# Abstracts

Title: *Modelling Topological Objects in Liquid Crystals*

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Title: *Recent results on phase diagrams of periodic systems using Nested Sampling*

Abstract: I will describe our recent efforts to compute pressure-temperature phase diagrams of materials using Nested Sampling. The unique nature of this approach is that we estimate the partition function as a function of temperature and pressure directly, and subsequently deriving the heat capacity from it, and thus need no a priori knowledge of the phases. Examples include hard spheres, various Lennard-Jones models and an Embedded Atom model of aluminium.

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Title: *Metastability for continuum interacting particle systems*

Abstract: We consider a system of point particles in a finite box in R2 that interact via a finite-range attractive pair potential, and move according to a Markov process that has the grand canonical Gibbs measure as a reversible measure. The chemical potential is such that the system favors a packed box, but has a nucleation barrier to overcome in order to go from an empty box to a filled box. We are interested in the nucleation time in the limit as the temperature tends to zero. We use the potential-theoretic approach to metastability. The results extend earlier work for lattice systems; the main difficulty lies in understanding the energy landscape of the continuum particle system, a problem of intrinsic interest in analysis, This talk reports on joint work in progress with Frank den Hollander.

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Title: *Cluster size distribution in classical many-body systems with Lennard-Jones potential*

Abstract: We study a classical many-body system with pair-interaction given by a stable Lennard-Jones potential. This interaction has an attractive term, which induces the formation of clusters of the particles. For fixed inverse temperature $\beta\in(0,\infty)$ and fixed particle density $\rho\in(0,\infty)$, we derive a large-deviation principle for the distribution of the cluster sizes in the thermodynamic limit. Afterwards, we show that the rate function Gamma-converges, in the low-temperature dilute limit $\beta\to\infty$ and $\rho\downarrow 0$ such that $-\beta^{-1}\log\rho\to\nu\in(0,\infty)$, towards some explicit rate function. This function has precisely one minimising cluster size configuration, which implies a law of large numbers for the cluster sizes in this decoupled limit. This is joint work with S. Jansen and B. Metzger. The limiting rate function appeared in earlier work with A. Collevecchio, P. Mörters and N. Sidorova, when the two limits (the thermodynamic and low-temperature dilute limit) were coupled with each other.

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Title: *From gradient Gibbs measure to nonlinear elasticity*

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Title:* A micro/macro parareal algorithm for a class of multiscale-in-time systems *

Abstract: The question of coarse-graining is ubiquitous in materials science. It often amounts to going from a high-dimensional description of a system, with degrees of freedom which possibly evolve on different characteristic times, to a low dimensional description, only retaining the "slow" degrees of freedom. Think e.g. of molecular simulation (where coarse-grained models are written in terms of low-dimensional reaction coordinates) or polymeric fluids (where coarse-grained models are used to circumvent or simplify the simulation of the evolving microstructure).

We introduce and analyze a micro/macro parareal algorithm for the time-parallel integration of this type of problems. The algorithm first computes a cheap macroscopic solution using a propagator of the coarse-grained model. This solution is iteratively corrected by using a propagator of the fine-scale model, in the parareal algorithm spirit. The iterative procedure converges to the solution of the fine-scale model.

The efficiency of the approach is demonstrated on the basis of numerical analysis arguments and numerical experiments on a simple class of problems.

Joint work with T. Lelievre and G. Samaey.

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Title: *Accurate and efficient molecular sampling with very large timestep*

Abstract: Molecular dynamics is a popular method for exploring the molecular conformations accessible at given conditions of state. It is widely used for applications in Chemistry, Physics, Biology and Engineering, with the number of atoms ranging from a few dozen to a few billion. In order to ensure sampling of the canonical ensemble some sort of thermal regulation device is needed, such as the Nose-Hoover thermostat or Langevin dynamics. The latter uses stochastic dynamics to provide a complete (ergodic) sampling of the distribution. The great challenge in using molecular dynamics to study materials at relevant scales lies in the need for small timesteps to stably integrate high frequency modes due to stiff components such as bond stretches and dipole oscillations.

In this talk I will discuss research in two directions on the design of integrators for molecular sampling. In the first (with C. Matthews, and lately with the help of G. Stoltz) we have been studying the accuracy of Langevin dynamics splitting methods, demonstrating that the choice of ordering of the steps of the algorithm can have profound effect on the attainable resolution of configurational averages. The ideal method allows simulation of a solvated, flexible biomolecule to be performed with stepsizes up to very near 3fs which is the upper limit for schemes that accurately resolve all dynamical modes.

