FrancescoCaravenna APolymerinaMulti-InterfaceMedium

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Introduction The Model Free Energy Path Behavior The Proof

A Polymer in a Multi-Interface Medium

Francesco Caravenna

Università degli Studi di Milano-Bicocca Joint work with Nicolas Pétrélis (Nantes)

Universit¨at Bonn ∼ December 16, 2010

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Outline

1. Introduction and motivations 2. Definition of the model 3. The free energy

4. Path results

5. Techniques from the proof

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Outline

4. Path results

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A polymer in a multi-interface medium

Single interface case is well understood.

Copolymerinteraction [den Hollander & W¨uthrich JSP 04]

Some path results for log log NTNlog N (N = polymer size) Focus on (homogeneous, attractive/repulsive)pinninginteraction.

Path behavior? Interplaybetween N and T = TN?

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A polymer in a multi-interface medium

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A polymer in a multi-interface medium

T + +

+ + +

+ + + +

+ + +

+++

+ + +

+ + + +

+

_

__ __

_ _ _

_ _ __

__ _ _ _ _

_ _ _ _

_ + _

_ + Water

Water Oil

__

_

+

Some path results for log log NTNlog N (N = polymer size)

Focus on (homogeneous, attractive/repulsive)pinninginteraction. Path behavior? Interplaybetween N and T = TN?

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A polymer in a multi-interface medium

T

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Stabilization of colloidal dispersions

The addition of polymers into acolloidcan prevent the aggregation of droplets via entropic repulsion (stericstabilizationof the colloid)

Polymer confinedbetween two walls andinteractingwith them: (polymer in a slit)

0 T

δ δ

Physics literature: [Brak et al. 2005], [Owkzarek et al. 2008], . . .

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Stabilization of colloidal dispersions

Polymer confinedbetween two walls andinteractingwith them: (polymer in a slit)

0 T

δ δ

Physics literature: [Brak et al. 2005], [Owkzarek et al. 2008], . . .

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Stabilization of colloidal dispersions

Polymer confinedbetween two walls andinteractingwith them:

(polymer in a slit)

0 T

δ δ

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Multi-interface medium vs. slit

0 0 T

T 2T 3T

−T δ

δ δ δ δ

δ + log 2 δ + log 2 Φ_T

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Outline

4. Path results

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Definition

Recall the situation we want to model

T

Polymer configurations ←→ Trajectories of a random process Random polymer model:probability measure P^T_N,δ on paths

I (1 + 1)-dimensionale model: {(i,S_i)}i ≥0 I P^T_N,δ absolutely continuous w.r.t. SRW{Si}i ≥0

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Definition

T

Polymer configurations ←→ Trajectories of a random process

Random polymer model:probability measure P^T_N,δ on paths

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Definition

T

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Definition

T

(1 + 1)-dimensionale model:{(i, )}

I P^T_N,δ absolutely continuous w.r.t. SRW{Si}i ≥0

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Definition

T

I (1 + 1)-dimensionale model:{(i,S_i)}i ≥0 I P^T_N,δ absolutely continuous w.r.t. SRW{Si}i ≥0

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Definition

Ingredients of P^T_N,δ:

I Simple symmetric random walk S ={Sn}n≥0 on Z:

S0 := 0 , Sn:= X1+ . . . + Xn, with {Xi}i i.i.d. and P(Xi = +1) = P(Xi =−1) = ¹₂.

I Polymer sizeN ∈ 2N (number of monomers)

I Interaction strength δ∈ R (> 0 attractive, < 0 repulsive)

I Interface distance T ∈ 2N (interfaces ≡ T Z) Polymer measure:

dP^T_N,δ

dP (S) := 1

Z_N,δ^T exp δLN(S)

= 1

Z_N,δ^T exp (

δ

N

X

i=1

1{S_i∈ T Z}

)

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Definition

dP^T_N,δ

dP (S) := 1

= 1

Z_N,δ^T exp (

δ

N

X

i=1

1{S_i∈ T Z}

)

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Definition

dP^T_N,δ

dP (S) := 1

= 1

Z_N,δ^T exp (

δ

N

X

i=1

1{S_i∈ T Z}

)

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Definition

I Interface distance T ∈ 2N (interfaces ≡ T Z)

