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Outline Transition Trajectory for Equilibrium Droplet Formation Andreas Nubaumer, Elmar Bittner, and Wolfhard Janke Computational Quantum Field Theory Institut fr Theoretische Physik Universitt Leipzig Computation of Transition

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Outline

Transition Trajectory for Equilibrium Droplet Formation

Andreas Nußbaumer, Elmar Bittner, and Wolfhard Janke

Computational Quantum Field Theory Institut für Theoretische Physik Universität Leipzig

Computation of Transition Trajectories and Rare Events in Non-Equilibrium Systems Centre Blaise Pascal, ENS de Lyon, 13 June 2012

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Outline

Outline

1

Motivation

2

Theory

3

Monte Carlo (MC) Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Outline

Outline

1

Motivation

2

Theory

3

Monte Carlo (MC) Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Outline

Outline

1

Motivation

2

Theory

3

Monte Carlo (MC) Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations

Evaporated Condensed Balancing fluctuations vs interface free energy, i.e.,

entropy vs energy

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations

Droplet formation: nucleation of “wrong” phase fluid droplet in gas phase, or “−” Ising droplet in “+” phase

magnetisation m distribution log P(m) 1 0.5

  • 0.5
  • 1

Fisher; Binder & Kalos; Furukawa & Binder; Pleimling & Selke; Neuhaus & Hager; . . . Biskup, L. Chayes & Kotecký, Europhys. Lett. 60 (2002) 21; Comm. Math. Phys. 242 (2003) 137

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations

Theory: Equilibrium Droplet Formation

2D Ising model formulation lattice gas: spin up = black = vacancy spin down = white = particle

m0 (V − vL) −m0vL

M = −m0 vL

  • droplet

+m0( V − vL

background

) ⇒ δM ≡ M − M0 = −2vLm0

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations

Gaussian fluctuations around peak: exp

  • −(δM)2

2Vχ

  • = exp
  • −(2m0vL)2

2Vχ

  • χ = χ(β) = βV
  • m2 − m2

= susceptibility Interface free energy of droplet: exp

  • −τW

√vL

  • τW = τW(β) = interfacial free energy per unit volume of optimal

Wulff shaped droplet

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations

2D Ising model Wulff shapes at various temperatures:

T = 0.005 T = 0.050 T = 0.300 T = 1.000 T = 1.500 T = 2.000 T = Tc

⇒ for 1.0 T ≤ Tc ≈ 2.27 the Wulff shape is almost isotropic

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations

Balancing the exponents of the two limiting cases: ∆ = (2m0vL)2/(2Vχ) τW √vL = 2 m2 χτW v3/2

L

V Terms are equally important for ∆ ! = 1: ⇒ vL ⇒ −δM = θV 2/3 with θ = 2χτW √2m0 2/3 −δM ≫θV 2/3: droplet dominates −δM ≪θV 2/3: fluctuations dominate “Isoperimetric reasoning” (Biskup et al.) shows that either of these two cases dominate – but no droplets of intermediate size can exist.

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations

In general, a single large droplet of size vd coexists with small fluctuations taking vL − vd of the total excess. large droplet costs e−τW

√vd, absorbs fraction

δMd = −2vdm0 of δM fluctuations cost e−(δM−δMd )2/(2Vχ) For large systems, probability for magnetization excess: e−τW

√vd− (δM−δMd)2

2Vχ

= e

−τW q

−δM 2m0 Φ∆(λ) ,

Φ∆(λ) = √ λ + ∆(1 − λ)2 where λ = δMd/δM is the fraction taken up by the droplet. ⇒ Optimize Φ∆(λ) in λ for given ∆

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations

Recall: λ = δMd/δM, ∆ = 2

m2 χτW v3/2

L

V

∆ < ∆c: λmin = 0 ∆ = ∆c = (1/2)(3/2)3/2 ≈ 0.92: λmin = λc = 2/3 ∆ > ∆c: λmin > 2/3

