Tensors for Physics

360 17 Tensor Dynamics

The quantityQi=−

√

6 Tμνiaνλaλμis explicitly given by

Q 0 =− 3 a 02 + 3 (a^21 +a^22 )−

3

2

(a^23 +a^24 ), (17.29)

Q 1 = 6 a 0 a 1 −

3

2

√

3 (a^23 −a 42 ), Q 2 = 6 a 0 a 2 − 3

√

3 a 3 a 4 ,

Q 3 =− 3 a 0 a 3 − 3

√

3 (a 1 a 3 +a 2 a 4 ), Q 4 =− 3 a 0 a 4 + 3

√

3 (a 1 a 4 −a 2 a 3 ).

TheΦioccurring in the relaxation equation (17.28) are the derivatives of the potential
functionΦwith respect to the componentsai,viz.Φi=∂Φ/∂a 1 , where

Φ=

1

2

θa^2 +Q+

1

2

(a^2 )^2 , Q=−

√

6 aμνaνλaλμ. (17.30)

Apart from the sign and a numerical factor,Qis the determinant of the alignment
tensor, cf. (5.44). In terms of theai, it is given by

Q=−a^30 + 3 a 0

(

a^21 +a^22 −

1

2

a^23 −

1

2

a^24

)

−

3

2

√

3 a 1

(

a 32 −a 42

)

− 3

√

3 a 2 a 3 a 4.
(17.31)
SinceQ, obviously, is not a function ofa^2 , the potentialΦis highly anisotropic in
the 5-dimensional space of theaicomponents.

17.2.4 Stability of Stationary Solutions

Letaμνst be a stationary solution of the inhomogeneous relaxation equation (16.149)
withΦμνgiven by (16.150). Insertion ofaμν=aμνst +δaμνinto the equation and
disregard of terms nonlinear in the small deviationδaμνfrom the stationary state
yields

∂

∂t

δaμν− 2 εμλκΩλδaκν− 2 κΓμκδaκν+Φμν,λκδaλκ= 0 , (17.32)

with the second derivative of the potential, viz.

Φμν,λκ=

∂

∂aλκ

Φμν=(θ+ 2 a^2 )Δμν,λκ+ 4 aμνaλκ− 6

√

6 Δ(μν,αβ,λκ^2 ,^2 ,^2 ) aαβ,(17.33)

evaluated at the stationary value for the alignment tensor. For the isotropic coupling
tensorΔ(···^2 ,^2 ,^2 )see (11.36).