The second direction is joint work with D. Margul and M. Tuckerman (NYU) where we have developed new stochastic-dynamical methods that incorporate isokinetic constraints in order to tame resonance-induced instabilities in multiple timestepping. These methods enable stable ergodic simulation to be performed with 'outer' timesteps of up to around 100fs in biomolecular simulation.

References

B. Leimkuhler and C. Matthews, AMRX, 2013, DOI:10.1093/amrx/abs010

B. Leimkuhler and C. Matthews, J. Chem. Phys., 2013, DOI:10.1063/1.4802990

B. Leimkuhler, C. Matthews and G. Stoltz, 2013, arXiv:1308.5814

B. Leimkuhler, D. Margul and M. Tuckerman, Mol. Phys., 2013, DOI:10.1080/00268976.2013.844369

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Title: *Mathematical analysis of accelerated dynamics*

Abstract: We will present some mathematical analysis of accelerated dynamics techniques which have been proposed by A.F. Voter in the late nineties: the parallel replica method, the hyperdynamics and the temperature accelerated

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Title: *Hyper-QC: A method to coarse-grain space and accelerate time*

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Title: *Hybrid models for liquid crystals and their applications*

Abstract: This talk focuses on the development, analysis and numerical implementation of mathematical models for a planar bistable nematic device reported in a paper by Tsakonas, Davidson, Brown and Mottram. We model this device within a continuum Landau-de Gennes framework and investigate the cases of strong and weak anchoring separately. In both cases, we find six distinct states and compute bifurcation diagrams as a function of the anchoring strength. We introduce the concept of an optimal boundary condition that prescribes the optimal interpolation between defects at the vertices. We discover a novel two-dimensional biaxial order reconstruction pattern connecting the vertices, as the device width becomes comparable to the biaxial correlation length. We develop a parallel lattice-based Landau-de Gennes interaction potential, by analogy with the Lebwohl-Lasher lattice-based model and study multistability within this discrete framework too by means of Monte Carlo methods. The different numerical approaches are compared and we conclude with a brief discussion on a multiscale modelling approach wherein a lattice-based interaction potential is coupled to a conventional continuum model.

This is joint work with Samo Kralj, Chong Luo and Radek Erban.

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Title: *Computational coarse-graining in configuration space as basis for a predictive ab initio thermodynamics*

Abstract: Blazej Grabowski, F. Körmann, Tilmann Hickel and Jörg Neugebauer Max-Planck-Institut für Eisenforschung, Düsseldorf, Germany

The combination of accurate first principles calculations with thermodynamic and/or kinetic concepts opens the door to tackle even advanced engineering problems such as the design novel alloys with superior mechanical properties. Key to these studies is the highly accurate determination of thermodynamic quantities at finite temperatures. In the first part of the talk it will be shown how novel sampling strategies allow to obtain a highly efficient coarse graining in configuration space resulting in a reduction from 107 to a few hundred configurations. This enormous reduction permits to employ highly converged density-functional theory calculations thus providing the basis for accurately determining all relevant temperature dependent free energy contributions such as harmonic and anharmonic vibrations, magnetic excitations or defect creation. The flexibility and the predictive power of this approach will be discussed in the second part of the talk for a few examples related to the design and understanding of novel structural materials.

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Title:* The Effects of polydispersity in liquid crystal systems*

Abstract: Two distinct approaches are typically used to describe nematic liquid crsystals: Maier Saupe theory, based on anisotropic London dispersion between molecules, and Onsager theory, based on steric repulsion of the constituent units. Orientational order depends primarily on temperature in the former, and on concentration in the latter. The effects of particle polydispersity have been considered in the context of both of these approaches. In many liquid crystal systems, both of attractive and repulsive interactions play an important role. Using a density functional approach which includes contributions from both attractive dispersive and repulsive steric interactions consistently, we examine the effects of polydispersity in nematic liquid crystals. We solve the self-consistent equation for the density, examine the stability of the solutions, and study the phase behavior as function of temperature and density.