Φ2.0 Φ∆c Φ0.5 λ Φ∆(λ) 2 1.5 1 0.5 2 1.5 1 0.5

General d: ∆c = 1

d

d+1

2

d+1

d , λc =

2 d+1

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations

Solution: λ = δMd/δM ≃ droplet size ∆ = 2 m2

χτW v3/2

L

V

≃ scaled magnetization (∆ = 0: peak location)

scaling parameter ∆ excess λ 0.5 ∆c 1.5 2 2.5 3

  • 0.2

0.2 0.4 2/3 0.8 1 1.2

Biskup et al., Europhys. Lett. 60 (2002) 21; Comm. Math. Phys. 242 (2003) 137

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations

Solution: λ = δMd/δM ≃ droplet size ∆ = 2 m2

χτW v3/2

L

V

≃ scaled magnetization (∆ = 0: peak location)

scaling parameter ∆ excess λ 0.5 ∆c 1.5 2 2.5 3

  • 0.2

0.2 0.4 2/3 0.8 1 1.2

magnetisation m distribution log P(m) 1 0.5

  • 0.5
  • 1

Biskup et al., Europhys. Lett. 60 (2002) 21; Comm. Math. Phys. 242 (2003) 137

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations

Numerical Studies

Suppressed two-phase region: use multicanonical type of simulation in magnetisation

  • 1.0
  • 0.5

0.0 0.5 1.0

M/V

2000 4000 6000

P(M) Pmuca Pcan

  • 1.0
  • 0.5

0.0 0.5 1.0

M/V

10

  • 7

10

  • 2

10

3

P(M) Pmuca Pcan

Clear NON-random-walk behaviour observed !

Neuhaus & Hager, J. Stat. Phys. 116 (2003) 47

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations

Numerical Studies

Suppressed two-phase region: use multicanonical type of simulation in magnetisation

  • 1.0
  • 0.5

0.0 0.5 1.0

M/V

2000 4000 6000

P(M) Pmuca Pcan

  • 1.0
  • 0.5

0.0 0.5 1.0

M/V

10

  • 7

10

  • 2

10

3

P(M) Pmuca Pcan

Clear NON-random-walk behaviour observed !

Neuhaus & Hager, J. Stat. Phys. 116 (2003) 47

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations

Reason: Two “hidden” barriers along transition trajectory

MC sweeps 100000 80000 60000 40000 20000 ∆c(L = 160) ∆c(L = ∞) distribution P(m) magnetisation m 1 10−5 10−10 10−15

  • 0.96
  • 0.965
  • 0.97
  • 0.975
  • 0.98
  • 0.985
  • 0.99

evaporation/ condensation

MC sweeps 100000 90000 80000 70000 60000 50000 40000 30000 20000 10000 −1/π distribution P(m) magnetisation m 1 10−10 10−20 10−30

  • 0.2
  • 0.4
  • 0.6
  • 0.8
  • 1

droplet/strip

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations

Reason: Two “hidden” barriers along transition trajectory

MC sweeps 100000 80000 60000 40000 20000 ∆c(L = 160) ∆c(L = ∞) distribution P(m) magnetisation m 1 10−5 10−10 10−15

  • 0.96
  • 0.965
  • 0.97
  • 0.975
  • 0.98
  • 0.985
  • 0.99

evaporation/ condensation

MC sweeps 100000 90000 80000 70000 60000 50000 40000 30000 20000 10000 −1/π distribution P(m) magnetisation m 1 10−10 10−20 10−30

  • 0.2
  • 0.4
  • 0.6
  • 0.8
  • 1

droplet/strip

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations

Droplet/strip barrier rather well understood in 2D:

strip: interface length = 2L circular droplet: Same length for radius R = L/π, area = L2/π # overturned spins = L2/π, hence (assuming isotropic interface tension) barrier located at about m = m0/π

But 2R = 2

πL < L: Leung & Zia, J. Phys. A 23 (1990) 4593

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

MC Simulations: Equilibrium Droplet Formation

Three goals: Test analytical prediction for the thermodynamic limit Investigate finite-size corrections Study lattice universality Simulation strategy: fix the total excess vL vL together with the “known constants” m0, χ, τW yields ∆(m0, χ, τW, vL). “micro-magnetical” simulation at: M = −m0vL + m0(V − vL) ⇒ M = m0V

  • 1 − 2vL

V

  • measure λ (≃ relative size of largest droplet)

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Algorithm: Kawasaki dynamics (M = const.) measure λ = vd/vL , the largest droplet size vd (i.e., second largest cluster), by using the Hoshen-Kopelman algorithm Difficulty: vd is the area of the second largest cluster ⇒ What is inside and what is outside?

