Physical Applications: Convexity and Legendretransforms

A.C. Maggs

CNRS+ESPCI, Paris

June 2016

Physical applications

I Landau theories for dielectric response

I Asymmetric electrolytes with finite volume

I numerical minimization

Exotic dielectric media, water �(q)

Figure: MD- Bopp, Kornyshev, Sutmann, 1995

Paradox: is � always positive? See Kirzhnits.

Non-local dielectrics

How to produce �(k)

I Constraintdiv D = ρ

I Field energy, (Landau-Ginzburg) for polarization P

U =(

E︷ ︸︸ ︷D− P)2

2+

P2

2χ+κp2

(div P)2

Water requires κp < 0

�(k) = 1 +χ0

1 + k2/k20

Variational equations

A =(Dr − Pr)2

2+

Prχ−1r,r′Pr′

2− φ(div D− ρ)

Take derivatives:

δP : − [D− P] + χ−1P = 0δD : + [D− P]︸ ︷︷ ︸

=E

+∇φ = 0

Or

δP : P(r) =

∫χr,r′E(r

′)

δD : E = −∇φ

Thus in Fourier space

�(q) = 1 + χ(q)

Stability

A =(Dr − Pr)2

2+

Pr χ−1r,r′ Pr′

2− φ(div D− ρ)

Consider coefficient of P2:

UP =Pq(1 + χ−1q )Pq

2≥ 0

Grey zone forbidden

�q = 1 + χq

Either �(q) > 1 Or �(q) < 0

Simplest Landau for negative �

G =κ

2P2−α

2(div P)2+

β

2(∇div P)2

with

minq

[1 + (κ− q2α + βq4)

]> 0

giving

� = 1 +1

κ− q2α + βq4 0 0.5 1 1.5 2−15−10

−5

0

5

10

15

20

25

Interpretation

Anti-correlation of hydrogen bond networks?

Asymmetric electrolytes

I Effects of finite volume - ionic liquids

I lattice models

I off-lattice formulation

Asymmetric Excluded volume in Poisson-Boltzmann

I Standard entropy function available for symmetric systemsbased on lattice theory

I Asymmetric generalizations have problematic limitsI Two solutions

I Recognise that Flory-Huggins entropy has good limitsI Construct theory for an off-lattice equation of state –

asymmetric Carnahan-Starling

More Legendre transforms, starting from general bulk freeenergy

Full coupled bulk-electrostatic problem:

F =∫Vd3r (f (c1, c2)− (µ1 − q1ψ)c1 − (µ2 + q2ψ)c2) +

+

∫Vd3r

(D2

2ε− ψ∇ ·D

)Double Legendre transform of free energy gives pressure

L12[f (c1, c2)]→ −p(µ1, µ2),

Gibbs-Duhem applied to the Grand potential.Giving generalized PB functional:

F [ψ] = −∫Vd3r

(12�(∇ψ)

2 + p(µ1 − q1ψ, µ2 + q2ψ)).

ApplicationsLattice gas, only log divergence at close packingFlory-Huggins -

βf (φ1, φ2) = φ1 log(φ1)/M + φ2 log(φ2)

Carnahan-Sterling (single component)

pV

NkBT=

1 + η + η2 − η3

(1− η)3η = c/c0

near close packing

f (c) =c0kBT

(1− c/c0)2

Legendre transform applied near close packing:

−p(ψ) = qc0ψ − γψ2/3

Compressibility gives very slow, power-law convergence to latticegas expression

Asymmetric hard spheres: with Podgornik

Asymmetric Carnahan-Starling equation of state. Potential or fieldformulation. p(ψ)

-50 -40 -30 -20 -10 0 10 20-20

0

20

40

60

80

100

9-3 -2 -1 0 1 2 3 4 5 6 7

L(F

)(9)

0

20

40

60

80

100

120

140

Full solution of p(ψ) : local charge density near electrode

−q = dp(ψ)/dψ ∼ 1− α/ψ1/3, compared with lattice gas

0 0.2 0.4 0.6 0.8 11=(-eA)1=3

0

0.2

0.4

0.6

0.8

1

1.2

(1=Z

2ec

0)d

p=dA

=!

q

Charged polymer: Numerical optimization

Model of charged polymer within a thin protein shell - Podgornik

βf (Ψ, φ) =a2

6(∇Ψ)2 + vΨ4 − φΨ2 + . . .

compare four minimizers

I

F1 = (∂f /∂φ)2 + (∂f /∂Ψ)2

I High order functional in (∇2φ)2

I Legendre convexification

I alternating minimization/maximization

Numerical results

N0 100 200 300 400

L1

10 -6

10 -4

10 -2

10 0

10 2

N10 1 10 2

t

10 0

10 1

10 2

10 3

10 4

10 5

Figure: residual errors and stopping time, blue:legendre , green squaredgradient, black alternating nested optimization, red generalized scalarfunctional.

Orland-Netz equation

∇2Φ− Λe−Ξc(r)/2 sinh Φ = −2ρf (r),

[∇2 − Λe−Ξc(r)/2 cosh Φ

]G (r, r′) = −4πδ(r − r′),

c(r) = limr→r′

[G (r, r′)− 1

|r − r′|

],

implicit matrix equations for a green function

Fluctuations in dual Poisson-Boltzmann

Fluctuation enhanced PB contains interesting physics - imagecharges, self-energies in non-homogeneous dielectric backgrounds

With Zhenli Xu we have a matrix/iterative solver for theself-consistent problem for Ξ < 5

Fluctuations in dual Poisson-Boltzmann

f = ρeφ− 2c0kT cosh (βqφ)−�

2(grad φ)2

One loop correction from: – includes Born energy,

log det{−div �grad + 2c0q2β cosh qβφ

}Equivalent form in terms of D

f =D2

2�+ L(cosh)[div D− ρ]

With L(cosh) the Legendre transform of CoshOne loop correction from

log det

{1

�(r)+∇∇L(2)(cosh)

}Are these related?

Determinant identities

At first sight these determinants seem very different they act onspaces of different dimensions N × N and 3N × 3N

Determinant identities

Compare |−div �(r)grad + c(r)| and∣∣∣ 1�(r) +−grad 1c(r)div ∣∣∣ |c | |�|

Use |c |∣∣AAT + 1∣∣ and ∣∣1 + ATA∣∣ ∣∣1� ∣∣

with A = 1√cdiv√� and AT = −

√�grad 1√

c

Proof: by singular values – A = UΣV∣∣1 + AAT ∣∣ = ∣∣1 + UΣΣTU∗∣∣ = ∣∣1 + ΣΣT ∣∣ = ∣∣1 + ΣTΣ∣∣Extra dimensions in matrix have unity on diagonal, samedeterminantSelf consistent one loop? relation between at one loop between φand D.

Dynamic casimir

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□ N=1000♢ N=2000○ N=4000

0.01 0.10 1 10 100

-0.20.00.2

0.4

0.6

0.8

1.0

v/Dm

〈±F(v,L)〉/F C