text: of the stable states, and so it is important to understand the overall conﬁgu- ration of critical points and how it responds to changes in temperature T as well as other parameters. → It is important to note that while the Hamiltonian H : L(E) R is invariant under the left action λ of G, there is no reason why it should be τ-invariant also and typically it is not. This symmetry-breaking role of the Hamiltonian, pointed out in [27],[32], is crucial to our analysis of bifurcation from isotropy in Section below. 3.3.1 Fixed-point formulation (cid:0) (cid:1) With the entropy − · S = S(η, W ) = k log Z(η) η W (20) regarded as a function of the two independent variables η and W , equa- ∇ ∇ tion (10) describes the locus where S(η, W ) = 0. The full gradient S ∇ η − on this manifold is therefore just S(η, W ) = kη. Hence ﬁnding critical W points of the free energy is equivalent to solving the problem ∇ H(W ) + kT η = 0. (21) Thus (as observed in [27]) the variable η does have a physical interpreta- − tion as (up to the factor kT ) the value of the gradient of the Hamiltonian at points of equilibrium. On substituting W = W (η) the equation (21) be- comes a ﬁxed point problem for η: it does not involve inverting the function W explicitly, and solutions η can then be substituted into W (η) to give the required solutions in terms of the order parameter W . In the particular choice of quadratic H(W ) as in (14) the equation (21) simpliﬁes to BW = kT η × which deﬁnes a linear subspace in L(E) L(E) of dimension n = dim L(E): its intersections with the graph of W as in (10) (typically isolated points) correspond to the critical points of the free energy. 3.4 Reduced models ⊂ If a linear subspace of D E is invariant under the action α then the above formalism applies equally well to the action α restricted to D. However, even if D is not α-invariant the key constructions can still be carried out 16

text: of

text: the

text: stable

text: states,

text: and

text: so

text: it

text: is

text: important

text: to

text: understand

text: the

text: overall

text: conﬁgu-

text: ration

text: of

text: critical

text: points

text: and

text: how

text: it

text: responds

text: to

text: changes

text: in

text: temperature

text: T

text: as

text: well

text: as

text: other

text: parameters.

text: It

text: is

text: important

text: to

text: note

text: that

text: while

text: the

text: Hamiltonian

text: H

text: L(E)

text: R

text: is

text: invariant

text: under

text: the

text: left

text: action

text: λ

text: of

text: G,

text: there

text: is

text: no

text: reason

text: why

text: it

text: should

text: be

text: τ-invariant

text: also

text: and

text: typically

text: it

text: is

text: not.

text: This

text: symmetry-breaking

text: role

text: of

text: the

text: Hamiltonian,

text: pointed

text: out

text: in

text: is

text: crucial

text: to

text: our

text: analysis

text: of

text: bifurcation

text: from

text: isotropy

text: in

text: Section

text: below.

text: Fixed-point

text: formulation

text: With

text: the

text: entropy

text: S

text: S(η,

text: W

text: k

text: log

text: Z(η)

text: η

text: W

text: regarded

text: as

text: a

text: function

text: of

text: the

text: two

text: independent

text: variables

text: η

text: and

text: W

text: equa-

text: tion

text: describes

text: the

text: locus

text: where

text: S(η,

text: W

text: The

text: full

text: gradient

text: S

text: η

text: on

text: this

text: manifold

text: is

text: therefore

text: just

text: S(η,

text: W

text: kη.

text: Hence

text: ﬁnding

text: critical

text: W

text: points

text: of

text: the

text: free

text: energy

text: is

text: equivalent

text: to

text: solving

text: the

text: problem

text: H(W

text: kT

text: η

text: Thus

text: (as

text: observed

text: in

text: the

text: variable

text: η

text: does

text: have

text: a

text: physical

text: interpreta-

text: tion

text: as

text: (up

text: to

text: the

text: factor

text: kT

text: the

text: value

text: of

text: the

text: gradient

text: of

text: the

text: Hamiltonian

text: at

text: points

text: of

text: equilibrium.

text: On

text: substituting

text: W

text: W

text: (η)

text: the

text: equation

text: be-

text: comes

text: a

text: ﬁxed

text: point

text: problem

text: for

text: η:

text: it

text: does

text: not

text: involve

text: inverting

text: the

text: function

text: W

text: explicitly,

text: and

text: solutions

text: η

text: can

text: then

text: be

text: substituted

text: into

text: W

text: (η)

text: to

text: give

text: the

text: required

text: solutions

text: in

text: terms

text: of

text: the

text: order

text: parameter

text: W

text: In

text: the

text: particular

text: choice

text: of

text: quadratic

text: H(W

text: as

text: in

text: the

text: equation

text: simpliﬁes

text: to

text: BW

text: kT

text: η

text: which

text: deﬁnes

text: a

text: linear

text: subspace

text: in

text: L(E)

text: L(E)

text: of

text: dimension

text: n

text: dim

text: L(E):

text: its

text: intersections

text: with

text: the

text: graph

text: of

text: W

text: as

text: in

text: (typically

text: isolated

text: points)

text: correspond

text: to

text: the

text: critical

text: points

text: of

text: the

text: free

text: energy.

text: Reduced

text: models

text: If

text: a

text: linear

text: subspace

text: of

text: D

text: E

text: is

text: invariant

text: under

text: the

text: action

text: α

text: then

text: the

text: above

text: formalism

text: applies

text: equally

text: well

text: to

text: the

text: action

text: α

text: restricted

text: to

text: D.

text: However,

text: even

text: if

text: D

text: is

text: not

text: α-invariant

text: the

text: key

text: constructions

text: can

text: still

text: be

text: carried

text: out


text: In (T, λ) coordinates the locus (44) is a circular cone K in R3 with axis parallel to the T -axis and vertex (T , u) with T = U /10 and u = 1(1, 1, 1). (cid:54) 0 0 0 3 For ﬁxed T = T the cone section in the λ-plane Λ is a circle K with centre 0 T u and radius R that increases linearly with T : explicitly we may write λ in T terms of ξ as (cid:113) (cid:0) (cid:1) − − (λ , λ , λ ) = 1(1, 1,(cid:113)1) + R 2 cos(2ξ + π), cos(2ξ π), cos 2ξ (47) 1 2 3 3 (cid:0) T 3(cid:1) 3 3 − and we ﬁnd R = 2 10T 1 . See Figure 1. T 3 U0 For parameter values (T, λ) inside the cone with T > T the origin 0 (isotropic state) in R4 is a nondegenerate minimum (stable), while outside − the cone the origin is a 2, 2 saddle point with the two directions of insta- bility depending on the direction of λ. Inside the cone with T < T the 0 isotropic state is unstable in all four directions. 6.3 Reduction of the free energy at bifurcation We now study more closely the bifurcations of critical points of the free energy function f = f(s, p, d, c)asthe parameter values(T, λ) cross the ∈ stability cone K. At a point (T, λ) where λ K we ﬁnd  T      −     sin ξ  0   −    0 sin ξ ker Hf(0) = span , . cos ξ 0 0 cos ξ       After the linear coordinate change (note the ordering of variables)             s  sin ξ cos ξ 0 0  x − d cos ξ sin ξ 0 0 y = (48) p 0 0 sin ξ cos ξ u − c 0 0 cos ξ sin ξ v the quadratic terms of f reduce to 2µ(y2 + v2). Next, from the Splitting Lemma or Reduction Lemma [8],[22],[24],[38] we know that there exists a local diﬀeomorphism (smooth invertible coordinate transformation with smooth inverse) in a neighbourhood of the origin in R4, 28 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

label: paragraph

text: (T,

text: λ)

text: coordinates

text: the

text: locus

text: is

text: a

text: circular

text: cone

text: K

text: in

text: with

text: axis

text: parallel

text: to

text: the

text: T

text: -axis

text: and

text: vertex

text: (T

text: u)

text: with

text: T

text: U

text: and

text: u

font: default

text: For

text: ﬁxed

text: T

text: T

text: the

text: cone

text: section

text: in

text: the

text: λ-plane

text: Λ

text: is

text: a

text: circle

text: K

text: with

text: centre

text: T

text: u

text: and

text: radius

text: R

text: that

text: increases

text: linearly

text: with

text: T

text: explicitly

text: we

text: may

text: write

text: λ

text: in

text: T

text: terms

text: of

text: ξ

text: as

label: equation

text: (λ

text: λ

text: λ

text: R

text: π),

text: π),

text: cos

text: 2ξ

text: T

text: and

text: we

text: ﬁnd

text: R

text: See

text: Figure

text: T

text: For

text: parameter

text: values

text: (T,

text: λ)

text: inside

text: the

text: cone

text: with

text: T

text: T

text: the

text: origin

text: (isotropic

text: state)

text: in

text: is

text: a

text: nondegenerate

text: minimum

text: (stable),

text: while

text: outside

text: the

text: cone

text: the

text: origin

text: is

text: a

text: saddle

text: point

text: with

text: the

text: two

text: directions

text: of

text: insta-

text: bility

text: depending

text: on

text: the

text: direction

text: of

text: λ.

text: Inside

text: the

text: cone

text: with

text: T

text: T

text: the

text: isotropic

text: state

text: is

text: unstable

text: in

text: all

text: four

text: directions.

text: Reduction

text: of

text: the

text: free

text: energy

text: at

text: bifurcation

text: We

text: now

text: study

text: more

text: closely

text: the

text: bifurcations

text: of

text: critical

text: points

text: of

text: the

text: free

text: energy

text: function

text: f

text: f(s,

text: p,

text: d,

text: c)asthe

text: parameter

text: values(T,

text: λ)

text: cross

text: the

text: stability

text: cone

text: K.

text: At

text: a

text: point

text: (T,

text: λ)

text: where

text: λ

text: K

text: we

text: ﬁnd

text: T

text: sin

text: ξ

text: sin

text: ξ

text: ker

text: span

text: cos

text: ξ

text: cos

text: ξ

text: After

text: the

text: linear

text: coordinate

text: change

text: (note

text: the

text: ordering

text: of

text: variables)

text: s

text: sin

text: ξ

text: cos

text: ξ

text: x

text: d

text: cos

text: ξ

text: sin

text: ξ

text: y

text: p

text: sin

text: ξ

text: cos

text: ξ

text: u

text: c

text: cos

text: ξ

text: sin

text: ξ

text: v

text: the

text: quadratic

text: terms

text: of

text: f

text: reduce

text: to

text: Next,

text: from

text: the

text: Splitting

text: Lemma

text: or

text: Reduction

text: Lemma

text: we

text: know

text: that

text: there

text: exists

text: a

text: local

text: diﬀeomorphism

text: (smooth

text: invertible

text: coordinate

text: transformation

text: with

text: smooth

text: inverse)

text: in

text: a

text: neighbourhood

text: of

text: the

text: origin

text: in

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: First we need the explicit expression for q˜(Z, W ) in terms of the original data for the function f(z, w). The following identities are straightforward to verify: f (z, w) = f (Z, W ) (58) 2 2 − f (z, w) = Cf (W, Z) S(cid:0)f (Z, W ) (cid:1) (59) 3 3 3 f (z, w) = f(cid:0) (Z, W ) (cid:1) (60) 4 4 (cid:61) − (cid:61) − (cid:61) − f (z, w) = (ZW ZW ) C (Z3 3ZW 2) + S (W 3 3W Z2) (61) 5 (cid:61) − (cid:61) − f (z, w) = C (Z3 3ZW 2) + S (W 3 3W Z2) 2 (62) 6 where (C, S) = (cos 3ξ, sin 3ξ) and S is of course not to be confused with the entropy in Section . Next, we deﬁne W := W + ϕ (63) 0 where ϕ = ϕ(Z) = ϕ (Z) + ϕ (Z) (64) 2 3 with ϕ (Z) a polynomial in (components of) Z of homogeneous degree j j for j = 2, 3, and then substitute for W into f (Z, W ) to obtain a function f0 ξ ξ of Z and W : 0 − f0(Z, W ) = f (Z, W ϕ(Z)) (65) ξ 0 ξ 0 | | = 2µ W 2 + q˜ (Z, W ). (66) 0 0 0 The key step is now to choose ϕ(Z) in such a way that the terms in q˜ (Z, W ) that contain W with degree 1 all vanish up to degree 5. In Table 1 0 0 0 we provide a list of all terms of degree at most 5 in f0 that involve a single ξ W , as well as a list of terms up to degree 6 that do not involve W . From 0 0 this it is evident that the result can be achieved by choosing ϕ(Z) to satisfy − − | | − 4µϕ¯ = 3a (Cϕ2 CZ2 2SZϕ) + 4a ( Z 2ϕ¯ Z2ϕ) 3 4 − | | − | | (cid:60) 4b ϕ¯ Z 2 3b C Z 2Z2 + 2a C (iZ3)iZ 4 5 5 34 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: First

text: we

text: need

text: the

text: explicit

text: expression

text: for

text: q˜(Z,

text: W

text: in

text: terms

text: of

text: the

text: original

text: data

text: for

text: the

text: function

text: f(z,

text: w).

