H2020Индивидуална стипендия2019–2021

SELEs · Stochastic Ericksen-Leslie Equations

„Хоризонт 2020“ — Действия „Мария Склодовска-Кюри“

Период: 2019-01-07 → 2021-01-07
Финансиране от ЕС: 195 455 €
Участници: 1
Схема: MSCA-IF

Линиите свързват координатора с партньорите.

Накратко на български

Нематичните течни кристали се изследват чрез математически модели, за да се разбере как шумът и скоростта на течността влияят върху пренареждането на молекулите им. Това помага за по-доброто разбиране на процесите, използвани при производството на LCD екрани.

Този кратък обзор е генериран от изкуствен интелект

Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.

Резултати накратко

Stochastic Ericksen-Leslie Equations

Molecules in nematic liquid crystals (NLC) are long and thin and have no positional order (like a fluid), but they tend to align along a direction called the optical director or director. This director can be easily distorted and aligned to form a specific pattern using an external control with intensity above a certain threshold value. This passage from one stable to another stable state, possibly with higher energy, caused by an external force or control is called the Fréedericksz transition and it plays an important role in many branches of applied sciences and the industry of LCDs. Although the effect of noise on the dynamics of the optical director has been the subject of numerous theoretical and experimental studies, there are still many questions that remain unsolved. For instance, in previous studies in physics the fluid velocity is assumed to be negligible, hence understanding of the simultaneous effects of the noise and the fluid velocity on the Fréedericksz transition is an open problem which is also challenging. One should notice that De Gennes and Prost pointed out that the fluid velocity plays an important role in the dynamics of the optical director. Recently, several mathematical papers such as the pioneering work of Hong, and Lin, Lin and Wang have taken into account the effects of the fluid velocity, but their equations contain neither deterministic nor stochastic external forces. It is the desire to solve these unchallenged problems that formed the impetus behind the present research project. The main objective of the project was to o give a sound mathematical description of the noise-induced Fréedericksz transition in Nematic Liquid Crystal (NLC) with general geometric configurations. In particular, we aimed to solve some important and difficult open mathematical problems related to the stochastic partial differential equations (SPDEs) describing the dynamics of noise driven NLC, and give a rigorous mathematical proof of the noise-induced Fréedericksz transition in NLC.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз

Цел на проекта

The objective of the research proposed in this project is to give a sound mathematical description of the noise-induced Fréedericksz transition in Nematic Liquid Crystal (NLC) with general geometry configurations. To this aim we will: 1) solve some important and difficult open mathematical problems related to the stochastic stochasic Ericksen-Leslie Equations (SELEs) which basically describe the dynamics of liquid crystals with stochastic perturbations, and 2) give a rigorous mathematical proof of the noise-induced Fréedericksz transition in NLC. In particular, we will establish the existence and uniqueness solution of the Ginzburg-Landau (GL) approximation of SELEs. By using Large Deviations Principle (LDP) theory and the de Giorgi Gamma-convergence we will prove that the action functional of the SELEs with small spatially converges to the action functional of the SELEs with spatially white noise. We will rigorously justify the probabilistic interpretation of the results in terms of the asymptotics of the mean exit time from a neighbourhood of an attracting stationary solution, a hint to noise-induced Fréedericksz transition. By using again LDP theory will rigorously show that in the presence of small noise there is a positive probability of transition between the attraction domains of the stationary solutions for the deterministic system; this is a rigorous mathematical proof of the noise-induced Freédricks’s transition. We will also prove the existence and uniqueness of an invariant measure which satisfies a LDP. The latter result confirms that in the long run the noise still induces transition between equilibria. Finally, we aim to prove the existence and uniqueness of solution of the SELEs, and if time permits transfer results obtained for the GL approximation of SELES to the original SELEs.

Оригинален текст от CORDIS (на английски).

Участници

UNIVERSITY OF YORK · YORK NORTH YORKSHIREКоординаторОбединеното кралство