Polymer measure: dP^T_N,δ

dP (S) := 1

= 1

Z_N,δ^T exp (

δ

N

X

i=1

1{S_i∈ T Z}

)

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Definition

dP^T_N,δ 1

= 1

Z_N,δ^T exp (

δ

N

X

i=1

1{S_i∈ T Z}

)

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Definition

dP^T_N,δ

dP (S) := 1

Z_N,δ^T exp δLN(S) = 1 Z_N,δ^T exp

( δ

N

X

i=1

1{S_i∈ T Z}

)

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Definition

Sn

N T

Polymer measure:

dP^T_N,δ 1

(S) = 1 ( _N

X

)

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Some questions

The law P^T_N,δ describes the statistical distributionof the configurations of the polymer, given the external conditions

We are interested in the properties of P^T_N,δ as N → ∞ (thermodynamic limit), forfixedδ ∈ R and for arbitraryT = TN

Some questions on P^T_N,δ^N for large N

I What is the typical size of SN?

I How many monomers touch the interfaces?

I How many different interfaces are visited?

I Interplay between the parametersδ and T ={TN}N? Penalization of the simple random walk

4. Path results

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The free energy

Thefree energyφ(δ,{Tn}n) encodes the exponential asymptotic behavior of thenormalization constantZ_N,δ^T^N (partition function)

φ(δ,{Tn}n) := lim

N→∞

1

N log Z_N,δ^T^N (super-additivity + . . . )

= lim

N→∞

1

N log E exp δL_N

LN := #{i ≤ N : Si ∈ T Z} number of visitsto the interfaces φ is a generating function:

∂

∂δφ(δ,{Tn}n) = lim

N→∞E^T_N,δ^N L_N N

LN ∼ ^∂φ_∂δ · N

φ(δ,{Tn}n) non-analytic at δ ←→ phase transition at δ

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The free energy

Thefree energyφ(δ,{Tn}n) encodes the exponential asymptotic behavior of the normalization constant Z_N,δ^T^N (partition function)

φ(δ,{Tn}n) := lim

N→∞

1

= lim

N→∞

1

N log E exp δL_N

LN := #{i ≤ N : Si ∈ T Z} number of visitsto the interfaces

φ is a generating function:

∂

LN ∼ ^∂φ_∂δ · N

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The free energy

φ(δ,{Tn}n) := lim

N→∞

1

= lim

N→∞

1

N log E exp δL_N

∂

L_N ∼ ^∂φ_∂δ · N

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The free energy

φ(δ,{Tn}n) := lim

N→∞

1

= lim

N→∞

1

N log E exp δL_N

∂

L_N ∼ ^∂φ_∂δ · N

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The free energy: characterization

We assume that TN → T ∈ 2N ∪ {∞}, i.e.,

I either TN ≡ T < ∞ is constant for N large

I or TN → ∞ as N → ∞ (T = ∞)

Let τ₁^T := infn > 0 : Sn∈ {±T , 0} and set Q_T(λ) := E e^−λτ¹^T

I if T <∞, QT : (−λT,∞) → (0, ∞) ^(λ^T ∼ −2T^π²²)

I if T =∞, Q^∞: [0,∞) → (0, 1] (first return to zero of SRW)

Theorem ([CP1]). Let TN → T . φ(δ,{Tn}n) = φ(δ, T ) =

( Q_T−1

(e^−δ) if T < +∞ Q∞

−1

(e^−δ ∧ 1) if T = +∞

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The free energy: characterization

I or TN → ∞ as N → ∞ (T = ∞)

I if T <∞, QT : (−λT,∞) → (0, ∞) ^(λ^T ∼ −2T^π²²)

( Q_T−1

(e^−δ) if T < +∞ Q∞

−1

(e^−δ ∧ 1) if T = +∞

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The free energy: characterization

I or TN → ∞ as N → ∞ (T = ∞)

I if T <∞, QT : (−λT,∞) → (0, ∞) ^(λ^T ∼ −2T^π²²)

( Q_T−1

(e^−δ) if T < +∞ Q∞

−1

(e^−δ ∧ 1) if T = +∞

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The free energy: characterization

I or TN → ∞ as N → ∞ (T = ∞)