Peter Palffy-Muhoray [1] and Xiaoyu Zheng [2]

[1] Liquid Crystal Institute, Kent State University

[2] Dept. of Mathematical Sciences, Kent State University

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Title:* Information-theoretic tools for coarse-graining of non-equilibrium extended systems*

Abstract: We present information-theoretic approach to parameterisation and error control of coarse-grained dynamics in molecular simulations. Rates of the coarse-grained process are parametrized and optimal parameters are selected by minimization of the relative entropy on the path space. This approach extends techniques also known as inverse Monte Carlo to models with non-equilibrium stationary states, for example systems driven by external parameters or reaction-diffusion systems in catalysis. The use of relative entropy also allows for error estimation and uncertainty quantification of the coarse-grained models

(joint work with M.A. Katsoulakis).

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Title: *Diagnostics and extensions for parallel replica dynamics*

Abstract: Parallel Replica Dynamics was introduced by A.F. Voter as a method for accelerating the numerical integration of molecular dynamics by exploiting parallel computation. The gain in performance is achieved by a suitable coarse graining, in time, of the trajectory. I will present the algorithm and discuss recent efforts to use Gelman - Rubin type statistics for convergence diagnostics, along with extensions of the algorithm to entropic problems. In addition, the algorithm can be extended to discrete in time processes, which permits a correction for time step discretization effects.

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Title: *On stability of the melting hedgehog solution in Landau-de Gennes model of liquid crystals*

Abstract: We study the stability of a radially symmetric solution, known as the “melting hedgehog”, in the framework of the Landau-de Gennes model of liquid crystals. We show local stability of melting hedgehog for small enough reduced temperature.

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Title: *The phase picture for a $q$-type Widom--Rowlinson model*

Abstract: The WR model with $q$ types is considered where particles of different types are repelling each other with given hard-core exclusion diameters. In the case of equal and large fugacities, for $q\leq 4$ we have a complete (rigorous) picture of pure phases. For larger values of $q$, there are conjectures which may require numerical simulations.

This is a joint work with A. Mazel and I Stuhl.

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Speaker: Dimitrios TSAGKAROGIANNIS

Title: *Estimation of coarse-grained potentials for polymeric systems*

Abstract: The main question we address is how to construct a ''good" coarse-grained approximation for polymeric systems. In recent years several methods have been proposed obtaining a good comparison of the atomistic and the suggested coarse-grained pair correlation functions via empirical/statistical adjustment of the coarse-grained parameters. Our intention is to propose a systematic way of constructing coarse-grained potentials in such a way that one also retains the correct thermodynamic character of the atomistic system. For that we use cluster expansion techniques in different temperature and density regimes and provide comparisons at the level of the pair correlation function with respect to the atomistic system. As a starting point we consider the case of methane in which the chains consist of only one atom of carbon and four atoms of hydrogen and use as a coarse-grained map the centre of mass of the atoms.

This is work in progress, jointly with V. Harmandaris and A. Tsourtis.

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Title: *Recent advances in hyperdynamics*

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Title: *Using multiscale preconditioning to accelerate the convergence of iterative molecular calculations*

Abstract: Many tasks in computational chemistry can be cast in terms of a repeated iterative procedure. Whether the goal is to find an optimized molecular geometry or to refine a reaction path, well developed computational tech- niques exist that are expected to progressively converge to a satisfactory solution when initiated from a reason- able starting guess. Nevertheless, the number of steps necessary for convergence of such calculations can rapidly become computationally prohibitive as the com- plexity of models increases to capture details of molecu- lar systems realistically. Most researchers are then faced with a painful choice: either represent the system with a model that is, in principle, capable of faithfully represent- ing the essential features of the system of interest but is too costly to be practical, or revert to a computationally inexpensive but more approximate model. I will describe an alternative, non-linear preconditioning, approach in which an inexpensive but possibly inaccurate, "coarse-grained (CG)" model is used to accelerate convergence to the solution corresponding to an accurate but possibly expensive "fine-grained (FG)" model. Care is taken to ensure that the approach preserves stability of the original FG solution, and is ro- bust to different choices of the CG model. The approach will be demonstrated in the contexts of simple geometry optimization and reaction path finding problems.

