Non-equilibrium phase transition for a system of - - PowerPoint PPT Presentation

Non-equilibrium phase transition for a system of diffusing-coalescing particles with deposition and evaporation

Oleg Zaboronski

Warwick University

Joint work with: Roger Tribe, Colm Connaughton and R. Rajesh

• J. Stat. Mech. P09016 (2010)

SM seminar. Warwick, 10.11.2011 – p. 1/18

Plan of the talk The evaporation-deposition model (EDM) Description of the phase transition Mathematics

Monotonicity Anti-correlation Existence of growing phase Existence of stationary phase

Theoretical physics

Growing phase: constant flux relation, multi-scaling Stationary phase: detailed balance

SM seminar. Warwick, 10.11.2011 – p. 2/18

The model

Configuration space: ZZd

+ ; IC: Mt=0 = 0

Diffusion-aggregation:

(Mt(x), Mt(y))

1/2d

− − − − − − → (0, Mt(x) + Mt(y)), x ∼ y

Deposition: Mt(x)

q

− → Mt(x) + 1

Evaporation: Mt(x)

pχ(Mt(x)>0)

− − − − − − − − − − − − − → Mt(x) − 1

SM seminar. Warwick, 10.11.2011 – p. 3/18

Phase transition

q > qc(p) ⇒ E(Mt(0))

t→∞

− − − − → ∞ (Growing phase) q < qc(p) ⇒ E(Mt(0))

t→∞

− − − − → ρM < ∞ (Stationary

phase) Discovered using numerical simulations and MF analysis Majumdar, Krishnamurthy, Barma, PRL 81 3691 (1998)

P(m) ∼ e−m/m∗, q < qc(p) P(m) ∼ m−τ, q > qc(p)

Conjecture: τ = 2d+2

d+2 , d ≤ 2, τ = 3/2, d > 2

SM seminar. Warwick, 10.11.2011 – p. 4/18

Mathematics of phase transition

SM seminar. Warwick, 10.11.2011 – p. 5/18

Definition of the transition point

• Def. qc(p) = inf{q ≥ 0 : limt→∞ E(Mt(0)) = ∞}
• Claim. E(Mt(0)) is increasing in q, t, decreasing in p

Therefore:

limt→∞ E(Mt(0)) = ∞ for q > qc(p) limt→∞ E(Mt(0)) < ∞ for q < qc(p)

SM seminar. Warwick, 10.11.2011 – p. 6/18

Existence of growing phase

Notations: Pn({(mk, xk)}n

k=1; t)=Prob. of finding particles

with masses m1, m2, . . . mn at lattice sites x1, x2, . . . xn at time t Moment equation: d

dtE(Mt(0)) = q − ps(t)

s(t) = ∞

m=1 P(m; t) ≤ 1 - occupation probability

Flux J(t) ≡ q − ps(t) ≥ q − p

q > p ⇒ limt→∞ E(Mt(0)) = ∞ ⇒ qc(p) ≤ p

Differential inequality estimate of P(m = 0; t):

qc(p) ≤ 1 2

• p − 2 +
• p2 + 2
• SM seminar. Warwick, 10.11.2011 – p. 7/18

Existence of stationary phase

Moment equation:

d dtE(Mt(0)2) = 2E(Mt(0)Mt(1))+q+ps(t)−2(p−q)E(Mt(1))

Monotonicity lemma[Ligget]: d

dtE(Mt(0)2) ≥ 0; d dtE(Mt(0)) ≥ 0 ⇒ q ≥ ps(t)

Anti-correlation lemma[van den Berg-Kesten-Reimers]:

E(Mt(0)Mt(1)) ≤ E(Mt(0))2

M1(t)2 − (p − q)M1(t) + q ≥ 0 for all t ≥ 0

If (p − q)2 − 4q > 0, M1(t) ≡ E(Mt(0)) ≤ C < ∞ ∀t

⇒ qc(p) ≥ p + 2 − 2√p + 1

Mean field phase curve is the rigorous lower bound

SM seminar. Warwick, 10.11.2011 – p. 8/18

Phase diagram of the model

0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 Evaporation rate, p Deposition rate, q Upper bound d=1 d=2 d=3 Long range hops Mean field bound

Growing phase, J>0 Stationary phase, J=0

limd→∞ qc(p) = qMF

c

(p) (Open problem)