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Idea: The Hoshen-Kopelman algorithm assigns to every cluster a unique number ⇒ interface between spins of the largest and second largest cluster Droplet algorithm: do Hoshen-Kopelman for every new cluster, mark positions of spins belonging to that cluster starting from a spin inside the second largest cluster, apply “flood-fill” flood-fill stops only at spins belonging to the largest cluster

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Idea: The Hoshen-Kopelman algorithm assigns to every cluster a unique number ⇒ interface between spins of the largest and second largest cluster Droplet algorithm: do Hoshen-Kopelman for every new cluster, mark positions of spins belonging to that cluster starting from a spin inside the second largest cluster, apply “flood-fill” flood-fill stops only at spins belonging to the largest cluster

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Outline

1

Motivation

2

Theory

3

Monte Carlo (MC) Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Square lattice NN Ising model (T = 1.5 ≈ 0.66Tc)

magnetisation m distribution log P(m) 1 0.8 0.6 0.4 0.2 L = 640 L = 320 L = 160 L = 80 L = 40 magnetisation m distribution P(m) 1 0.995 0.99 0.985 0.98 0.975 0.97 10-50 10-40 10-30 10-20 10-10 1

m range between arrows scanned with Kawasaki dynamics

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Square lattice NN Ising model

m0(β) =

  • 1 − sinh−4 (2β)

1/8 χ(β) = β n

i=0 ciu2i

with u = 1/(2 sinh(2β)) with c = {0, 0, 4, 16, 104, 416, 2224, 8896, 43840, 175296, 825648, 3300480, 15101920, ...} up to order 323 (last term at T = 1.5: 10−158). 2008–10: order 2000 . . . τW(β) = 2 √ W with W =

4 β2

βσ0 dx cosh−1

cosh2(2β) sinh(2β) − cosh(x)

  • and σ0 = 2 + ln[tanh(β)]/β the tension of an (1,0) interface.

Note: At T = 1.5 ≈ 0.66Tc, assuming isotropy: τW ≈ 2√πσ0 = 4.219; exact τW = 4.245 (0.6% difference).

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Square lattice NN Ising model (T/Tc ≈ 0.66)

L = 640 L = 320 L = 160 L = 80 L = 40 analytic ∆ = 2 m2 χτW v3/2

L

L2 λ 3 2.5 2 1.5 ∆c 0.5 1 0.8 2/3 0.4 0.2

Nußbaumer, Bittner, Neuhaus & WJ, Europhys. Lett. 75 (2006) 716

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Square lattice NN Ising model

Numerical problem: “Hidden” barrier is reflected in simulations

MC sweeps 100000 80000 60000 40000 20000 ∆c(L = 160) ∆c(L = ∞) distribution P(m) magnetisation m 1 10−5 10−10 10−15

  • 0.96
  • 0.965
  • 0.97
  • 0.975
  • 0.98
  • 0.985
  • 0.99

evaporation/ condensation Similar “hidden droplet barriers” in spin glasses?

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Evaporation/condensation barrier:

magnetisation m multimagnetic distribution h(m)

  • 0.96
  • 0.965
  • 0.97
  • 0.975
  • 0.98
  • 0.985
  • 0.99

0.002 0.0015 0.001 0.0005

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Evaporation/condensation barrier:

evaporated condensed both magnetisation m multimagnetic distribution h(m)

  • 0.96
  • 0.965
  • 0.97
  • 0.975
  • 0.98
  • 0.985
  • 0.99

0.002 0.0015 0.001 0.0005

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Square lattice NN Ising model

Distribution of fraction λ = vd/vL for ∆ close to ∆c

L = 320 L = 160 L = 80 L = 40
  • distribution
P () 1 0.8 0.6 0.4 0.2 3 2.5 2 1.5 1 0.5

⇒ Clear coexistence signal !