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Title: *Dynamic statistical scaling in Landau-de Gennes theory*

Abstract: In this talk, we consider the behaviour of thermotropic nematic liquid crystals at the onset of the isotropic-nematic phase transition. There is evidence in the physics literature that those regions where the material is in the nematic phase invade the ambient isotropic phase in a self-similar manner as time progresses. We perform a rigorous investigation of this phenomenon. The model of the isotropic-nematic phase transition we employ is the $L^{2}(\mathbb{R}^{3})$-gradient flow of the well-known Landau-de Gennes energy. By studying measures which evolve under the gradient flow dynamics, we prove for suitable initial data that the structure of solutions is self-similar {\em in an average sense} as time $t\rightarrow \infty$.

This is joint work with Eduard Kirr and Arghir Zarnescu.

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Title: *Computer simulation of liquid crystals using atomistic and coarse-grained molecular models*

Abstract: This talk describes progress in the use of atomistic and coarse-grained models to study liquid crystalline systems.

At the atomistic level of modelling, results are presented for nematic and smectic phases of thermotropic liquid crystals, and for molecular self-assembly and meosphase formation in chromonic systems.

Design of suitable coarse-grained models for liquid crystals is discussed, including the use of systematic coarse-graining methods to parameterise models for liquid crystalline systems, and the use of generic coarse-grained molecular models for liquid crystals.

Results from the use of advanced simulation methods to enhance phase space sampling in liquid crystal simulations are presented, including the use of statistical temperature molecular dynamics, and hamiltonian replica exchange.

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Title: *Atomistic and coarse grained computer simulations of liquid crystals: descriptions and predictions*

Abstract: Understanding the behaviour of currently known liquid crystals (LC) or designing novel ones with specifically tailored functionalities relies on establishing a robust link between microscopic features and observable properties. The properties of interest range from detailed single molecules properties (e.g. order parameters, NMR dipolar couplings) to molecular organizations in the various mesophases, macroscopic properties (e.g. transition temperatures) and topological defects. Corresponding to this range properties and their different length scales, the problem of modelling and simulating liquid crystals (e.g. using Monte Carlo and Molecular Dynamics) [1,2] has also to be tackled at different resolutions, deploying atomistic, or coarser grain models. In this talk we plan to show some recent examples of computer simulation applications to various liquid crystals, involving e.g. actuation in swollen LC elastomers [3]. nanodroplets [4] predictive atomistic simulations of properties like transition temperatures [5], anchoring and molecular organization at interfaces [6,7].

1. See, e.g.,D. Frenkel and B. Smit, Understanding Molecular Simulations. From Algorithms to Applications (Academic Press, San Diego, 2002)

2. P. Pasini and C. Zannoni, eds., Advances in the Computer Simulations of Liquid Crystals (Kluwer, Dordrecht, 2000).

3. G. Skacej, C. Zannoni, Molecular simulations elucidate electric field actuation in swollen liquid crystal elastomers, PNAS, 109, 10193–10198 (2012)

4. D. Vanzo, M. Ricci, R. Berardi, C. Zannoni, Shape, chirality and internal order of freely suspended nematic nanodroplets, Soft Matter, 8, 11790-11800 (2012)

5. G. Tiberio, L. Muccioli, R. Berardi and C. Zannoni, Towards in silico liquid crystals. Realistic transition temperatures and physical properties for n-cyanobiphenyls via molecular dynamics simulations, ChemPhysChem 10, 125 (2009)

6. A. Pizzirusso, L. Muccioli , M. Ricci and C. Zannoni, Predicting surface anchoring: molecular organization across a thin film of 5CB liquid crystal on silicon, Chem. Sci., 3, 573 (2012)

7. O. M. Roscioni, L. Muccioli, R. G. Della Valle, A. Pizzirusso, M. Ricci, C. Zannoni, Predicting the anchoring of liquid crystals at a solid surface: 5-cyanobiphenyl on cristobalite and glassy silica surfaces of increasing roughness, Langmuir, 29, 8950-8958 (2013)

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Title: *The interplay between temperature and statistical constraints in Q-tensor models of nematic liquid crystals*

Abstract: Obtaining the Q-tensor description of nematics out of statistical mechanics considerations imposes certain constraints on the size of eigenvalues for the Q-tensors. One can model these constraints using a singular potential that blows up when the Q-tensor does not satisfy these constraints. Depending on how the temperature is associated with this singular potential one obtains different types of equations, whose analysis poses different types of challenges. We will discuss these issues on two specific models.

This is joint work with E. Fereisl, E.Rocca and G. Schimperna.