2 Here, A = A 0 (1 − T/T $^{∗}$) T ∗ denoting the supercooling transition temperature, A 0 a constant, L 1 is an elas- tic constant and α, β, γ denote the Cartesian directions. From the inequality 1 $_{6}$( T r Q $^{2}$)3 ≥ ( T r Q $^{3}$)$^{2}$, higher pow- ers of T r Q 3 can be excluded for the description of the uniaxial phase. Uniaxial phases are described by E ′ = 0 while biaxial phases require E ′ = 0. For the nematic phase (rod-like molecules) B < 0 whereas for the dis- cotic phase (plate-like molecules) B > 0. The quantities C and E ′ must always be positive to ensure bounded- ness and stability of the free energy in all phases. We omit other symmetry-allowed gradient terms in this pa- per, thus working in the limit where all three Frank con- stants are assumed to be equal. Such symmetry-allowed terms, as also total derivative surface terms, can be ac- counted for without essential change, using the numerical method described below. In the limit that hydrodynamic interactions may be neglected, i.e. the Rouse or free-draining limit, the dy- namical fluctuations of Q$_{αβ}$ are not coupled to other hy- drodynamic variables. The Langevin equations are those appropriate to a non-conserved order parameter with an overdamped, relaxational dynamics of the form We expand the orientational tensor in a basis of sym- metric traceless matrices T i αβ as with T 1 = √ 3 / 2 ˆ zˆ z , T 2 = √ 1 / 2 ( ˆ x ˆ x − ˆ y ˆ y ) , T 3 = √$_{2}$ ˆ x ˆ y , T 4 = √$_{2}$ ˆ x ˆ z and T 5 = √$_{2}$ ˆ y ˆ z . The com- plete basis of matrices is orthogonal in the sense that T i $_{αβ}$T j αβ = δ$_{ij}$ . In previous work we have presented explic- itly the equations for the basis coefficients a$_{i}$ ( x , t ) that follow from the deterministic part of the relaxational ki- netics ∂$_{t}$Q$_{αβ}$ = − Γ$_{αβµν}$ δF/δQ$_{µν}$ [11]. (These differ from the equations derived by others in that we include all non-linearities as well as an additional symmetry-allowed gradient ( L 2 ) term.) Here we focus on how an explicit construction of the noise can be implemented by expand- ing in the same basis. We expand the noise as Here the kinetic coefficients Γ$_{αβµν}$, defined as Γ$_{αβµν}$ = Γ[ δ$_{αµ}$δ$_{βν}$ + δ$_{αν}$δ$_{βµ}$ − 2 $_{3}$δ$_{αβ}$ δ$_{µν}$ ], ensure that the dynam- ics preserves the symmetry and tracelessness property of the order parameter. In the absence of long-range forces, a local approximation for the kinetic coefficients is ad- equate and Γ can be taken as constant. The ξ$_{αβ}$ are symmetric, traceless Gaussian white noises, which sat- isfy a fluctuation-dissipation relation at equilibrium of the form where each ξ$_{i}$ ( x , t ) is a zero-mean Gaussian white noise. From the orthogonality of the basis the inverse relation is From this, and the fluctuation-dissipation relation it fol- lows that Here k$_{B}$ is the Boltzmann constant, T the temperature and 〈〉 denotes the average over the probability distri- bution of the noise. These Langevin equations, together with the fluctuation-dissipation relation for the noise, en- sure that the stationary one-point probability distribu- tion of Q$_{αβ}$ , P [ Q$_{αβ}$ ], converges to Boltzmann equilibrium with P [ Q$_{αβ}$ ] ∼ exp( − F/k$_{B}$T ). The equations above are five coupled, non-linear stochastic partial differential equations, with a noise term which has a tensorial structure. A numerical method of solution must maintain the symmetry and traceless of Q$_{αβ}$ . To ensure equilibrium dynamics, it must also maintain the balance between fluctuation and dissipa- tion. These two stringent requirements may be satisfied by transforming to a basis in which Q$_{αβ}$ is traceless and symmetric by construction. Symmetry and tracelessness of Q$_{αβ}$ is automatic. As we show below, the noise can be constructed out of independent Gaussian noises. This shows that the non-trivially correlated noise ξ$_{αβ}$ can be constructed from uncorrelated noises ξ$_{i}$ . Thus, by construction, the noise ξ$_{αβ}$ is symmetric, traceless and satisfies the fluctuation-dissipation relation. When anharmonic terms are ignored in the Ginzburg- Landau-de Gennes functional, the Langevin equations are linear and correlation functions can be calculated ex- plicitly in the T basis. Then, the Langevin equations of motion in terms of are From this, the static and dynamic correlations follow im- mediately, Q$_{αβ}$ ( x , t ) = 5 ∑ i =1 a$_{i}$ ( x , t ) T i αβ (6) ξ$_{αβ}$ ( x , t ) = 5 ∑ i =1 ξ$_{i}$ ( x , t ) T i $_{αβ}$, (7) ξ$_{i}$ ( x , t ) = ∑ α,β ξ$_{αβ}$ ( x , t ) T i $_{αβ}$. (8) 〈 ξ$_{i}$ ( x , t ) ξ$_{j}$ ( x $^{′}$, t$^{′}$) 〉 = ∑ $_{αβµν}$〈 ξ$_{αβ}$ ( x , t ) ξ$_{µν}$ ( x $^{′}$, t$^{′}$) 〉 T i $_{αβ}$T j $_{µν}$, (9) = ∑ αβµν 2 k$_{B}$T Γ$_{αβµν}$ T i $_{αβ}$T j $_{µν}$δ ( x − x $^{′}$) δ ( t − t $^{′}$) , = 2 k$_{B}$T Γ δ$_{ij}$δ ( x − x $^{′}$) δ ( t − t $^{′}$) . a$_{i}$ ( q , t ) = ∫ d $^{3}$x exp( − i q · x ) a$_{i}$ ( x , t ) , (10) ∂$_{t}$a$_{i}$ ( q , t ) = − Γ( A + L 1 q $^{2}$) a$_{i}$ ( q , t ) + ξ$_{i}$ ( q , t ) . (11) C$_{ij}$ ( q ) = 〈 a$_{i}$ ( q ) a$_{j}$ ( − q ) 〉 = k$_{B}$T A + L 1 q $_{2}$δ$_{ij}$ , (12) 〈 ξ$_{αβ}$ ( x , t ) 〉 = 0 , (4) 〈 ξ$_{αβ}$ ( x , t ) ξ$_{µν}$ ( x $^{′}$, t$^{′}$) = 2 k$_{B}$T Γ$_{αβµν}$ δ ( x − x $^{′}$) δ ( t − t $^{′}$) . (5) ∂$_{t}$Q$_{αβ}$ = − Γ$_{αβµν}$ δF δQ$_{µν}$ + ξ$_{αβ}$, (3)