SM seminar. Warwick, 10.11.2011 – p. 9/18

Physics of phase transition

SM seminar. Warwick, 10.11.2011 – p. 10/18

Mass flux due to aggregation, d = 1

0.1 0.2 0.3 0.4 0.5 100 101 102 103 104 105 106 Jagg m q=1.00 q=0.75 q=0.50 q≈qc

Simulation: t = ∞, p = 1.0, qc ≈ 0.31 Observe: for q > qc, limm→∞ Jagg(m) = J > 0

SM seminar. Warwick, 10.11.2011 – p. 11/18

Balance equation

Mass 1 m I

m

J

agg

(m) J

agg

(1) J

ev

(m)

Im + J(m)

agg = J(1) agg + J(m) ev

J(1)

agg = qP(0)

J(m)

(ev) = pmP(m + 1)

Im = p m

µ=1 P(µ) − q m−1 µ=1 P(µ)

J(m)

agg = 2 m µ=1 µP(µ) − m µ≥µ′ µP(µ′, µ − µ′) + qmP(m)

SM seminar. Warwick, 10.11.2011 – p. 12/18

J > 0 phase: constant flux relation

J(m)

ev

= pmP(m + 1)

m→∞

− − − − → 0 Im

m→∞

− − − − → (p − q)s

Balance equation ⇒ J(m)

agg = (q − ps) + O(m−α)

Constant flux J = q − ps Solve BE w.r.t. P(m1, m2) ⇒

P(m1, m2) =

1 (m1m2)3/2Φ

m1

m2

• , m1, m2 >> 1

Holds in all dimensions

SM seminar. Warwick, 10.11.2011 – p. 13/18

On the nature of J > 0 phase

Result: J(m)

agg = (q − ps) + O(m−α)

Conjecture: Large m-limit of correlation functions in the

growing phase of EDM(p, q) is given by EDM(0, q − ps) Multi-scaling in 1d EDM(0, J):

Pn ([m, 0]; [m, 1]; . . . ; [m, n]; t = ∞) ∼ m−γn γn = 4n

3 + n(n−1) 6

(PRL 94 194503 (2005), RG) Rigorous proof for A + A → A: Commun. Math. Phys. Vol 268, p. 717 (2006)

SM seminar. Warwick, 10.11.2011 – p. 14/18

J > 0: non-linear scaling of occupation probabilities

• 100
• 90
• 80
• 70
• 60
• 50
• 40
• 30
• 20
• 10

2 4 6 8 10 12 14 ln[Pk(m)] ln(m) k=1 k=2 k=3

• 1.33
• 3.00
• 5.04

1 2 3 4 5 0.5 1 1.5 2 2.5 3 γn n

k=1

Simulation Theory

Nonlinear scaling in 1d: γ1 = 4/3, γ2 = 3, γ3 = 5. Is J > 0 phase equivalent to p = 0 aggregation model? (Open

problem)

SM seminar. Warwick, 10.11.2011 – p. 15/18

On the nature of J = 0 phase for m >> 1

J(m)

ev

≈ pmP(m) J(1)

agg − Im ≈ pP(0) − (p − q) m µ=1 P(µ) ≈ (q − p) ∞ m P(µ)

Mean field distribution: P(m) ∼

A m3/2e−m/m∗

|J(1)

agg − Im| << J(m) ev

⇒

Scale-by-scale balance: J(m)

agg ≈ J(m) ev

SM seminar. Warwick, 10.11.2011 – p. 16/18

J = 0: Scale-by-scale balance in 1d

10-12 10-10 10-8 10-6 10-4 10-2 100 5 10 15 20 25 30 35 40 45 m Jagg P(m)

10-10 10-8 10-6 10-4 10-2 100 10 20 30 40 50 m

Jagg pmP(m+1)

The relation limm→∞

J(m)

agg

J(m)

ev

= 1 seems to hold in one

dimension even though mean field theory does not apply (Open problem)

SM seminar. Warwick, 10.11.2011 – p. 17/18

Conclusions

It is possible to rigorously establish the existence of phase transition for EDM in all dimensions. Local information: moment equations Global information: monotonicity, anti-correlation lemma

J > 0 phase: equivalence to p = 0 model with q′ = q − ps?

Constant flux relation? Non-linear scaling of multi-particle correlation functions?

J = 0 phase:

Exponential tails of mass distribution? Scale-by-scale balance between aggregation and evaporation?

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