I if T <∞, QT : (−λT,∞) → (0, ∞) ^(λ^T ∼ −2T^π²²)

( Q_T−1

(e^−δ) if T < +∞ Q∞

−1

(e^−δ ∧ 1) if T = +∞

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The free energy: characterization

I or TN → ∞ as N → ∞ (T = ∞)

I if T <∞, QT : (−λT,∞) → (0, ∞) ^(λ^T ∼ −2T^π²²)

( Q_T−1

(e^−δ) if T < +∞ Q∞

−1

(e^−δ ∧ 1) if T = +∞

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The free energy: sumup

Case TN ≡ T < ∞

I φ(δ, T ) is analytic on R:no phase transitions

I ∂

∂δφ(δ, T ) > 0∀δ ∈ R: positive density of contacts L_N ∼ c · N

I Path behavior: diffusive scalingof SN

Case TN → ∞

I Phase transition (only) at δ = 0

I If δ≤ 0 then φ(δ, ∞) ≡ _∂δ^∂φ(δ,∞) ≡ 0 LN = o(N)

I If δ > 0 then _∂δ^∂φ(δ,∞) > 0 LN ∼ c · N

I Every{TN}N → ∞ yieldsthe same free energy as if TN ≡ ∞ (homogeneous pinning model) same density of visits

I Same path behavior? NO!

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The free energy: sumup

as N → ∞ . (?)

What if both N, T → ∞?

For δ < 0, φ(δ, T ) =− π² 2T² + c_δ

T³ + o

1 T³

with cδ > 0 explicit. [Owczarek et al. 2008] prove (for thepolymer in a slit) that

Z_N,δ^T ≈ exp

− π²

2T²N+ o N T²

as N, T → ∞ .

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Beyond the free energy

as N → ∞ . (?)

For δ < 0, φ(δ, T ) =− π² 2T² + c_δ

T³ + o

1 T³

with cδ> 0 explicit.

[Owczarek et al. 2008] prove (for thepolymer in a slit) that Z_N,δ^T ≈ exp

− π²

2T²N+ o N T²

as N, T → ∞ .

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Beyond the free energy

as N → ∞ . (?)

For δ < 0, φ(δ, T ) =− π² 2T² + c_δ

T³ + o

1 T³

− π²

2T²N+ o N T²

as N, T → ∞ .

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Beyond the free energy

as N → ∞ . (?)

For δ < 0, φ(δ, T ) =− π² 2T² + c_δ

T³ + o

1 T³

− π²

2T²N+ o N T²

as N, T → ∞ . We show that φ(δ, T ) can be developed at wish in (?).

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Beyond the free energy

as N → ∞ . (?)

For δ < 0, φ(δ, T ) =− π² 2T² + c_δ

T³ + o

1 T³

− π²

2T²N+ o N T²

as N, T → ∞ .

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Beyond the free energy

as N → ∞ . (?)

For δ < 0, φ(δ, T ) =− π² 2T² + c_δ

T³ + o

1 T³

− π²

2T²N+ o N T²

as N, T → ∞ . We show that φ(δ, T ) can be developed at wish in (?).

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Outline

4. Path results

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The attractive case δ > 0: heuristics

HenceforthT_N → ∞.

Forδ > 0 positive density of contacts with the interfaces: LN ∼ Cδ· N with Cδ= ^∂φ_∂δ(δ,∞) > 0.

Simple homogeneous pinning model(one interface at zero):

I If TN log N nothing changes: polymer localized at zero

I If TN log N different interfaces worth visiting

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The attractive case δ > 0: heuristics

Forδ > 0 positive density of contacts with the interfaces:

LN ∼ Cδ· N with Cδ= ^∂φ_∂δ(δ,∞) > 0.

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The attractive case δ > 0: heuristics

LN ∼ Cδ· N with Cδ= ^∂φ_∂δ(δ,∞) > 0.

N max|Sⁱ| ≈log N

SN= O(1)

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The attractive case δ > 0: heuristics

The attractive case δ > 0: heuristics

Notation

S_N TN

r N

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The attractive case δ > 0: heuristics

Notation

S_N TN

r N

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The attractive case δ > 0: heuristics

Notation

S_N TN

r N