text: The

text: following

text: identities

text: are

text: straightforward

text: to

text: verify:

text: f

text: (z,

text: w)

text: f

text: (Z,

text: W

text: f

text: (z,

text: w)

text: Cf

text: (W,

text: Z)

text: (Z,

text: W

text: f

text: (z,

text: w)

text: (Z,

text: W

text: f

text: (z,

text: w)

text: (ZW

text: ZW

text: C

text: S

text: (W

text: f

text: (z,

text: w)

text: C

text: S

text: (W

text: where

text: (C,

text: S)

text: (cos

text: sin

text: and

text: S

text: is

text: of

text: course

text: not

text: to

text: be

text: confused

text: with

text: the

text: entropy

text: in

text: Section

text: Next,

text: we

text: deﬁne

text: W

text: W

text: ϕ

text: where

text: ϕ

text: ϕ(Z)

text: ϕ

text: (Z)

text: ϕ

text: (Z)

text: with

text: ϕ

text: (Z)

text: a

text: polynomial

text: in

text: (components

text: of)

text: Z

text: of

text: homogeneous

text: degree

text: j

text: j

text: for

text: j

text: and

text: then

text: substitute

text: for

text: W

text: into

text: f

text: (Z,

text: W

text: to

text: obtain

text: a

text: function

text: ξ

text: ξ

text: of

text: Z

text: and

text: W

text: W

text: f

text: (Z,

text: W

text: ϕ(Z))

text: ξ

text: ξ

text: 2µ

text: W

text: q˜

text: (Z,

text: W

text: The

text: key

text: step

text: is

text: now

text: to

text: choose

text: ϕ(Z)

text: in

text: such

text: a

text: way

text: that

text: the

text: terms

text: in

text: q˜

text: (Z,

text: W

text: that

text: contain

text: W

text: with

text: degree

text: all

text: vanish

text: up

text: to

text: degree

text: In

text: Table

text: we

text: provide

text: a

text: list

text: of

text: all

text: terms

text: of

text: degree

text: at

text: most

text: in

text: that

text: involve

text: a

text: single

text: ξ

text: W

text: as

text: well

text: as

text: a

text: list

text: of

text: terms

text: up

text: to

text: degree

text: that

text: do

text: not

text: involve

text: W

text: From

text: this

text: it

text: is

text: evident

text: that

text: the

text: result

text: can

text: be

text: achieved

text: by

text: choosing

text: ϕ(Z)

text: to

text: satisfy

text: 4µϕ¯

text: Z

text: 2ϕ¯

text: ϕ¯

text: Z

text: C

text: Z

text: C

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: new symbols m, n respectively, to emphasise that they are assumed nonzero unlike (possibly) the coeﬃcients e , . . . , e . (We do not need to suppose as 2 5 in [32] that the coeﬃcients of the highest-order terms considered are inde(cid:101)- pendent of the temperature T : as we see, the T -dependence can be absorbed into lower-order coeﬃcients.) Then, regarding the free energy function f in (49) as a perturbation of the function f (x, y, u, v) := 2µ(y2 + v2) + q (x, u) (89) 0 0 where q (x, u) consists of the degree-6 terms 0 − q (x, u) := mX3 + nY 2 = m(x2 + u2)3 + n(x3 3xu2)2 0 (cid:54) and it is assumed mn = 0, we invoke the theory of versal deformation (or universal unfolding) to exhibit a 7-parameter family f of perturbations of q e 0 that captures all possible local bifurcation behaviour for (89). More precisely, any other multiparameter perturbation of (89) can be written in terms of f e by an appropriate local coordinate-mapping (invertible in the x, u-variables) that respects the D -symmetry. For further explanation and a sketch of the 3 proof of the Proposition see Appendix C. Stating the result in more formal terms: Proposition 14 The 7-parameter family of polynomial functions q (x, u) := e + e X + e Y + e X2 + e XY + e X3 + e X4 + q (x, u) (90) e 0 2 3 4 5 6 8 0 R is a D - -versal deformation of q at the origin in R2. (cid:50) 3 0 The same arguments extend easily to show also Proposition 15 The 7-parameter family of polynomial functions f (x, y, u, v) := 2µ(y2 + v2) + q (x, u) (91) e e R is a D - -versal deformation of the free energy function 3 f (x, y, u, v) := 2µ(y2 + v2) + q (x, u) 0 0 at the origin in R4. (cid:50) Therefore, in view of Proposition 15 and the fact that the constant e is 0 irrelevant for locating critical points and for assigning relative function values to those points, we have the following consequence. 41

text: new

text: symbols

text: m,

text: n

text: respectively,

text: to

text: emphasise

text: that

text: they

text: are

text: assumed

text: nonzero

text: unlike

text: (possibly)

text: the

text: coeﬃcients

text: e

text: e

text: (We

text: do

text: not

text: need

text: to

text: suppose

text: as

text: in

text: that

text: the

text: coeﬃcients

text: of

text: the

text: highest-order

text: terms

text: considered

text: are

text: pendent

text: of

text: the

text: temperature

text: T

text: as

text: we

text: see,

text: the

text: T

text: -dependence

text: can

text: be

text: absorbed

text: into

text: lower-order

text: coeﬃcients.)

text: Then,

text: regarding

text: the

text: free

text: energy

text: function

text: f

text: in

text: as

text: a

text: perturbation

text: of

text: the

text: function

text: f

text: (x,

text: y,

text: u,

text: v)

text: q

text: (x,

text: u)

text: where

text: q

text: (x,

text: u)

text: consists

text: of

text: the

text: terms

text: q

text: (x,

text: u)

text: nY

text: and

text: it

text: is

text: assumed

text: mn

text: we

text: invoke

text: the

text: theory

text: of

text: versal

text: deformation

text: (or

text: universal

text: unfolding)

text: to

text: exhibit

text: a

text: family

text: f

text: of

text: perturbations

text: of

text: q

text: e

text: that

text: captures

text: all

text: possible

text: local

text: bifurcation

text: behaviour

text: for

text: More

text: precisely,

text: any

text: other

text: multiparameter

text: perturbation

text: of

text: can

text: be

text: written

text: in

text: terms

text: of

text: f

text: e

text: by

text: an

text: appropriate

text: local

text: coordinate-mapping

text: (invertible

text: in

text: the

text: x,

text: u-variables)

text: that

text: respects

text: the

text: D

text: -symmetry.

text: For

text: further

text: explanation

text: and

text: a

text: sketch

text: of

text: the

text: proof

text: of

text: the

text: Proposition

text: see

text: Appendix

text: C.

text: Stating

text: the

text: result

text: in

text: more

text: formal

text: terms:

text: Proposition

text: The

text: family

text: of

text: polynomial

text: functions

text: q

text: (x,

text: u)

text: e

text: e

text: X

text: e

text: Y

text: e

text: e

text: XY

text: e

text: e

text: q

text: (x,

text: u)

text: e

text: R

text: is

text: a

text: D

text: -versal

text: deformation

text: of

text: q

text: at

text: the

text: origin

text: in

text: The

text: same

text: arguments

text: extend

text: easily

text: to

text: show

text: also

text: Proposition

text: The

text: family

text: of

text: polynomial

text: functions

text: f

text: (x,

text: y,

text: u,

text: v)

text: q

text: (x,

text: u)

text: e

text: e

text: R

text: is

text: a

text: D

text: -versal

text: deformation

text: of

text: the

text: free

text: energy

text: function

text: f

text: (x,

text: y,

text: u,

text: v)

text: q

text: (x,

text: u)

text: at

text: the

text: origin

text: in

text: Therefore,

text: in

text: view

text: of

text: Proposition

text: and

text: the

text: fact

text: that

text: the

text: constant

text: e

text: is

text: irrelevant

text: for

text: locating

text: critical

text: points

text: and

text: for

text: assigning

text: relative

text: function

text: values

text: to

text: those

text: points,

text: we

text: have

text: the

text: following

text: consequence.


text: · · ∗ ∈ Au v = u A v for every u, v E. In terms of (real) matrices this is simply ∈ ∗ the transpose, and the map A L(E) is symmetric if A = A. We denote the set of symmetric elements of L(E) by Sym(E), and the subset of those with zero trace by Sym (E). 0 ∈ ∗ The linear map A L(E) is orthogonal if A A is the identity map, so that A preserves the scalar product on E. The action α is orthogonal if ∈ ∈ α(g) L(E) is orthogonal for every g G, in which case · · α(g)u α(g)v = u v ∈ ∈ for every g G and u, v E. By a standard process of averaging it is easy to show that an inner product can always be chosen to make a given action α orthogonal: see [6, Theorem II(1.7)] or [11, Theorem 4.4.3] for example. This greatly simpliﬁes the handling of matrices describing symmetries. In particular, if α is orthogonal then Sym(E) and Sym (E) are preserved by 0 the conjugacy action α˜ on L(E). A natural scalar product on the space L(E) arises from that on E and given by · ∗ A B := tr(A B) where tr denotes trace. The following identities are easily veriﬁed: Proposition 1 · · ∗ · ∗ ∗ · ∗ A B = B A = A B = B A ∈ for all A, B L(E). (cid:50) A further elementary result is important for our purposes: Proposition 2 If the action α on E is orthogonal then the action α˜ on L(E) is also orthogonal. (cid:0) (cid:1) Proof. By deﬁnition (cid:0) (cid:1) · ∗ α˜(g)(B) α˜(g)(C) = tr(cid:0)(α˜(g)(B)) α˜(g)(C(cid:1) ) − ∗ − = tr (α(g)Bα(g) 1) α(g)Cα(g) 1 ∗ − = tr α(g) B C α(g) 1 ∗ · = tr(B C) = B C ∗ − using Proposition 1 and the fact that α(g) = α(g) 1 by orthogonality. (cid:50) 6

text: Au

text: v

text: u

text: A

text: v

text: for

text: every

text: u,

text: v

text: E.

