5 FIG. 4: (Color online) Autocorrelation C ij ( q , τ ) for the linear Langevin equation, calculated for small wavenumbers. Nu- merical parameters are k B T = A = 0 . 05 , L 1 = 0 . 5 , Γ = 1 . 0. The integration is performed for 4 . 2 × 10 3 time steps on a 16 2 grid. The time average is taken over 4 × 10 3 time steps and an ensemble average is taken over 40 independent realizations. The relaxation time scale is τ = (Γ A ) $_{−}$1 = 20, the diffusion time scale of the shortest Fourier mode τ$_{d}$ = L 2 $_{x}$/ (4 π $^{2}$L 1 Γ) ∼ 13 and the correlation length λ = √ L 1 /A = 3 . 16. tor C ( q ) = ∑ | q | = $_{q}$C$_{ij}$ ( q 〉 for comparison with the nu- merical simulation. We now compare theoretical and numerical results: In Fig. (2) we show the histogram of the a$_{i}$ ( q ) for a par- ticular Fourier mode. This is normally distributed, as expected, with zero mean and variance as required by thermal equilibrium. Similarly, all Fourier modes exam- ined have correct normal distributions. The variances obtained are compared in Fig. 3(a) with the analytical values by plotting contours of C$_{ij}$ ( q ). There is excel- lent agreement. A close inspection reveals some degree of anisotropy in both the analytical and numerical results at high wavenumbers. This is attributed to the lack of isotropy of the nearest-neighbour finite-difference Lapla- cian mentioned earlier. However, the anisotropies are removed upon angular averaging, as shown in Fig. 3(b). Thus, the present discretisation should be adequate in most cases, unless highly accurate isotropies are required from the simulation. From these results, we conclude that correlations in thermal equilibrium are accurately captured by the stochastic method of lines approach. We next compare the dynamics of fluctuations at equi- librium, by comparing two-point autocorrelation func- tions calculated analytically and numerically. Fig.(4) shows C$_{ij}$ ( q , τ ) for three sets of Fourier modes. The ex- ponential decay of the autocorrelation function is repro- duced accurately within the numerics and fit the theo- retical curve very closely. We conclude, therefore, that the stochastic method of lines accurately reproduces both static and dynamic fluctuations in a harmonic theory. Finally, we compare theory and simulation in a situ- ation where a linearization of the Q$_{αβ}$ equations about Q$_{αβ}$ = 0 is inapplicable, that of director fluctuations within the nematic phase. In the T basis, the equations of motion are where, Q 2 αβ is the traceless symmetric projection of Q 2 $_{αβ}$.These equations of motion are anharmonic, and the analytical solutions obtained earlier within the harmonic expansion are no longer available for comparison. We therefore extract the fluctuations of the angular displace- ments from the uniform nematic ground state using an approach based on the Frank free energy. Consider an uniform uniaxial nematic with director n 0 with small fluctuations δ n ( x ). Decomposing the fluctu- ations into parts parallel and perpendicular to n 0 and imposing the normalization of the director, we find from the Frank free energy that, where K = (9 S $^{2}$/ 2) L 1 is a Frank constant [4]. Since fluc- tuations in the plane perpendicular to n 0 can be charac- terized through a single angle θ , an equivalent result is 〈| δ n$_{⊥}$ ( q ) | $^{2}$〉 = k$_{B}$T Kq $_{2}$, (22) ∂$_{t}$a$_{i}$ = − Γ [( A + CTrQ $^{2}$) a$_{i}$ (21) + ( B + 6 E $^{′}$TrQ $^{3}$) T i αβ Q 2 αβ − L 1 ∇ $^{2}$a$_{i}$ ] + ξ$_{i}$. FIG. 5: (Color online) Angularly averaged structure factor of the nematic phase. The numerical parameters are k B T = 0 . 05 , A = − 3 . 5 , B = − 10 k B T, C = 2 . 67 , L 1 = 32 . 0 , Γ = 0 . 01 on a 64 2 grid. The relaxation time scale is τ = (Γ A ) $_{−}$1 = 28.57, the diffusion time scale of the shortest Fourier mode is τ$_{d}$ = L 2 $_{x}$/ (4 π $^{2}$L 1 Γ) = 324 . 23 and the correlation length is λ = √ L 1 /A = 3 . 02. The time average is taken over 5 × 10 4 time steps and an ensemble average is taken over 20 independent realizations. 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 < a i ( q , t) a i (− q , t+ ) > q$_{x}$ = q$_{y}$ = 2  Analytic q$_{x}$ = q$_{y}$ = 3 q$_{x}$ = 2, q$_{y}$ = 3 0 1 2 3 4 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 −3 q C  (q) Numeric Analytic