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Dynamic Statistical Scaling in the Landau–de Gennes Theory of Nematic Liquid Crystals

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Abstract

In this article, we investigate the long time behaviour of a correlation function \(c_{\mu _{0}}\) which is associated with a nematic liquid crystal system that is undergoing an isotropic-nematic phase transition. Within the setting of Landau–de Gennes theory, we confirm a hypothesis in the condensed matter physics literature on the average self-similar behaviour of this correlation function in the asymptotic regime at time infinity, namely

$$\begin{aligned} \left\| c_{\mu _{0}}(r, t)-e^{-\frac{|r|^{2}}{8t}}\right\| _{L^{\infty }(\mathbb {R}^{3}, \,dr)}=\mathcal {O}(t^{-\frac{1}{2}}) \quad \mathrm as \quad t\longrightarrow \infty . \end{aligned}$$

In the final sections, we also pass comment on other scaling regimes of the correlation function.

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Notes

  1. For the definition and basic properties of \(L^{p}\)-spaces, we advise the reader to consult Chapter 2 of Lieb and Loss [14].

  2. We write \(m_{k}(x)^{k}\) as shorthand for \(m_{k}(x, \ldots , x)\) with the argument \(x\) repeated \(k\) times.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, Champaign, IL, USA

    Eduard Kirr

  2. Département de Mathématiques et Applications, École Normale Supérieure, Paris, France

    Mark Wilkinson

  3. Department of Mathematics, University of Sussex, Brighton, UK

    Arghir Zarnescu

Authors
  1. Eduard Kirr
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  2. Mark Wilkinson
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  3. Arghir Zarnescu
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Correspondence to Mark Wilkinson.

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Kirr, E., Wilkinson, M. & Zarnescu, A. Dynamic Statistical Scaling in the Landau–de Gennes Theory of Nematic Liquid Crystals. J Stat Phys 155, 625–657 (2014). https://doi.org/10.1007/s10955-014-0970-6

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