text: In

text: terms

text: of

text: (real)

text: matrices

text: this

text: is

text: simply

text: the

text: transpose,

text: and

text: the

text: map

text: A

text: L(E)

text: is

text: symmetric

text: if

text: A

text: A.

text: We

text: denote

text: the

text: set

text: of

text: symmetric

text: elements

text: of

text: L(E)

text: by

text: Sym(E),

text: and

text: the

text: subset

text: of

text: those

text: with

text: zero

text: trace

text: by

text: Sym

text: (E).

text: The

text: linear

text: map

text: A

text: L(E)

text: is

text: orthogonal

text: if

text: A

text: A

text: is

text: the

text: identity

text: map,

text: so

text: that

text: A

text: preserves

text: the

text: scalar

text: product

text: on

text: E.

text: The

text: action

text: α

text: is

text: orthogonal

text: if

text: α(g)

text: L(E)

text: is

text: orthogonal

text: for

text: every

text: g

text: G,

text: in

text: which

text: case

text: α(g)u

text: α(g)v

text: u

text: v

text: for

text: every

text: g

text: G

text: and

text: u,

text: v

text: E.

text: By

text: a

text: standard

text: process

text: of

text: averaging

text: it

text: is

text: easy

text: to

text: show

text: that

text: an

text: inner

text: product

text: can

text: always

text: be

text: chosen

text: to

text: make

text: a

text: given

text: action

text: α

text: orthogonal:

text: see

text: Theorem

text: or

text: Theorem

text: for

text: example.

text: This

text: greatly

text: simpliﬁes

text: the

text: handling

text: of

text: matrices

text: describing

text: symmetries.

text: In

text: particular,

text: if

text: α

text: is

text: orthogonal

text: then

text: Sym(E)

text: and

text: Sym

text: (E)

text: are

text: preserved

text: by

text: the

text: conjugacy

text: action

text: α˜

text: on

text: L(E).

text: A

text: natural

text: scalar

text: product

text: on

text: the

text: space

text: L(E)

text: arises

text: from

text: that

text: on

text: E

text: and

text: given

text: by

text: A

text: B

text: tr(A

text: B)

text: where

text: tr

text: denotes

text: trace.

text: The

text: following

text: identities

text: are

text: easily

text: veriﬁed:

text: Proposition

text: A

text: B

text: B

text: A

text: A

text: B

text: B

text: A

text: for

text: all

text: A,

text: B

text: L(E).

text: A

text: further

text: elementary

text: result

text: is

text: important

text: for

text: our

text: purposes:

text: Proposition

text: If

text: the

text: action

text: α

text: on

text: E

text: is

text: orthogonal

text: then

text: the

text: action

text: α˜

text: on

text: L(E)

text: is

text: also

text: orthogonal.

text: Proof.

text: By

text: deﬁnition

text: α˜(g)(B)

text: α˜(g)(C)

text: tr

text: (α(g)Bα(g)

text: α(g)Cα(g)

text: tr

text: α(g)

text: B

text: C

text: α(g)

text: tr(B

text: C)

text: B

text: C

text: using

text: Proposition

text: and

text: the

text: fact

text: that

text: α(g)

text: α(g)

text: by

text: orthogonality.


text: pictures/fig09.eps Figure 9: Combined bifurcation set for e = 0 and ﬁxed e > 0, showing 5 4 uniaxial equilibria on the x -axis and biaxial equilibria oﬀ the x -axis lying (cid:48) ∩ on Γ C as in Figure 7 for representative choices of e and e . 2 3 exhibits a fold singularity [8, 22, 38], which has the local geometric property that almost any smooth curve in M crossing Σ projects by π to a curve S S ∩ tangent to ∆ . Consequently at the points of B S that correspond to S∩ points of M0 Σ the bifurcation sets B and S will be mutually tangent. B S 52 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: Combined

label: caption

text: bifurcation

text: set

text: for

text: e

text: and

text: ﬁxed

text: e

text: showing

text: uniaxial

text: equilibria

text: on

text: the

text: x

text: -axis

text: and

text: biaxial

text: equilibria

text: oﬀ

text: the

text: x

text: -axis

text: lying

text: on

text: Γ

text: C

text: as

text: in

text: Figure

text: for

text: representative

text: choices

text: of

text: e

text: and

text: e

text: exhibits

text: a

text: fold

text: singularity

text: which

text: has

text: the

text: local

text: geometric

text: property

text: that

text: almost

text: any

text: smooth

text: curve

text: in

text: M

text: crossing

text: Σ

text: projects

text: by

text: π

text: to

text: a

text: curve

text: S

text: S

text: tangent

text: to

text: Consequently

text: at

text: the

text: points

text: of

text: B

text: S

text: that

text: correspond

text: to

text: S∩

text: points

text: of

text: Σ

text: the

text: bifurcation

text: sets

text: B

text: and

text: S

text: will

text: be

text: mutually

text: tangent.

text: B

text: S

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:48) together with their proﬁles in the T, x -plane which show more clearly the order in which branching occurs as T decreases. Note that it remains the case (cid:48) that biaxial equilibria occur in pairs with equal and opposite values of u . The lower diagram (a) indicates that ﬁrst a pair of uniaxial equilibria is (cid:48) created at a saddle-node (fold) bifurcation with x > 0, and then one these passes through zero (isotropic state) which loses stability. As T decreases a (cid:48) pair of biaxial equilibria branches from the uniaxial equilibrium with x > 0, (cid:48) and for these the value of x tends to zero as T decreases. Of course each branch corresponds to three symmetrically-placed branches in the original (x, u) coordinates. For the lower diagram (b) the sequence is more complicated. First, two pairs of biaxial equilibria are created as T decreases at a (double) saddle-node bifurcation. Next, a pair of uniaxial equilibria is created at a saddle-node (cid:48) bifurcation with x > 0. The ‘inner’ pair of biaxial equilibria then coalesces with one of these uniaxial equilibria. As T decreases further a second saddle- (cid:48) node creation of uniaxial equilibria occurs with x < 0, while the origin remains a locally stable equilibrium throughout. However, a uniaxial equi- (cid:48) librium with x > 0 then passes through the origin as the latter equilibrium becomes unstable, and two uniaxial equilibria mutually annihilate at a saddle (cid:48) (cid:48) node with x < 0. Meanwhile the biaxial equilibria have persisted, with x values tending to zero as T decreases. Figure 14 being rather congested, we reproduce a zoom image of its central portion in Figure 16. This indicates the point of tangency of the uniaxial bi- furcation set with the biaxial bifurcation set, corresponding (as T decreases) to the simultaneous annihilation of a pair of biaxial equilibria at a point of creation of a pair of uniaxial equilibria. As T decreases the bifurcation se- quence along the path (c) is a follows (we omit repetition of ‘equilibrium’): creation of a uniaxial pair, creation of two biaxial pairs, annihilation of two biaxials at a uniaxial, loss of stability of the origin, creation of a uniaxial pair, annihilation of a uniaxial pair. Along the path (d) the order of the ﬁrst two transitions is interchanged, as is the order of the penultimate two. 7.1 Further symmetry-breaking The preceding study of the swallowtail and bluebird bifurcations assumed e = 0 since, up to certain nonlinear coordinate changes as described in sec- 5 tion , there is no loss of generality in doing so as far as overall conﬁgurations of critical points of the free energy are concerned. However, in the present con- text of phase transitions for liquid crystals the allowed coordinate changes are somewhat too general since the distinguished role of the temperature parameter T may be compromised. This diﬃculty may be addressed by 59

text: together

text: with

text: their

text: proﬁles

text: in

text: the

text: T,

text: x

text: -plane

text: which

text: show

text: more

text: clearly

text: the

text: order

text: in

text: which

text: branching

text: occurs

text: as

text: T

text: decreases.

text: Note

text: that

text: it

text: remains

text: the

text: case

text: that

text: biaxial

text: equilibria

text: occur

text: in

text: pairs

text: with

text: equal

text: and

text: opposite

text: values

text: of

text: u

text: The

text: lower

text: diagram

text: (a)

text: indicates

text: that

text: ﬁrst

text: a

text: pair

text: of

text: uniaxial

text: equilibria

text: is

text: created

text: at

text: a

text: saddle-node

text: (fold)

text: bifurcation

text: with

text: x

text: and

text: then

text: one

text: these

text: passes

text: through

text: zero

text: (isotropic

text: state)

text: which

text: loses

text: stability.

text: As

text: T

text: decreases

text: a

text: pair

text: of

text: biaxial

text: equilibria

text: branches

text: from

text: the

text: uniaxial

text: equilibrium

text: with

text: x

text: and

text: for

text: these

text: the

text: value

text: of

text: x

text: tends

text: to

text: zero

text: as

text: T

text: decreases.

text: Of

text: course

text: each

text: branch

text: corresponds

text: to

text: three

text: symmetrically-placed

text: branches

text: in

text: the

text: original

text: (x,

text: u)

text: coordinates.

text: For

text: the

text: lower

text: diagram

text: (b)

text: the

text: sequence

text: is

text: more

text: complicated.

text: First,

text: two

text: pairs

text: of

text: biaxial

text: equilibria

text: are

text: created

text: as

text: T

text: decreases

text: at

text: a

text: (double)

text: saddle-node

text: bifurcation.

text: Next,

text: a

text: pair

text: of

text: uniaxial

text: equilibria

text: is

text: created

text: at

text: a

text: saddle-node

text: bifurcation

text: with

text: x

text: The

text: ‘inner’

text: pair

text: of

text: biaxial

text: equilibria

text: then

text: coalesces

text: with

text: one

text: of

text: these

text: uniaxial

text: equilibria.

text: As

text: T

text: decreases

text: further

text: a

text: second

text: saddle-

text: node

text: creation

text: of

text: uniaxial

text: equilibria

text: occurs

text: with

text: x

text: while

text: the

text: origin

text: remains

text: a

text: locally

text: stable

text: equilibrium

text: throughout.

text: However,

text: a

text: uniaxial

text: equi-

text: librium

text: with

text: x

text: then

text: passes

text: through

text: the

text: origin

text: as

text: the

text: latter

text: equilibrium

text: becomes

text: unstable,

text: and

text: two

text: uniaxial

text: equilibria

text: mutually

text: annihilate

text: at

text: a

text: saddle

text: node

text: with

text: x

text: Meanwhile

text: the

text: biaxial

text: equilibria

text: have

text: persisted,

text: with

text: x

text: values

text: tending

text: to

text: zero

text: as

text: T

text: decreases.

text: Figure

text: being

text: rather

text: congested,

text: we

text: reproduce

text: a

text: zoom

text: image

text: of

text: its

text: central

text: portion

text: in

text: Figure

text: This

text: indicates

text: the

text: point

text: of

text: tangency

text: of

text: the

text: uniaxial

text: bi-

text: furcation

text: set

text: with

text: the

text: biaxial

text: bifurcation

text: set,

text: corresponding

text: (as

text: T

text: decreases)

text: to

text: the

text: simultaneous

text: annihilation

text: of

text: a

text: pair

text: of

text: biaxial

text: equilibria

text: at

text: a

text: point

text: of

text: creation

text: of

text: a

text: pair

text: of

text: uniaxial

text: equilibria.

text: As

text: T

text: decreases

text: the

text: bifurcation

text: se-

text: quence

text: along

text: the

text: path

text: (c)

text: is

text: a

text: follows

text: (we

text: omit

text: repetition

text: of

text: ‘equilibrium’):

text: creation

text: of

text: a

text: uniaxial

text: pair,

text: creation

text: of

text: two

text: biaxial

text: pairs,

text: annihilation

text: of

text: two

text: biaxials

text: at

text: a

text: uniaxial,

text: loss

text: of

text: stability

text: of

text: the

text: origin,

text: creation

text: of

text: a

text: uniaxial

text: pair,

text: annihilation

text: of

text: a

text: uniaxial

text: pair.

