changes, and at a critical attractive interaction strength, thermotropic behavior of the Maier-Saupe type is attained.

cond-mat.soft↗

arXiv2018

Convex Integration Arising in the Modelling of Shape-Memory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations

We study convex integration solutions in the context of the modelling of shape-memory alloys. The purpose of the article is two-fold, treating both rigidity and flexibility properties: Firstly, we relate the maximal regularity of convex integration solutions to the presence of lower bounds in variational models with surface energy. Hence, variational models with surface energy could be viewed as a selection mechanism allowing for or excluding convex integration solutions. Secondly, we present the first numerical implementations of convex integration schemes for the model problem of the geometrically linearised two-dimensional hexagonal-to-rhombic phase transformation. We discuss and compare the two algorithms from [RZZ16] and [RZZ17].

math.AP↗

arXiv2017

Density functional theory for dense nematics with steric interactions

The celebrated work of Onsager on hard particle systems, based on the truncated second order virial expansion, is valid at relatively low volume fractions for large aspect ratio particles. While it predicts the isotropic-nematic phase transition, it fails to provide a realistic equation of state in that the pressure remains finite for arbitrarily high densities. In this work, we derive a mean field density functional form of the Helmholtz free energy for nematics with hard core repulsion. In addition to predicting the isotropic-nematic transition, the model provides a more realistic equation of state. The energy landscape is much richer, and the orientational probability distribution function in the nematic phase possesses a unique feature: it vanishes on a nonzero measure set in orientational space.

cond-mat.soft↗

arXiv2017

Oseen-Frank-type theories of ordered media as the $Γ$-limit of a non-local mean-field free energy

In this work we recover the Oseen-Frank theory of nematic liquid crystals as a $Γ$-limit of a particular mean-field free energy as the sample size becomes infinitely large. The Frank constants are not necessarily all equal. Our analysis takes place in a broader framework however, also providing results for more general systems such as biaxial or polar molecules. We start from a mesoscopic model describing a competition between entropy and a non-local pairwise interactions between molecules. We assume the interaction potential is separable so that the energy can be reduced to a model involving a macroscopic order parameter. We assume only integrability and decay properties of the macroscopic interaction, but no regularity assumptions are required. In particular, singular interactions are permitted. The analysis becomes significantly simpler on a translationally invariant domain, so we first consider periodic domains with increasing domain of periodicity. Then we tackle the problem on a Lipschitz domain with non-local boundary conditions by considering an asymptotically equivalent problem on periodic domains. We conclude by applying the results to a case which reduces to the Oseen-Frank model of elasticity, and give expressions for the Frank constants in terms of integrals of the interaction kernel.

math.AP↗

arXiv2017

An analysis of equilibria in dense nematic liquid crystals

This paper is concerned with the rigorous analysis of a recently proposed model of Zheng et. al. for describing nematic liquid crystals within the dense regime, with the orientation distribution function as the variable. A key feature of the model is that in high density regimes all non-trivial minimisers are zero on a set of positive measure so that L^infinity variations cannot generally be taken about minimisers. In particular, it is unclear if the Euler-Lagrange equation is well defined, and if local minimisers satisfy it. It will be shown that there exists an analogue of the Euler-Lagrange equation that is satisfied by L^p local minimisers by reducing the minimisation problem to an equivalent finite-dimensional saddle-point problem, obtained by observing that on certain subsets of the domain the free-energy functional is convex so that duality methods can be applied. This analogue of the Euler-Lagrange equation is then shown to be equivalent to a vanishing variation criteria on a certain family of non-linear curves on which the free- energy functional is sufficiently smooth. All critical points of the finite-dimensional saddle-point problem also correspond to all probability distributions where these non-linear variations vanish. Furthermore, the analysis provides results on some qualitative phase behaviour of the model.

math.AP↗

arXiv2016

Maximum entropy methods as the bridge between macroscopic and microscopic theory

This paper investigates a function of macroscopic variables known as the singular potential, building on previous work by Ball and Majumdar. The singular potential is a function of the admissible statistical averages of probability distributions on a state space, defined so that it corresponds to the maximum possible entropy given known observed statistical averages, although non-classical entropy-like objective functions will also be considered. First the set of admissible moments must be established, and under the conditions presented in this work the set is open, bounded and convex allowing a description in terms of supporting hyperplanes, which provides estimates on the development of singularities for related probability distributions. Under appropriate conditions it is shown that the singular potential is strictly convex, as differentiable as the microscopic entropy and blows up uniformly as the macroscopic variable tends to the boundary of the set of admissible moments. Applications of the singular potential are then discussed, and particular consideration will be given to certain free-energy functionals typical in mean-field theory, demonstrating an equivalence between certain microscopic and macroscopic free-energy functionals. This allows statements about L^1-local minimisers of Onsager's free energy to be obtained which cannot be given by two-sided variations, and overcomes the need to ensure local minimisers are bounded away from zero and infinity before taking bounded variations. The analysis also permits the definition of a dual order parameter for which Onsager's free energy allows an explicit representation. Also the difficulties in approximating the singular potential by everywhere defined functions, in particular by polynomials, are addressed with examples demonstrating the failure of the Taylor approximation to preserve shape properties of the singular potential.

math.AP↗

arXiv2016

Non-Gaussian chain statistics and finite extensibility in liquid crystal elastomers

In this work we will derive an anisotropic generalisation of the finitely extensible chain model, due to Kuhn and Grün, which is well known in rubber elasticity. This provides a chain energy that couples elastic behaviour to a probability distribution describing the orientations of liquid crystal monomers within a main chain elastomer. The key point is to invoke a maximum relative entropy assumption on the distribution of bond angles in an observed chain. The chain energy's fourth order Taylor expansion is also given, which couples to the second and fourth moments of the nematic distribution function only.

cond-mat.soft↗