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# Fluctuating Nematodynamics using the Stochastic Method of Lines 

A. K. Bhattacharjee, Gautam I. Menon, and R. Adhikari 

The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600013, India 

(Dated: November 16, 2018) 

We construct Langevin equations describing the fluctuations of the tensor order parameter Q$_{αβ}$ in nematic liquid crystals by adding noise terms to time-dependent variational equations that follow from the Ginzburg-Landau-de Gennes free energy. The noise is required to preserve the symmetry and tracelessness of the tensor order parameter and must satisfy a fluctuation-dissipation relation at thermal equilibrium. We construct a noise with these properties in a basis of symmetric traceless matrices and show that the Langevin equations can be solved numerically in this basis using a stochastic version of the method of lines. The numerical method is validated by comparing equi- librium probability distributions, structure factors and dynamic correlations obtained from these numerical solutions with analytic predictions. We demonstrate excellent agreement between numer- ics and theory. This methodology can be applied to the study of phenomena where fluctuations in both the magnitude and direction of nematic order are important, as for instance in the nematic swarms which produce enhanced opalescence near the isotropic-nematic transition or the problem of nucleation of the nematic from the isotropic phase. 

Fluctuation phenomena in nematic liquid crystals are typically studied within Ericksen-Leslie theory, which as- sumes that the orientation of the normalized nematic di- rector is the only fluctuating variable[1]. This approxi- mation is adequate deep within the nematic phase, where the strength of nematic order is not significantly affected by thermal fluctuations. However, in the vicinity of the weakly first-order isotropic-nematic transition, sig- nificant fluctuations in the nematic order are observed, suggesting that the phase-only approximation embodied in Leslie-Ericksen theory is inadequate [2]. The study of nucleation in quenches from the isotropic to the nematic phase involves the growth of one phase within another, mandating the use of descriptions capable of describing both isotropic and nematic phases on the same foot- ing. In these and similar situations, a tensorial descrip- tion of nematic order which uses the symmetric, trace- less quadrupole moment tensor Q$_{αβ}$ , is appropriate, as first clarified by de Gennes in his Ginzburg-Landau the- ory of the isotropic-nematic transition[3]. The Ginzburg- Landau-de Gennes (GLdG) approach provides a simple, but accurate phenomenological description of nematic fluctuations in the static case[4]. 

The description of the fluctuating dynamics of the ori- entation tensor within GLdG theory has received consid- erably less attention[5, 6]. Understanding the results of inelastic scattering experiments on nematic systems[7], the description of the rate of nucleation into the ne- matic phase[8], the modeling of nontrivial stresses aris- ing from Casimir interactions[9] and the calculation of the spectrum of capillary waves on the isotropic-nematic interface[10] are all problems which require a dynami- cal theory of fluctuations in the orientation tensor. This problem is addressed in this paper, in which we present and solve the Langevin equations for dynamical fluctu- 

PACS numbers: 05.10.Gg, 02.50.Ey, 02.60.Cb, 64.70.mf 

## I. INTRODUCTION 

ations at equilibrium for the nematic orientation ten- sor. These are stochastic non-linear partial differential equations for the five components of the orientation ten- sor. Analytical solutions can be obtained when these equations are linearized. For the solution of the gen- eral non-linear equations, we propose an efficient numer- ical method, based on a stochastic generalization of the method of lines. We compare our results with analytic results where such calculations are possible, finding ex- cellent agreement. 

## II. FLUCTUATING NEMATODYNAMICS 

Orientational order in the nematic phase is described by a second-rank, symmetric traceless tensor Q$_{αβ}$ ( x , t ). This is the second moment of the microscopic orienta- tional distribution function. The tensor can be expanded as 

<img src='content_image/380033.jpg'>

The three principal axes of this tensor, obtained by di- agonalizing Q$_{αβ}$ in a local frame, specify the direction of nematic ordering n , the codirector l and the joint nor- mal to these, labeled by m . The principal values S and T represent the strength of ordering in the direction of n and m , quantifying, respectively, the degree of uniaxial and biaxial nematic order. 

The static fluctuations of Q$_{αβ}$ can be calculated from a Ginzburg-Landau functional, first proposed by de Gennes, based on an expansion in rotationally invariant combinations of Q$_{αβ}$ and its gradients. The Ginzburg- Landau-de Gennes functional F is 

<img src='content_image/380034.jpg'>