text: Along

text: the

text: path

text: (d)

text: the

text: order

text: of

text: the

text: ﬁrst

text: two

text: transitions

text: is

text: interchanged,

text: as

text: is

text: the

text: order

text: of

text: the

text: penultimate

text: two.

text: Further

text: symmetry-breaking

text: The

text: preceding

text: study

text: of

text: the

text: swallowtail

text: and

text: bluebird

text: bifurcations

text: assumed

text: e

text: since,

text: up

text: to

text: certain

text: nonlinear

text: coordinate

text: changes

text: as

text: described

text: in

text: sec-

text: tion

text: there

text: is

text: no

text: loss

text: of

text: generality

text: in

text: doing

text: so

text: as

text: far

text: as

text: overall

text: conﬁgurations

text: of

text: critical

text: points

text: of

text: the

text: free

text: energy

text: are

text: concerned.

text: However,

text: in

text: the

text: present

text: con-

text: text

text: of

text: phase

text: transitions

text: for

text: liquid

text: crystals

text: the

text: allowed

text: coordinate

text: changes

text: are

text: somewhat

text: too

text: general

text: since

text: the

text: distinguished

text: role

text: of

text: the

text: temperature

text: parameter

text: T

text: may

text: be

text: compromised.

text: This

text: diﬃculty

text: may

text: be

text: addressed

text: by


text: 2.2 Invariant functions → A real-valued function f : E R is said to be invariant under the action α of G on E when it is the case that f(α(g)x) = f(x) ∈ ∈ for all g G and all x E. It is a classical theorem due to Hilbert that if G is ﬁnite or compact then there is a ﬁnite set of G-invariant polynomials (a Hilbert basis or integrity basis) such that every G-invariant polynomial func- tion on E can be expressed as a polynomial function of these basic polynomi- als. This result was extended by Schwarz [45] to show that every G-invariant ∞ smooth (i.e. C ) function on E can be expressed as a smooth function of the polynomials in a Hilbert basis. Thus, for example, every smooth func- → tion V = Sym (R3) R invariant under the conjugacy action of SO(3) 0 can be written as a smooth function of the two invariants tr Q2 and tr Q3 for ∈ Q V . This result, well known in liquid crystal theory, follows as a simple application of a powerful and elegant general technique in invariant theory that we now describe for later use. 2.3 Molien series The Hilbert basis for a given group action is not unique, and ﬁnding one in practice can be a considerable challenge [47]. However, a valuable tool exists enabling us to calculate in advance how many elements a Hilbert basis will have. The Molien series [11],[35],[(cid:88)47] for the given action α of G on E is a formal power series ∞ P (t) = r td α d d=0 where the coeﬃcient r is the number of linearly independent G-invariant d polynomial functions on E of homogeneous degree d . Remarkably, in the (cid:88) P case of a ﬁnite group G there is an explicit expression for (t) as α P 1 1 (t) = | | − (3) α G det(I tα(g)) ∈ g G | | where G is the number of elements in G, with an analogous expression for an inﬁnite compact group using an integral rather than a sum. Furthermore, P the series (t) is expressible as a rational function which gives more detailed α information about the algebraic relationships between the invariants [35],[47]. Example Action of D on R2 3 7 ##LTLine## ##LTLine##

label: section

text: Invariant

text: functions

text: A

text: real-valued

text: function

text: f

text: E

text: R

text: is

text: said

text: to

text: be

text: invariant

text: under

text: the

text: action

text: α

text: of

text: G

text: on

text: E

text: when

text: it

text: is

text: the

text: case

text: that

text: f(α(g)x)

text: f(x)

text: for

text: all

text: g

text: G

text: and

text: all

text: x

text: E.

text: It

text: is

text: a

text: classical

text: theorem

text: due

text: to

text: Hilbert

text: that

text: if

text: G

text: is

text: ﬁnite

text: or

text: compact

text: then

text: there

text: is

text: a

text: ﬁnite

text: set

text: of

text: G-invariant

text: polynomials

text: (a

text: Hilbert

text: basis

text: or

text: integrity

text: basis)

text: such

text: that

text: every

text: G-invariant

text: polynomial

text: func-

text: tion

text: on

text: E

text: can

text: be

text: expressed

text: as

text: a

text: polynomial

text: function

text: of

text: these

text: basic

text: polynomi-

text: als.

text: This

text: result

text: was

text: extended

text: by

text: Schwarz

text: to

text: show

text: that

text: every

text: G-invariant

text: smooth

text: (i.e.

text: C

text: function

text: on

text: E

text: can

text: be

text: expressed

text: as

text: a

text: smooth

text: function

text: of

text: the

text: polynomials

text: in

text: a

text: Hilbert

text: basis.

text: Thus,

text: for

text: example,

text: every

text: smooth

text: func-

text: tion

text: V

text: Sym

text: R

text: invariant

text: under

text: the

text: conjugacy

text: action

text: of

text: can

text: be

text: written

text: as

text: a

text: smooth

text: function

text: of

text: the

text: two

text: invariants

text: tr

text: and

text: tr

text: for

text: Q

text: V

text: This

text: result,

text: well

text: known

text: in

text: liquid

text: crystal

text: theory,

text: follows

text: as

text: a

text: simple

text: application

text: of

text: a

text: powerful

text: and

text: elegant

text: general

text: technique

text: in

text: invariant

text: theory

text: that

text: we

text: now

text: describe

text: for

text: later

text: use.

text: Molien

text: series

text: The

text: Hilbert

text: basis

text: for

text: a

text: given

text: group

text: action

text: is

text: not

text: unique,

text: and

text: ﬁnding

text: one

text: in

text: practice

text: can

text: be

text: a

text: considerable

text: challenge

text: However,

text: a

text: valuable

text: tool

text: exists

text: enabling

text: us

text: to

text: calculate

text: in

text: advance

text: how

text: many

text: elements

text: a

text: Hilbert

text: basis

text: will

text: have.

text: The

text: Molien

text: series

text: for

text: the

text: given

text: action

text: α

text: of

text: G

text: on

text: E

text: is

text: a

text: formal

text: power

text: series

text: P

text: (t)

text: r

text: td

text: α

text: d

text: where

text: the

text: coeﬃcient

text: r

text: is

text: the

text: number

text: of

text: linearly

text: independent

text: G-invariant

text: d

text: polynomial

text: functions

text: on

text: E

text: of

text: homogeneous

text: degree

text: d

text: Remarkably,

text: in

text: the

text: P

text: case

text: of

text: a

text: ﬁnite

text: group

text: G

text: there

text: is

text: an

text: explicit

text: expression

text: for

text: (t)

text: as

text: α

text: P

text: (t)

text: α

text: G

text: det(I

text: tα(g))

text: g

text: G

text: where

text: G

text: is

text: the

text: number

text: of

text: elements

text: in

text: G,

text: with

text: an

text: analogous

text: expression

text: for

text: an

text: inﬁnite

text: compact

text: group

text: using

text: an

text: integral

text: rather

text: than

text: a

text: sum.

text: Furthermore,

text: P

text: the

text: series

text: (t)

text: is

text: expressible

text: as

text: a

text: rational

text: function

text: which

text: gives

text: more

text: detailed

text: α

text: information

text: about

text: the

text: algebraic

text: relationships

text: between

text: the

text: invariants

text: Example

text: Action

text: of

text: D

text: on

text: ##LTLine##

text: ##LTLine##


text: pictures/fig17a_17b.eps Figure 17: The eﬀect of (a) nonzero (positive) e on the swallowtail and 5 bluebird conﬁgurations and (b) nonzero (positive) x-term on the swallowtail conﬁguration, both for negative e . 4 We have attempted to chart a rigorous path from the molecular ﬁeld theory KKLS model to the bifurcation geometry of a D -invariant function 3 on R2, using the language and tools of group actions and equivariant singu- larity theory, since although many of these ideas and methods are implicit in the literature, sometimes expressed in terms of physics, there has not ap- peared to be any clear mathematical account of the whole framework. The results presented here provide a geometric setting in which to study the com- plete bifurcation behaviour of critical points of the free energy function close to isotropy. The further inclusion into this picture of the Maxwell set ge- ometry, characterising the competition between critical points at the same energy value, will then yield a full map of the associated phase transitions. Acknowledgements. The author is much indebted to Paolo Biscari, Giovanni De Matteis, Chuck Gartland, Lech Longa, Nigel Mottram, Stefano Turzi, Epifanio Virga and many other liquid crystal experts for numerous valuable and instructive conversations, and especially to Tim Sluckin for highlighting liquid crystal phase transitions as a problem in symmetric bifurcation and for continual inspiration over at least a decade. Thanks are also due to Reiner Lauterbach, Cormac Long and Gareth Jones for helpful discussions about the (cid:111) 72-element group D Z and its representation theory, and to Terry Wall 3 2 for help with G-determinacy. The author is grateful to the Isaac Newton Institute, Cambridge for supporting the Programme on the Mathematics of Liquid Crystals in 2013, to the Leverhulme Trust for an Emeritus Research 63 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: The

text: eﬀect

text: of

text: (a)

text: nonzero

text: (positive)

text: e

text: on

text: the

text: swallowtail

text: and

text: bluebird

text: conﬁgurations

text: and

text: (b)

text: nonzero

text: (positive)

text: x-term

text: on

text: the

text: swallowtail

text: conﬁguration,

text: both

text: for

text: negative

text: e

text: We

text: have

text: attempted

text: to

text: chart

text: a

text: rigorous

text: path

text: from

text: the

text: molecular

text: ﬁeld

text: theory

text: KKLS

text: model

text: to

text: the

text: bifurcation

text: geometry

text: of

text: a

text: D

text: -invariant

text: function

text: on

text: using

text: the

text: language

text: and

text: tools

text: of

text: group

text: actions

text: and

text: equivariant

text: singu-

text: larity

text: theory,

text: since

text: although

text: many

text: of

text: these

text: ideas

text: and

text: methods

text: are

text: implicit

text: in

text: the

text: literature,

text: sometimes

text: expressed

text: in

text: terms

text: of

text: physics,

text: there

text: has

text: not

text: ap-

text: peared

text: to

text: be

text: any

text: clear

text: mathematical

text: account

text: of

text: the

text: whole

text: framework.

text: The

text: results

text: presented

text: here

text: provide

text: a

text: geometric

text: setting

text: in

text: which

text: to

text: study

text: the

text: com-

text: plete

text: bifurcation

text: behaviour

text: of

text: critical

text: points

text: of

text: the

text: free

text: energy

text: function

text: close

text: to

text: isotropy.

text: The

text: further

text: inclusion

text: into

text: this

text: picture

text: of

text: the

text: Maxwell

text: set

text: ge-

text: ometry,

text: characterising

text: the

text: competition

text: between

text: critical

text: points

text: at

text: the

text: same

text: energy

text: value,

text: will

text: then

text: yield

text: a

text: full

text: map

text: of

text: the

text: associated

text: phase

text: transitions.

text: Acknowledgements.

