content_image/443808.jpg content_image/443809.jpg content_image/443804.jpg content_image/443806.jpg content_image/443801.jpg content_image/443802.jpg content_image/443803.jpg content_image/443805.jpg content_image/443807.jpg
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 

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 

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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. 

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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 

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 

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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 

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From this, the static and dynamic correlations follow im- mediately, 

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