Rigorously computing symmetric stationary states of the Ohta-Kawasaki problem in three dimensions

In this paper we develop a symmetry preserving method for the rigorous computation of stationary states of the Ohta-Kawasaki partial differential equation in three space dimensions. By preserving the relevant symmetries we achieve an enormous reduction in computational cost. This makes it feasible to construct computer-assisted proofs of complex three-dimensional structures. In particular, we provide the first existence proofs for both the double gyroid and body centered cubic packed sphere solutions to this problem

Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model

We present a computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a fourth-order parabolic partial differential equation subject to homogeneous Neumann boundary conditions, which contains as a special case the celebrated Cahn-Hilliard equation. While the attractor structure of the latter model is completely understood for one-dimensional domains, the diblock copolymer extension exhibits considerably richer long-term dynamical behavior, which includes a high level of multistability. In this paper, we establish the existence of certain heteroclinic connections between the homogeneous equilibrium state, which represents a perfect copolymer mixture, and all local and global energy minimizers. In this way, we show that not every solution originating near the homogeneous state will converge to the global energy minimizer, but rather is trapped by a stable state with higher energy. This phenomenon can not be observed in the one-dimensional Cahn-Hillard equation, where generic solutions are attracted by a global minimizer

Towards computational Morse-Floer homology: forcing results for connecting orbits by computing relative indices of critical points

To make progress towards better computability of Morse-Floer homology, and thus enhance the applicability of Floer theory, it is essential to have tools to determine the relative index of equilibria. Since even the existence of nontrivial stationary points is often difficult to accomplish, extracting their index information is usually out of reach. In this paper we establish a computer-assisted proof approach to determining relative indices of stationary states. We introduce the general framework and then focus on three example problems described by partial differential equations to show how these ideas work in practice. Based on a rigorous implementation, with accompanying code made available, we determine the relative indices of many stationary points. Moreover, we show how forcing results can be then used to prove theorems about connecting orbits and traveling waves in partial differential equations.Comment: 30 pages, 4 figures. Revised accepted versio

Stability of Lamellar Configurations in a Nonlocal Sharp Interface Model

Equilibrium models based on a free energy functional deserve special interest in recent investigations, as their critical points exhibit various pattern structures. These systems are characterized by the presence of coexisting phases, whose distribution results from the competition between short and long-range interactions. This article deals with an energy-driven sharp interface model with long-range interaction being governed by a screened Coulomb kernel. We investigate a number of criteria for the stability of lamellar configurations, as they are indeed strict local minimizers. We also give a sufficient condition to ensure a nontrivial periodic 2D minimal energy configuration

Theory of domain patterns in systems with long-range interactions of Coulomb type

We develop a theory of the domain patterns in systems with competing short-range attractive interactions and long range repulsive Coulomb interactions. We take an energetic approach, in which patterns are considered as critical points of a mean-field free energy functional. Close to the microphase separation transition, this functional takes on a universal form, allowing to treat a number of diverse physical situations within a unified framework. We use asymptotic analysis to study domain patterns with sharp interfaces. We derived an interfacial representation of the pattern's free energy which remains valid in the fluctuating system, with a suitable renormalization of the Coulomb interaction's coupling constant. We also derived integrodifferential equations describing the stationary domain patterns of arbitrary shapes and their thermodynamic stability, coming from the first and second variation of the interfacial free energy. We showed that the length scale of a stable domain pattern must obey a certain scaling law with the strength of the Coulomb interaction. We analyzed existence and stability of localized (spots, stripes, annuli) and periodic (lamellar, hexagonal) patterns in two dimensions. We showed that these patterns are metastable in certain ranges of the parameters and that they can undergo morphological instabilities leading to the formation of more complex patterns. We discuss nucleation of the domain patterns by thermal fluctuations and pattern formation scenarios for various thermal quenches. We argue that self-induced disorder is an intrinsic property of the domain patterns in the systems under consideration.Comment: 59 pages (RevTeX 4), 9 figures (postscript), to be published in the Phys. Rev.

Systems of Points with Coulomb Interactions

Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and give rise to a variety of questions pertaining to calculus of variations, Partial Differential Equations and probability. We will review these as well as "the mean-field limit" results that allow to derive effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order beyond the mean-field limit, giving information on the system at the microscopic level. In the setting of statistical mechanics, this allows for instance to observe the effect of the temperature and to connect with crystallization questions.Comment: 30 pages, to appear as Proceedings of the ICM201

Complex-to-Real Mapping for Polymer Field Theories

Polymer field theory is a valuable tool for studying the behavior of dense polymer systems at near-atomistic length scales. In its most successful form, it is a complex-valued, d+1–dimensional theory that can be studied using fully-fluctuating simulations or at the level of self-consistent field theory (SCFT). This theory requires calculations using fields that depend on d spatial variables as well as a contour variable indicating the position along the polymer chain. In principle, it can be reduced to a simpler, real-valued, d-dimensional theory that is more efficient to simulate numerically—a phase field model—but these are typically less accurate due to approximations invoked in their derivations. We introduce and refine a new method for constructing phase field models, phase field mapping, that systematically parametrizes an optimized phase field (OPF) model using the output of inexpensive SCFT simulations. We develop an OPF model for the diblock copolymer melt and characterize its performance in terms of speed, accuracy, and transferability. Then we modify and generalize the model to produce a weakly compressible model suitable for running simulations in confined templates. In bulk and in confinement, the OPF model is faster to simulate than SCFT and more accurate than other phase field models. With these advantages, the OPF model is a useful alternative to complex-valued field theories