text: The

text: author

text: is

text: much

text: indebted

text: to

text: Paolo

text: Biscari,

text: Giovanni

text: De

text: Matteis,

text: Chuck

text: Gartland,

text: Lech

text: Longa,

text: Nigel

text: Mottram,

text: Stefano

text: Turzi,

text: Epifanio

text: Virga

text: and

text: many

text: other

text: liquid

text: crystal

text: experts

text: for

text: numerous

text: valuable

text: and

text: instructive

text: conversations,

text: and

text: especially

text: to

text: Tim

text: Sluckin

text: for

text: highlighting

text: liquid

text: crystal

text: phase

text: transitions

text: as

text: a

text: problem

text: in

text: symmetric

text: bifurcation

text: and

text: for

text: continual

text: inspiration

text: over

text: at

text: least

text: a

text: decade.

text: Thanks

text: are

text: also

text: due

text: to

text: Reiner

text: Lauterbach,

text: Cormac

text: Long

text: and

text: Gareth

text: Jones

text: for

text: helpful

text: discussions

text: about

text: the

text: group

text: D

text: Z

text: and

text: its

text: representation

text: theory,

text: and

text: to

text: Terry

text: Wall

text: for

text: help

text: with

text: G-determinacy.

text: The

text: author

text: is

text: grateful

text: to

text: the

text: Isaac

text: Newton

text: Institute,

text: Cambridge

text: for

text: supporting

text: the

text: Programme

text: on

text: the

text: Mathematics

text: of

text: Liquid

text: Crystals

text: in

text: to

text: the

text: Leverhulme

text: Trust

text: for

text: an

text: Emeritus

text: Research

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: it can be placed in a family of functions with ﬁnitely many parameters which captures all possible local perturbations of f in some neighbourhood of that point: this applies in a wide range of contexts for functions and mappings with various notions of equivalence up to coordinate change, and in particular applies to G-invariant functions under G-R-equivalence [12],[37]. We now formulate this a little more precisely. All functions and maps are taken to ∞ be C . A real-valued function H deﬁned on a neighbourhood of the origin in × · Rn Rs is called an s-parameter deformation of the function h := H( , 0) on a neighbourhood of the origin in Rn. Let K be an (cid:96)-parameter deformation ∈ of the same function h, where δ = (δ , . . . , δ ) Rt. Then K is said to be 1 t induced by H if K can be expressed in terms of H, that is to say there is a map (cid:55)→ (x, δ) (x˜(x, δ), ε˜(δ)) such that ≡ H(x˜, ε˜) K(x, δ) (cid:55)→ close to the origin, the map x (x˜, δ) being a local diﬀeomorphism for δ = 0 R and hence for all small δ. The deformation H of h is called -versal if every deformation of h is induced by H in this way. Moreover, if h is invariant with respect to the action of a group G on Rn, and versality holds even when the → map x (x˜, δ) is required to be equivariant with respect to this G-action, R then H is G- -versal. Remarkably, versal deformations typically exist, and there are explicit R algebraic criteria for recognising and constructing them. For -versality and without the G-action this is the material of elementary catastrophe theory (where versal deformations are more often called universal unfoldings): see [8],[22],[38] for example. Expositions in the G-invariant setting are generally highly technical; the result we need can be found in [12] [37]. { } Theorem 1 Let v , . . . , v be a set of G-invariant polynomials with the 1 r property that every G-invariant polynomial p can be written in the form · · · p = w + ε v + + ε v 1 1 r r ∈ ∈ W for some coeﬃcients ε , . . . , ε R, where w (h) as in Section . Then 1 r · · · H := k + ε v + + ε v 1 1 r r R is a G- -versal deformation of h. 70

text: it

text: can

text: be

text: placed

text: in

text: a

text: family

text: of

text: functions

text: with

text: ﬁnitely

text: many

text: parameters

text: which

text: captures

text: all

text: possible

text: local

text: perturbations

text: of

text: f

text: in

text: some

text: neighbourhood

text: of

text: that

text: point:

text: this

text: applies

text: in

text: a

text: wide

text: range

text: of

text: contexts

text: for

text: functions

text: and

text: mappings

text: with

text: various

text: notions

text: of

text: equivalence

text: up

text: to

text: coordinate

text: change,

text: and

text: in

text: particular

text: applies

text: to

text: G-invariant

text: functions

text: under

text: G-R-equivalence

text: We

text: now

text: formulate

text: this

text: a

text: little

text: more

text: precisely.

text: All

text: functions

text: and

text: maps

text: are

text: taken

text: to

text: be

text: C

text: A

text: real-valued

text: function

text: H

text: deﬁned

text: on

text: a

text: neighbourhood

text: of

text: the

text: origin

text: in

text: Rn

text: Rs

text: is

text: called

text: an

text: s-parameter

text: deformation

text: of

text: the

text: function

text: h

text: H(

text: on

text: a

text: neighbourhood

text: of

text: the

text: origin

text: in

text: Rn.

text: Let

text: K

text: be

text: an

text: deformation

text: of

text: the

text: same

text: function

text: h,

text: where

text: δ

text: (δ

text: δ

text: Rt.

text: Then

text: K

text: is

text: said

text: to

text: be

text: t

text: induced

text: by

text: H

text: if

text: K

text: can

text: be

text: expressed

text: in

text: terms

text: of

text: H,

text: that

text: is

text: to

text: say

text: there

text: is

text: a

text: map

text: (x,

text: δ)

text: (x˜(x,

text: δ),

text: ε˜(δ))

text: such

text: that

text: H(x˜,

text: ε˜)

text: K(x,

text: δ)

text: close

text: to

text: the

text: origin,

text: the

text: map

text: x

text: (x˜,

text: δ)

text: being

text: a

text: local

text: diﬀeomorphism

text: for

text: δ

text: R

text: and

text: hence

text: for

text: all

text: small

text: δ.

text: The

text: deformation

text: H

text: of

text: h

text: is

text: called

text: -versal

text: if

text: every

text: deformation

text: of

text: h

text: is

text: induced

text: by

text: H

text: in

text: this

text: way.

text: Moreover,

text: if

text: h

text: is

text: invariant

text: with

text: respect

text: to

text: the

text: action

text: of

text: a

text: group

text: G

text: on

text: Rn,

text: and

text: versality

text: holds

text: even

text: when

text: the

text: map

text: x

text: (x˜,

text: δ)

text: is

text: required

text: to

text: be

text: equivariant

text: with

text: respect

text: to

text: this

text: G-action,

text: R

text: then

text: H

text: is

text: G-

text: -versal.

text: Remarkably,

text: versal

text: deformations

text: typically

text: exist,

text: and

text: there

text: are

text: explicit

text: R

text: algebraic

text: criteria

text: for

text: recognising

text: and

text: constructing

text: them.

text: For

text: -versality

text: and

text: without

text: the

text: G-action

text: this

text: is

text: the

text: material

text: of

text: elementary

text: catastrophe

text: theory

text: (where

text: versal

text: deformations

text: are

text: more

text: often

text: called

text: universal

text: unfoldings):

text: see

text: for

text: example.

text: Expositions

text: in

text: the

text: G-invariant

text: setting

text: are

text: generally

text: highly

text: technical;

text: the

text: result

text: we

text: need

text: can

text: be

text: found

text: in

text: Theorem

text: Let

text: v

text: v

text: be

text: a

text: set

text: of

text: G-invariant

text: polynomials

text: with

text: the

text: r

text: property

text: that

text: every

text: G-invariant

text: polynomial

text: p

text: can

text: be

text: written

text: in

text: the

text: form

text: p

text: w

text: ε

text: v

text: ε

text: v

text: r

text: r

text: W

text: for

text: some

text: coeﬃcients

text: ε

text: ε

text: R,

text: where

text: w

text: (h)

text: as

text: in

text: Section

text: Then

text: r

text: H

text: k

text: ε

text: v

text: ε

text: v

text: r

text: r

text: R

text: is

text: a

text: G-

text: -versal

text: deformation

text: of

text: h.


text: The Molien function (3) for the natural action δ of D on R2 as the 3 symmetries of an equilateral triangle is easily found to be P 1 (t) = − − (4) δ (1 t2)(1 t3) which conveys the information that a Hilbert basis consists of one poly- nomial of degree 2 and one of degree 3, there being no algebraic relation between them as the terms t2, t3 appear only in the denominator. For suit- able coordinates in R2 the basis elements may be taken to be x2 + y2 and − x3 3xy2. Example Action of SO(3) on V = Sym (R3) 0 ∈ × An element g SO(3) is a 3 3 orthogonal matrix with eigenvalues { ± } 1, e iθ for some angle θ, and for the conjugacy action α˜ the eigenvalues { ± ± } of α˜(g) are 1, e iθ, e 2iθ . To evaluate the Molien series using the integral version of the formula (3) we make use of the Weyl Integral Formula (as described in [6] for example) which shows that such an integral can be de- composed into an integral over a maximal torus in G and over the quotient of the group by this torus. In the case when the integrand is a class function (invariant under conjugation in the group) then the latter integral becomes trivial, and the whole integral reduces to an integral over the maximal torus only, at the cost of introducing a further term into the integrand that we call the Weyl factor. The integral over the circle is expressed in terms of angular variables and can be evaluated using residue calculus.   In our case the maximal ‘torus’ is a circle given by the matrices   1 0 0 − 0 cos θ sin θ 0 sin θ cos θ ≤ for 0 θ < 2π, and integration is over the unit circle regarded as the circle | | z = 1 in the complex plane. The Weyl factor can be found to be − − − 1(1 z)(1 z¯) = (1 cos θ) 2 (cid:90) (cf. [29]) and so the Molien integral (3) for α˜ becomes (cid:81) − P 1 i (1 z)2 dz (t) = − . α˜ 2 2π | | 2 − (1 tzj) z2 z =1 j= 2 P Since the aim is to ﬁnd (t) as an inﬁnite series in t we regard t as a small α˜ | | real variable and in any case with t < 1, this restriction to be exploited 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: Molien

text: function

text: for

text: the

text: natural

text: action

text: δ

text: of

text: D

text: on

text: as

text: the

text: symmetries

text: of

text: an

text: equilateral

text: triangle

text: is

text: easily

text: found

text: to

text: be

text: P

text: (t)

text: δ

text: which

text: conveys

text: the

text: information

text: that

text: a

text: Hilbert

text: basis

text: consists

text: of

text: one

text: poly-

text: nomial

text: of

text: degree

text: and

text: one

text: of

text: degree

text: there

text: being

text: no

text: algebraic

text: relation

text: between

text: them

text: as

text: the

text: terms

text: appear

text: only

text: in

text: the

text: denominator.

text: For

text: suit-

text: able

text: coordinates

text: in

text: the

text: basis

text: elements

text: may

text: be

text: taken

text: to

text: be

text: and

text: Example

text: Action

text: of

text: on

text: V

text: Sym

text: An

text: element

text: g

text: is

text: a

text: orthogonal

text: matrix

text: with

text: eigenvalues

text: e

text: iθ

text: for

text: some

text: angle

text: θ,

text: and

text: for

text: the

text: conjugacy

text: action

text: α˜

text: the

text: eigenvalues

text: of

text: α˜(g)

text: are

text: e

text: iθ,

text: e

text: To

text: evaluate

text: the

text: Molien

text: series

text: using

text: the

text: integral

text: version

text: of

text: the

text: formula

text: we

text: make

text: use

text: of

text: the

text: Weyl

text: Integral

text: Formula

text: (as

text: described

text: in

text: for

text: example)

text: which

text: shows

text: that

text: such

text: an

text: integral

text: can

text: be

text: de-

text: composed

text: into

text: an

text: integral

text: over

text: a

text: maximal

text: torus

text: in

text: G

text: and

text: over

text: the

text: quotient

text: of

text: the

text: group

text: by

text: this

text: torus.

