4 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0 50 100 150 200 250 300 350 400 450 a$_{1}$ ( q ) Frequency n$_{x}$ n y −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15 Numeric Analytic (a) 0 0.5 1 1.5 2 2.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 q C(q) Numeric Analytic (b) FIG. 2: (Color online) Histogram of the real part of a i ( q ) for n x = n y = 6 in a box of dimension L x = L y = 16, using a harmonic free energy, with k B T = A = 0 . 05 and L 1 = 0 . 5, Γ = 1 . 0. The fluctuations are Gaussian with the expected variance. The histogram is obtained from 20 independent realizations, each realization contributing 4000 time steps. FIG. 3: (Color online) (a) Contour plot of the structure factor C ij ( q ) and (b) angular average of C ij ( q ). The parameters are k B T = A = 0 . 05 , L 1 = 0 . 5 , Γ = 1 . 0. The time averaging is over 5 × 10 4 time steps and ensemble averaging is over 20 independent realizations in a box with L x = L y = 64. The relaxation time scale is τ = (Γ A ) $_{−}$1 = 20, the diffusion time scale of the smallest Fourier mode is τ$_{d}$ = L 2 $_{x}$/ (4 π $^{2}$L 1 Γ) = 207 . 51 and the correlation length is λ = √ L 1 /A = 3 . 16. accurately describes fluctuations about the isotropic phase. We then consider expansions about the ordered state, comparing static correlations obtained analytically within the Frank approximation with our numerical re- sults. We use a finite-difference discretisation with nearest- neighbour stencils for gradients and the Laplacian. We implement periodic boundary conditions. Specifically, in three dimensions, we consider a box of dimension L$_{x}$, L$_{y}$ and L$_{z}$ along the Cartesian directions, and grid these lengths with equal grid spacing Δ x = Δ y = Δ z = 1. The latter defines lattice units for the spatial coordi- nate. We define corresponding discrete time units for the temporal variables by choosing Δ t = 1. Fourier modes are labelled by the wave-vector q = ( q$_{x}$, q$_{y}$, q$_{z}$ ), where each component is of the form q$_{α}$ = 2 πn$_{α}$/L$_{α}$ , with n$_{α}$ = 0 , 1 , 2 , . . . , ( L$_{α}$ − 1). With this discretisation, the Laplacian in Fourier space is given by The nearest-neighbour finite difference stencil suffers from lack of isotropy at high wavenumbers. This can be improved through the use of higher-point stencils [17–19]. Applying the method of lines discretisation to Eq. (3) reduces it to a system of stochastic ordinary differential equations, whose Fourier representation in the harmonic approximation of Eq. (11) is The Fourier representation of the drift-diffusion dynam- ics is encoded in the linear operator D , D = A − L 1 L ( q ) . (18) ∂$_{t}$a$_{i}$ ( q , t ) = − Γ D a$_{i}$ ( q , t ) + ξ$_{i}$ ( q , t ) . (17) L ( q ) = 2[cos( q$_{x}$ ) + cos( q$_{y}$ ) + cos( q$_{z}$ ) − 3] . (16) C$_{ij}$ ( q ) = k$_{B}$T D δ$_{ij}$, (19) C$_{ij}$ ( q , τ ) = C$_{ij}$ ( q ) exp( − Γ D τ ) . (20) Fourier representations of the one and two-dimensional method of lines discretisations are obtained by setting the corresponding wavenumbers to zero. The static and dynamic autocorrelations in Fourier space follow in a straightforward manner though the replacement of q 2 by its discrete Laplacian representation. The results are It is also useful to define an angle-averaged structure fac-