text: In

text: the

text: case

text: when

text: the

text: integrand

text: is

text: a

text: class

text: function

text: (invariant

text: under

text: conjugation

text: in

text: the

text: group)

text: then

text: the

text: latter

text: integral

text: becomes

text: trivial,

text: and

text: the

text: whole

text: integral

text: reduces

text: to

text: an

text: integral

text: over

text: the

text: maximal

text: torus

text: only,

text: at

text: the

text: cost

text: of

text: introducing

text: a

text: further

text: term

text: into

text: the

text: integrand

text: that

text: we

text: call

text: the

text: Weyl

text: factor.

text: The

text: integral

text: over

text: the

text: circle

text: is

text: expressed

text: in

text: terms

text: of

text: angular

text: variables

text: and

text: can

text: be

text: evaluated

text: using

text: residue

text: calculus.

text: In

text: our

text: case

text: the

text: maximal

text: ‘torus’

text: is

text: a

text: circle

text: given

text: by

text: the

text: matrices

text: cos

text: θ

text: sin

text: θ

text: sin

text: θ

text: cos

text: θ

text: for

text: θ

text: and

text: integration

text: is

text: over

text: the

text: unit

text: circle

text: regarded

text: as

text: the

text: circle

text: z

text: in

text: the

text: complex

text: plane.

text: The

text: Weyl

text: factor

text: can

text: be

text: found

text: to

text: be

text: z¯)

text: cos

text: θ)

text: (cf.

text: and

text: so

text: the

text: Molien

text: integral

text: for

text: α˜

text: becomes

text: P

text: i

text: dz

text: (t)

text: α˜

text: 2π

text: tzj)

text: z

text: j=

text: P

text: Since

text: the

text: aim

text: is

text: to

text: ﬁnd

text: (t)

text: as

text: an

text: inﬁnite

text: series

text: in

text: t

text: we

text: regard

text: t

text: as

text: a

text: small

text: α˜

text: real

text: variable

text: and

text: in

text: any

text: case

text: with

text: t

text: this

text: restriction

text: to

text: be

text: exploited

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Therefore if − M := 1( L + L + L ) 1 2 1 2 3 we see that M (B ) = B while M takes A , A , B , B to zero, so that 1 1 1 1 1 2 2 3 M represents orthogonal projection of L(V ) onto the B -axis. Likewise we 1 1 construct M , M representing orthogonal projections to the B - and B - 2 3 2 3 axes r(cid:101)espectively. (cid:101) ∈ Since B and A have the same(cid:101)eigen(cid:101)values there exists N SO(3) such 2 1 that NB = A . As N is an orthogonal transformation of V (Proposition 2) 2 1 − ∈ it follows that the operator N := NM N 1 L(V ) is orthogonal projection 2 2 onto the A -axis that therefore annihilates B , B , B but does not annihilate 1 − 1 2 3 A : in fact N A = 1A . However, we can also construct the analogous op- 2 2 2 2 1 erator N that exchanges the roles of A , A , so suitable linear combinations 1 1 2 of N , N r(cid:101)epresent orthogonal projections to the A - and A -axes. Finally, 1 2 1 2 since all of A , . . . , B are conjugate to e(cid:101)ach other, a linear combination of 1 ∈ 3 operators R for R SO(3) can be found to take any one of these ﬁve to any (cid:101) { ∈ } other while annihilating the rest. Thus R : R SO(3) spans L(V ). { ∈ } It remains to show that the set R : R SO(3) does not lie in a proper aﬃne subspace of L(V). For this, we observe that if         1 0 0 0 1 0 0 0 1 I = K = , I = , I = 1 0 0 1 0 2 0 0 1 3 1 0 0 0 0 1 1 0 0 0 1 0 (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) then (cid:101) (K + K + K + K )(I + I + I ) = 0 0 1 2 3 1 2 3 ∈ and so there exists a linear combination of elements N L(V ) with positive coeﬃcients that gives zero. This completes the proof. References [1] Allender, D, and Longa, L., Landau-de Gennes theory of biaxial nemat- ics reexamined, Phys. Rev. E 78 (2008), 11704. [2] Arnold, V.I., Wave front evolution and equivariant Morse lemma, Comm. Pure Appl. Math. 29 (1976), 557–582. [3] Arnold, V. I., Goryunov, V. V., Lyashko, O. V. and Vasiliev, V. A., Sin- gularity Theory II. Classiﬁcation and Applications Dynamical systems VIII, Encyclopædia Math. Sci, 39, Springer 1993. [4] Arnold, V.I., Gusein-Zade, S.M. and Varchenko, A.N., Singularities of Diﬀerentiable Maps, Vol.I, Birkh¨auser 1985. 72 ##LTLine## ##LTLine##

text: Therefore

text: if

text: M

text: L

text: L

text: L

text: we

text: see

text: that

text: M

text: (B

text: B

text: while

text: M

text: takes

text: A

text: A

text: B

text: B

text: to

text: zero,

text: so

text: that

text: M

text: represents

text: orthogonal

text: projection

text: of

text: L(V

text: onto

text: the

text: B

text: -axis.

text: Likewise

text: we

text: construct

text: M

text: M

text: representing

text: orthogonal

text: projections

text: to

text: the

text: B

text: and

text: B

text: axes

text: Since

text: B

text: and

text: A

text: have

text: the

text: there

text: exists

text: N

text: such

text: that

text: NB

text: A

text: As

text: N

text: is

text: an

text: orthogonal

text: transformation

text: of

text: V

text: (Proposition

text: it

text: follows

text: that

text: the

text: operator

text: N

text: NM

text: N

text: L(V

text: is

text: orthogonal

text: projection

text: onto

text: the

text: A

text: -axis

text: that

text: therefore

text: annihilates

text: B

text: B

text: B

text: but

text: does

text: not

text: annihilate

text: A

text: in

text: fact

text: N

text: A

text: However,

text: we

text: can

text: also

text: construct

text: the

text: analogous

text: op-

text: erator

text: N

text: that

text: exchanges

text: the

text: roles

text: of

text: A

text: A

text: so

text: suitable

text: linear

text: combinations

text: of

text: N

text: N

text: orthogonal

text: projections

text: to

text: the

text: A

text: and

text: A

text: -axes.

text: Finally,

text: since

text: all

text: of

text: A

text: B

text: are

text: conjugate

text: to

text: other,

text: a

text: linear

text: combination

text: of

text: operators

text: R

text: for

text: R

text: can

text: be

text: found

text: to

text: take

text: any

text: one

text: of

text: these

text: ﬁve

text: to

text: any

text: other

text: while

text: annihilating

text: the

text: rest.

text: Thus

text: R

text: R

text: spans

text: L(V

text: It

text: remains

text: to

text: show

text: that

text: the

text: set

text: R

text: R

text: does

text: not

text: lie

text: in

text: a

text: proper

text: aﬃne

text: subspace

text: of

text: L(V).

text: For

text: this,

text: we

text: observe

text: that

text: if

text: I

text: K

text: I

text: I

text: then

text: (K

text: K

text: K

text: K

text: )(I

text: I

text: I

text: and

text: so

text: there

text: exists

text: a

text: linear

text: combination

text: of

text: elements

text: N

text: L(V

text: with

text: positive

text: coeﬃcients

text: that

text: gives

text: zero.

text: This

text: completes

text: the

text: proof.

text: References

text: Allender,

text: D,

text: and

text: Longa,

text: L.,

text: Landau-de

text: Gennes

text: theory

text: of

text: biaxial

text: nemat-

text: ics

text: reexamined,

text: Phys.

text: Rev.

text: E

text: Arnold,

text: V.I.,

text: Wave

text: front

text: evolution

text: and

text: equivariant

text: Morse

text: lemma,

text: Comm.

text: Pure

text: Appl.

text: Math.

text: Arnold,

text: V.

text: I.,

text: Goryunov,

text: V.

text: V.,

text: Lyashko,

text: O.

text: V.

text: and

text: Vasiliev,

text: V.

text: A.,

text: Sin-

text: gularity

text: Theory

text: II.

text: Classiﬁcation

text: and

text: Applications

text: Dynamical

text: systems

text: VIII,

text: Encyclopædia

text: Math.

text: Sci,

text: Springer

text: Arnold,

text: V.I.,

text: Gusein-Zade,

text: S.M.

text: and

text: Varchenko,

text: A.N.,

text: Singularities

text: of

text: Diﬀerentiable

text: Maps,

text: Vol.I,

text: Birkh¨auser

text: ##LTLine##

text: ##LTLine##


text: [49] Turzi, S.S. and Sluckin, T.J., Symmetry adapted molecular-ﬁeld theory for thermotropic biaxial nematic liquid crystals and its expansion at low temperature, SIAM J. Appl. Math. 73 (2013), 1139–1163. [50] Wasserman, A., Equivariant diﬀerential topology, Topology 8 (1969), 127–150. [51] Wasserman, G., (r, s)-unfolding theory, Stability of unfoldings in space and time, Acta Mathematica 135 (1975), 57–128. [52] Zheng, X. and Palﬀy-Muhoray, P., One order parameter tensor mean ﬁeld theory for biaxial liquid crystals, Discrete Contin. Dyn. Syst. Ser. B 15 (2011), 475–490. 76

label: reference

text: Turzi,

text: S.S.

text: and

text: Sluckin,

text: T.J.,

text: Symmetry

text: adapted

text: molecular-ﬁeld

text: theory

text: for

text: thermotropic

text: biaxial

text: nematic

text: liquid

text: crystals

text: and

text: its

text: expansion

text: at

text: low

text: temperature,

text: SIAM

text: J.

text: Appl.

text: Math.

text: Wasserman,

text: A.,

text: Equivariant

text: diﬀerential

text: topology,

text: Topology

text: Wasserman,

text: G.,

text: (r,

text: s)-unfolding

text: theory,

text: Stability

text: of

text: unfoldings

text: in

text: space

text: and

text: time,

text: Acta

text: Mathematica

text: Zheng,

text: X.

text: and

text: Palﬀy-Muhoray,

text: P.,

text: One

text: order

text: parameter

text: tensor

text: mean

text: ﬁeld

text: theory

text: for

text: biaxial

text: liquid

text: crystals,

text: Discrete

text: Contin.

text: Dyn.

text: Syst.

text: Ser.

text: B


text: √ when evaluating residues. In our case the relevant residues are at z = 0 and ± at z = t, t and easy calculations give P 1 (t) = − − (5) α˜ (1 t2)(1 t3) which shows that the ring of α˜-invariant functions on V has two generators of homogeneous degree 2, 3 respectively. These may be taken to be tr Q2, tr Q3 although the latter could be replaced by det Q since the Cayley-Hamilton equation shows that tr Q3 = 3 det Q when trace Q = 0. P It is no coincidence that the Molien function (t) coincides with that of α˜ the natural representation δ of D on R2. This is because of special properties 3 of α˜: every α˜-invariant function on V is uniquely determined by its values on the 2-dimensional subspace D of V consisting of diagonal matrices, and the δ-orbit of a point in D is precisely the intersection of its α˜ orbit with D. 3 The KKLS model We now turn to the main focus of this paper which is the KKLS model [27, 32] for the construction of a free energy function on an appropriate space of order parameters. Following [27, 32] we begin with the key assumption that properties of a molecule relevant to its interactions with its neighbours are characterised by an n-tuple of real numbers (orientational order or ‘shape’ ∈ parameters) Q = (Q , . . . , Q ) Rn, and that each element Ω of the rotation 1 n group SO(3) when applied to a given molecule induces a linear transforma- tion Q(Ω) of Rn that is a measure of the interaction between a molecule and its rotated counterpart. This places us immediately in the mathematical setting of an action α of G = SO(3) on E = Rn as discussed in Section , ∈ although given Ω SO(3) we write α(Ω)Q rather than Q(Ω). Here and throughout we assume homogeneity of the medium, so that local properties of the liquid crystal are the same everywhere and spatial derivatives of Q do not come into play. A well-established technique to ﬁnesse the intractable task of calculating mutual interactions of huge numbers of molecules is to apply instead a mean ﬁeld theory where each particular molecule is seen as reacting to the average inﬂuence of all the other molecules. Evaluating this average corresponds to integrating α(Ω) with respect to an appropriate probability distribution on the group SO(3), this being determined by the relevant physical theory. The 9 ##LTLine## ##LTLine##

text: when

text: evaluating

text: residues.

text: In

text: our

text: case

text: the

text: relevant

text: residues

text: are

text: at

text: z

text: and

text: at

text: z

text: t,

text: t

text: and

text: easy

text: calculations

text: give

text: P

text: (t)

text: α˜

text: which

text: shows

text: that

text: the

text: ring

text: of

text: α˜-invariant

text: functions

text: on

text: V

text: has

text: two

text: generators

text: of

text: homogeneous

text: degree

text: respectively.

text: These

text: may

text: be

text: taken

text: to

text: be

text: tr

text: tr

text: although

text: the

text: latter

text: could

text: be

text: replaced

text: by

text: det

text: Q

text: since

text: the

text: Cayley-Hamilton

text: equation

text: shows

text: that

text: tr

text: det

text: Q

text: when

text: trace

text: Q

text: P

text: It

text: is

text: no

text: coincidence

text: that

text: the

text: Molien

text: function

text: (t)

text: coincides

text: with

text: that

text: of

text: α˜

text: the

text: natural

text: representation

text: δ

text: of

text: D

text: on

text: This

text: is

text: because

text: of

text: special

text: properties

text: of

text: α˜:

text: every

text: α˜-invariant

text: function

text: on

text: V

text: is

text: uniquely

text: determined

text: by

text: its

text: values

text: on

text: the

text: subspace

text: D

text: of

text: V

text: consisting

text: of

text: diagonal

text: matrices,

text: and

text: the

text: δ-orbit

text: of

text: a

text: point

text: in

text: D

text: is

text: precisely

text: the

text: intersection

text: of

text: its

text: α˜

text: orbit

text: with

text: D.

text: The

text: KKLS

text: model

text: We

text: now

text: turn

text: to

text: the

text: main

text: focus

text: of

text: this

text: paper

text: which

text: is

text: the

text: KKLS

text: model

text: for

text: the

text: construction

text: of

text: a

text: free

text: energy

text: function

text: on

text: an

text: appropriate

text: space

text: of

text: order

text: parameters.

text: Following

text: we

text: begin

text: with

text: the

text: key

text: assumption

text: that

text: properties

text: of

text: a

text: molecule

text: relevant

text: to

text: its

text: interactions

text: with

text: its

text: neighbours

text: are

text: characterised

text: by

text: an

text: n-tuple

text: of

text: real

text: numbers

text: (orientational

text: order

text: or

text: ‘shape’

text: parameters)

text: Q

text: (Q

text: Q

text: Rn,

text: and

text: that

text: each

text: element

text: Ω

text: of

text: the

text: rotation

text: n

text: group

text: when

text: applied

text: to

text: a

text: given

text: molecule

text: induces

text: a

text: linear

text: transforma-

text: tion

text: Q(Ω)

text: of

text: Rn

text: that

text: is

text: a

text: measure

text: of

text: the

text: interaction

text: between

text: a

text: molecule

text: and

text: its

text: rotated

text: counterpart.

text: This

text: places

text: us

text: immediately

text: in

text: the

text: mathematical

text: setting

text: of

text: an

text: action

text: α

text: of

text: G

text: on

text: E

text: Rn

text: as

text: discussed

text: in

text: Section

text: although

text: given

text: Ω

text: we

text: write

text: α(Ω)Q

text: rather

text: than

text: Q(Ω).

text: Here

text: and

text: throughout

text: we

text: assume

text: homogeneity

text: of

text: the

text: medium,

text: so

text: that

text: local

text: properties

text: of

text: the

text: liquid

text: crystal

text: are

text: the

text: same

text: everywhere

text: and

text: spatial

text: derivatives

text: of

text: Q

text: do

text: not

text: come

text: into

text: play.

text: A

text: well-established

text: technique

text: to

text: ﬁnesse

text: the

text: intractable

text: task

text: of

text: calculating

text: mutual

text: interactions

text: of

text: huge

text: numbers

text: of

text: molecules

text: is

text: to

text: apply

text: instead

text: a

text: mean

text: ﬁeld

text: theory

text: where

text: each

text: particular

text: molecule

text: is

text: seen

text: as

text: reacting

text: to

text: the

text: average

text: inﬂuence

text: of

text: all

text: the

text: other

text: molecules.

text: Evaluating

text: this

text: average

text: corresponds

text: to

text: integrating

text: α(Ω)

text: with

text: respect

text: to

text: an

text: appropriate

text: probability

text: distribution

text: on

text: the

text: group

text: this

text: being

text: determined

text: by

text: the

text: relevant

text: physical

text: theory.

text: The

text: ##LTLine##

text: ##LTLine##


text: (cid:104) (cid:105) ∈ components of the average α L(Rn) with respect to a suitably chosen basis for Rn are the order parameters for the liquid crystal. In an equilibrium state the alignment of a particular molecule must neces- sarily be consistent with the creation of the mean ﬁeld in the ﬁrst place, thus posing a signiﬁcant mathematical problem. We ﬁrst formulate this problem in a general coordinate-free setting, while following the paradigm of [27, 32], and we then apply it in the context of molecules with eﬀective D symme- 2h try (Scho¨nﬂies notation) or Z3 symmetry (algebraic notation) where each Z 2 2 factor acts by rotation through angle π about one of the coordinate axes. As far as interactions are concerned the molecule can be represented by an ellipsoid whose departure from sphericity is given by a∼real symmetric × ∈ tracele∈ss 3 3∼susceptibility matrix Q V = Sym0(R3) = R5, therefore α(Ω) L(V ) = R25. As we shall see, however, further assumptions can often be made that enable reduction to much lower dimensions. In the KKLS and other models for equilibrium states (phases) of liq- uid crystals, an equilibrium corresponds to a critical point of a suitable free energy function on the space of order parameters, with stable phases corre- sponding to global or absolute minima. The KKLS model is characterised by the fact that the free energy has two components: a molecular interaction or Hamiltonian term and an entropy term. We focus ﬁrst on the latter and return afterwards to the former. 3.1 The entropy component We follow the general formulation of [27, 32], expressed here in the more abstract context of Section . Consider an arbitrary compact group G with orthogonal action α on a linear space E, together with a probability distribution φ on G expressed in terms of the action α: thus φ has the form φ(g) = ψ(α(g)) for some suitable → ψ : L(E) R which we assume smooth. There are two important physic(cid:90)al quantities to be derived from φ. The ﬁrst is the ensemble average (cid:104) (cid:105) ∈ W := α = α(g) φ(g) L(E) (6) φ φ G ∈ which represents the average α-action of g G with respect to the par- ticular probability distribution φ, the integral being taken over the natural normalised invariant (Haar) measure on G. This is a quantity that in princi- ple might be measured in the laboratory, whereas the probability distribution 10

text: components

text: of

text: the

text: average

text: α

text: L(Rn)

text: with

text: respect

text: to

text: a

text: suitably

text: chosen

text: basis

text: for

text: Rn

text: are

text: the

text: order

text: parameters

text: for

text: the

text: liquid

text: crystal.

text: In

text: an

text: equilibrium

text: state

text: the

text: alignment

text: of

text: a

text: particular

text: molecule

text: must

text: neces-

text: sarily

text: be

text: consistent

text: with

text: the

text: creation

text: of

text: the

text: mean

text: ﬁeld

text: in

text: the

text: ﬁrst

text: place,

text: thus

text: posing

text: a

text: signiﬁcant

text: mathematical

text: problem.

text: We

text: ﬁrst

text: formulate

text: this

text: problem

text: in

text: a

text: general

text: coordinate-free

text: setting,

text: while

text: following

text: the

text: paradigm

text: of

text: and

text: we

text: then

text: apply

text: it

text: in

text: the

text: context

text: of

text: molecules

text: with

text: eﬀective

text: D

text: symme-

text: try

text: (Scho¨nﬂies

text: notation)

text: or

text: symmetry

text: (algebraic

text: notation)

text: where

text: each

text: Z

text: factor

text: acts

text: by

text: rotation

text: through

text: angle

text: π

text: about

text: one

text: of

text: the

text: coordinate

text: axes.

text: As

text: far

text: as

text: interactions

text: are

text: concerned

text: the

text: molecule

text: can

text: be

text: represented

text: by

text: an

text: ellipsoid

text: whose

text: departure

text: from

text: sphericity

text: is

text: given

text: by

text: a∼real

text: symmetric

text: tracele∈ss

text: matrix

text: Q

text: V

text: therefore

text: α(Ω)

text: L(V

text: As

text: we

text: shall

text: see,

text: however,

text: further

text: assumptions

text: can

text: often

text: be

text: made

text: that

text: enable

text: reduction

text: to

text: much

text: lower

text: dimensions.

text: In

text: the

text: KKLS

text: and

text: other

text: models

text: for

text: equilibrium

text: states

text: (phases)

text: of

text: liq-

text: uid

text: crystals,

text: an

text: equilibrium

text: corresponds

text: to

text: a

text: critical

text: point

text: of

text: a

text: suitable

text: free

text: energy

text: function

text: on

text: the

text: space

text: of

text: order

text: parameters,

text: with

text: stable

text: phases

text: corre-

text: sponding

text: to

text: global

text: or

text: absolute

text: minima.

text: The

text: KKLS

text: model

text: is

text: characterised

text: by

text: the

text: fact

text: that

text: the

text: free

text: energy

text: has

text: two

text: components:

text: a

text: molecular

text: interaction

text: or

text: Hamiltonian

text: term

text: and

text: an

text: entropy

text: term.

text: We

text: focus

text: ﬁrst

text: on

text: the

text: latter

text: and

text: return

text: afterwards

text: to

text: the

text: former.

text: The

text: entropy

text: component

text: We

text: follow

text: the

text: general

text: formulation

text: of

text: expressed

text: here

text: in

text: the

text: more

text: abstract

text: context

text: of

text: Section

text: Consider

text: an

text: arbitrary

text: compact

text: group

text: G

text: with

text: orthogonal

text: action

text: α

text: on

text: a

text: linear

text: space

text: E,

text: together

text: with

text: a

text: probability

text: distribution

text: φ

text: on

text: G

text: expressed

text: in

text: terms

text: of

text: the

text: action

text: α:

text: thus

text: φ

text: has

text: the

text: form

text: φ(g)

text: ψ(α(g))

text: for

text: some

text: suitable

text: ψ

text: L(E)

text: R

text: which

text: we

text: assume

text: smooth.

text: There

text: are

text: two

text: important

text: quantities

text: to

text: be

text: derived

text: from

text: φ.

text: The

text: ﬁrst

text: is

text: the

text: ensemble

text: average

text: W

text: α

text: α(g)

text: φ(g)

text: L(E)

text: φ

text: φ

text: G

text: which

text: represents

text: the

text: average

text: α-action

text: of

text: g

text: G

text: with

text: respect

text: to

text: the

text: par-

text: ticular

text: probability

text: distribution

text: φ,

text: the

text: integral

text: being

text: taken

text: over

text: the

text: natural

text: normalised

text: invariant

text: (Haar)

text: measure

text: on

text: G.

text: This

text: is

text: a

text: quantity

text: that

text: in

text: princi-

text: ple

text: might

text: be

text: measured

text: in

text: the

text: laboratory,

text: whereas

text: the

text: probability

text: distribution


text: ∗ Predicting Multi-actor collaborations using Hypergraphs † † † Ankit Sharma Jaideep Srivastava Abhishek Chandra May 26, 2020 Abstract In the domain of Social Science, a lot of studies have Social networks are now ubiquitous and most of them con- been conducted to understand how groups form, their tain interactions involving multiple actors (groups) like au- static as well as dynamic attributes and structures, and thor collaborations, teams or emails in an organizations, etc. how they evolve over time [2]. The research collabora- Hypergraphs are natural structures to eﬀectively capture multi-actor interactions which conventional dyadic graphs tions in scientiﬁc community are an excellent example of fail to capture. In this work the problem of predicting col- social networks in which individuals of various expertise laborations is addressed while modeling the collaboration collaborate to solve a research problem. Collaboration network as a hypergraph network. The problem of predict- ing future multi-actor collaboration is mapped to hyperedge networks from scientiﬁc research community have been prediction problem. Given that the higher order edge predic- extensively used for studying team dynamics [3][4][5]. tion is an inherently hard problem, in this work we restrict Group dynamics has real life applications as well for ex- to the task of predicting edges (collaborations) that have already been observed in past. In this work, we propose ample in building emergency response teams for natural a novel use of hyperincidence temporal tensors to capture disasters management, automation of team selection for time varying hypergraphs and provides a tensor decompo- military operations, etc. sition based prediction algorithm. We quantitatively com- pare the performance of the hypergraphs based approach with the conventional dyadic graph based approach. Our hypothesis that hypergraphs preserve the information that simple graphs destroy is corroborated by experiments using author collaboration network from the DBLP dataset. Our results demonstrate the strength of hypergraph based ap- proach to predict higher order collaborations (size>4) which is very diﬃcult using dyadic graph based approach. More- over, while predicting collaborations of size>2 hypergraphs in most cases provide better results with an av∈er{age increas}e of approx. 45% in F-Score for diﬀerent sizes 3, 4, 5, 6, 7 (Figure 6). Figure 2: Bipartite of Keywords. Collaboration networks, social networks, Figure 1: Hypergraph hypergraph (Fig.1) link prediction, tensors, hypergraphs, team formation. The above examples reveal that there can be mul- 1 Introduction tiple overlapping collaborations which form a network of collaborations. Modeling such collaborations in dy- The problem of understanding group dynamics is namic settings where relationship between actors is central to the ﬁeld of Social Sciences. Moreover, the evolving over time is a challenging task. Unfortunately, increasing use of internet has led to an exponential in- most of the prior research in social network analy- crease in amount of online interaction data. As exam- sis deals with dyadic interactions or small well-deﬁned ples, social networking sites like Facebook or Twitter, groups [6] rather than at the group level. There are group communication tools like Skype, Google Hangout, some studies that have dealt with group interactions by Google Docs, Massive Online multi-player games such collapsing the group into dyadic links [4] and therefore, as World of Warcraft, etc., are generating social net- fail to keep the group level information intact. Ghoshal working data at a massive scale. These social datasets et. al. [7] have used tripartite regular hypergraph whcih provides minute by minute account of interaction along captures folksonomy data but is too restrictive to cap- with the structure and the content of these relationships ture variable size social collaborations. Guimer`a et al. [´azquez28122004aa´azquez28122004]. [8] attempts attempts to model group using node and group attributes which can explain the network struc- ∗ ture but fails to deal with individual group evolution. Supported by NSF grant EAGER 1248169. † Bipartite graphs (ﬁgure 2) as network models have also Dept. of Computer Science, University of Minnesota, { } ankit,srivasta,chandra @cs.umn.edu been used to capture groups [9] where one set of nodes ##LTFigure## ##LTFigure## ##LTLine##

label: title

text: Predicting

text: Multi-actor

text: collaborations

text: using

text: Hypergraphs

label: author

text: Ankit

text: Sharma

text: Jaideep

text: Srivastava

text: Abhishek

text: Chandra

text: May

text: Abstract

text: In

text: the

text: domain

text: of

text: Social

text: Science,

text: a

text: lot

text: of

text: studies

text: have

text: Social

label: abstract

text: networks

text: are

text: now

text: ubiquitous

text: and

text: most

text: of

text: them

text: con-

text: been

text: conducted

text: to

text: understand

text: how

text: groups

text: form,

text: their

text: tain

text: interactions

text: involving

text: multiple

text: actors

text: (groups)

text: like

text: au-

text: static

text: as

text: well

text: as

text: dynamic

text: attributes

text: and

text: structures,

text: and

text: thor

text: collaborations,

text: teams

text: or

text: emails

text: in

text: an

text: organizations,

text: etc.

text: how

text: they

text: evolve

text: over

text: time

text: The

text: research

text: collabora-

text: Hypergraphs

text: are

text: natural

text: structures

text: to

text: eﬀectively

text: capture

text: multi-actor

text: interactions

text: which

text: conventional

text: dyadic

text: graphs

text: tions

text: in

text: scientiﬁc

text: community

text: are

text: an

text: excellent

text: example

text: of

text: fail

text: to

text: capture.

text: In

text: this

text: work

text: the

text: problem

text: of

text: predicting

text: col-

text: social

text: networks

text: in

text: which

text: individuals

text: of

text: various

text: expertise

text: laborations

text: is

text: addressed

text: while

text: modeling

text: the

text: collaboration

text: collaborate

text: to

text: solve

text: a

text: research

text: problem.

text: Collaboration

text: network

text: as

text: a

text: hypergraph

text: network.

text: The

text: problem

text: of

text: predict-

text: ing

text: future

text: multi-actor

text: collaboration

text: is

text: mapped

text: to

text: hyperedge

text: networks

text: from

text: scientiﬁc

text: research

text: community

text: have

text: been

text: prediction

text: problem.

text: Given

text: that

text: the

text: higher

text: order

text: edge

text: predic-

text: extensively

text: used

text: for

text: studying

text: team

text: dynamics

text: tion

text: is

text: an

text: inherently

text: hard

text: problem,

text: in

text: this

text: work

text: we

text: restrict

text: Group

text: dynamics

text: has

text: real

text: life

text: applications

text: as

text: well

text: for

text: ex-

text: to

text: the

text: task

text: of

text: predicting

text: edges

text: (collaborations)

text: that

text: have

text: already

text: been

text: observed

text: in

text: past.

text: In

text: this

text: work,

text: we

text: propose

text: ample

text: in

text: building

text: emergency

text: response

text: teams

text: for

text: natural

text: a

text: novel

text: use

text: of

text: hyperincidence

text: temporal

text: tensors

text: to

text: capture

text: disasters

text: management,

text: automation

text: of

text: team

text: selection

text: for

text: time

text: varying

text: hypergraphs

text: and

text: provides

text: a

text: tensor

text: decompo-

text: military

text: operations,

text: etc.

text: sition

text: based

text: prediction

text: algorithm.

text: We

text: quantitatively

text: com-

text: pare

text: the

text: performance

text: of

text: the

text: hypergraphs

text: based

text: approach

text: with

text: the

text: conventional

text: dyadic

text: graph

text: based

text: approach.

text: Our

text: hypothesis

text: that

text: hypergraphs

text: preserve

text: the

text: information

text: that

text: simple

text: graphs

text: destroy

text: is

text: corroborated

text: by

text: experiments

text: using

text: author

text: collaboration

text: network

text: from

text: the

text: DBLP

text: dataset.

text: Our

text: results

text: demonstrate

text: the

text: strength

text: of

text: hypergraph

text: based

text: ap-

text: proach

text: to

text: predict

text: higher

text: order

text: collaborations

text: which

text: is

text: very

text: diﬃcult

text: using

text: dyadic

text: graph

text: based

text: approach.

text: More-

text: over,

text: while

text: predicting

text: collaborations

text: of

text: hypergraphs

text: in

text: most

text: cases

text: provide

text: better

text: results

text: with

text: an

text: av∈er{age

text: increas}e

text: of

text: approx.

text: in

text: F-Score

text: for

text: diﬀerent

text: sizes

text: (Figure

text: Figure

text: Bipartite

text: of

text: Keywords.

text: Collaboration

text: networks,

text: social

text: networks,

text: Figure

text: Hypergraph

text: hypergraph

text: link

text: prediction,

text: tensors,

text: hypergraphs,

text: team

text: formation.

text: The

text: above

text: examples

text: reveal

text: that

text: there

text: can

text: be

text: mul-

text: Introduction

text: tiple

text: overlapping

text: collaborations

text: which

text: form

text: a

text: network

text: of

text: collaborations.

text: Modeling

text: such

text: collaborations

text: in

text: dy-

text: The

text: problem

text: of

text: understanding

text: group

text: dynamics

text: is

text: namic

text: settings

text: where

text: relationship

text: between

text: actors

text: is

text: central

text: to

text: the

text: ﬁeld

text: of

text: Social

text: Sciences.

text: Moreover,

text: the

text: evolving

text: over

text: time

text: is

text: a

text: challenging

text: task.

text: Unfortunately,

text: increasing

text: use

text: of

text: internet

text: has

text: led

text: to

text: an

text: exponential

text: in-

text: most

text: of

text: the

text: prior

text: research

text: in

text: social

text: network

text: analy-

text: crease

text: in

text: amount

text: of

text: online

text: interaction

text: data.

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text: sis

text: deals

text: with

text: dyadic

text: interactions

text: or

text: small

text: well-deﬁned

text: ples,

text: social

text: networking

text: sites

text: like

text: Facebook

text: or

text: Twitter,

text: groups

text: rather

text: than

text: at

text: the

text: group

text: level.

text: There

text: are

text: group

text: communication

text: tools

text: like

text: Skype,

text: Google

text: Hangout,

text: some

text: studies

text: that

text: have

text: dealt

text: with

text: group

text: interactions

text: by

text: Google

text: Docs,

text: Massive

text: Online

text: multi-player

text: games

text: such

text: collapsing

text: the

text: group

text: into

text: dyadic

text: links

text: and

text: therefore,

text: as

text: World

text: of

text: Warcraft,

text: etc.,

text: are

text: generating

text: social

text: net-

text: fail

text: to

text: keep

text: the

text: group

text: level

text: information

text: intact.

text: Ghoshal

text: working

text: data

text: at

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text: massive

text: scale.

text: These

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text: datasets

text: et.

text: al.

text: have

text: used

text: tripartite

text: regular

text: hypergraph

text: whcih

text: provides

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text: account

text: of

text: interaction

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text: captures

text: folksonomy

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text: but

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text: size

text: social

text: collaborations.

text: Guimer`a

text: et

text: al.

text: attempts

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text: model

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text: using

text: node

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text: attributes

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text: Supported

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text: NSF

text: grant

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text: Dept.

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text: Science,

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text: Minnesota,

text: ankit,srivasta,chandra

text: @cs.umn.edu

text: been

text: used

text: to

text: capture

text: groups

text: where

text: one

text: set

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text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##