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Square lattice NN Ising model

Free-energy barrier resp. autocorrelation time scaling with system size

+
  • ln(L)
  • 0:1522L
2=3 +
  • ln(L)
  • 1
L 2=3
  • nst-M
, equal heigh t L ln
  • P
min =P max
  • 300
250 200 150 100 50 1 0.5
  • 0.5
  • 1
  • 1.5
  • 2
  • 2.5
  • 3
+
  • ln(L)
+ 0:1522L 2=3 +
  • ln(L)
+ 1 L 2=3
  • nst-M
, at
  • L
ln( in t ) 300 250 200 150 100 50 7 6 5 4 3 2 1

Theory: β∆F ≈ 0.1522 L2/3 (at T = 1.5)

Nußbaumer, Bittner & WJ, Prog. Theor. Phys. Suppl. 184 (2010) 400

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Outline

1

Motivation

2

Theory

3

Monte Carlo (MC) Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Triangular lattice Ising model (T/Tc ≈ 0.66)

L = 640 L = 320 L = 160 L = 80 L = 40 analytic ∆ = 2α m2 χτW v3/2

L

L2 λ 3 2.5 2 1.5 ∆c 0.5 1 0.8 2/3 0.4 0.2

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Outline

1

Motivation

2

Theory

3

Monte Carlo (MC) Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Square lattice NNN Ising model (T/Tc ≈ 0.66)

L = 640 L = 320 L = 160 L = 80 L = 40 analytic ∆ = 2 m2 χτW v3/2

L

L2 λ 3 2.5 2 1.5 ∆c 0.5 1 0.8 2/3 0.4 0.2 Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Check lattice universality (L = 640, T/Tc ≈ 0.66)

n.n.n. triangular n.n. analytic ∆ λ 3 2.5 2 1.5 ∆c 0.5 1 0.8 2/3 0.4 0.2

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Summary

Analytical predictions for the asymptotic behaviour of the evaporation/condensation transition of the 2D square lattice NN Ising model confirmed numerically Finite-size corrections investigated Universality tested by studying triangular lattice and square lattice NNN Ising models

Nußbaumer, Bittner, Neuhaus & WJ, Europhys. Lett. 75 (2006) 716; Nußbaumer, Bittner & WJ, Phys. Rev. E 77 (2008) 041109; Prog. Theor. Phys. Suppl. 184 (2010) 400

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Acknowledgements

Collaboration with Thomas Neuhaus (JSC/IAS, FZ Jülich, Germany) Many thanks to Kurt Binder, Roman Kotecký, Michel Pleimling, Royce Zia Supported by Deutsche Forschungsgemeinschaft DFH–UFA Graduate School Nancy–Leipzig EU–RTN Network “ENRAGE” Graduate School “BuildMoNa” Computer time NIC, Forschungszentrum Jülich

THANK YOU !

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Oops – wait another minute, it’s football time

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Footballphysics How probable is the next goal in soccer?

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Football fever: goal distributions and non-Gaussian statistics

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

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Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

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Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

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Transition Trajectory for Equilibrium Droplet Formation

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Motivation Theory MC Simulations Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model

Bio-, Econo-, Sociophysics, . . . , Footballphysics

Bundesliga Weltmeisterschaften Tore Wahrscheinlichkeit in % 14 12 10 8 6 4 2 100.00 10.00 1.00 0.10 0.01

Is all that true?

Disclaimer: No NIC supercomputer (JUMP , JUBL,. . . ) time was used

  • E. Bittner, A. Nußbaumer, M. Weigel & WJ, Eur. Phys. J. B 67 (2009)

459

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation

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Bio-, Econo-, Sociophysics, . . . , Footballphysics

Bundesliga Weltmeisterschaften Tore Wahrscheinlichkeit in % 14 12 10 8 6 4 2 100.00 10.00 1.00 0.10 0.01

Is all that true? Watch the EURO2012 games and check yourself !!!

Disclaimer: No NIC supercomputer (JUMP , JUBL,. . . ) time was used

  • E. Bittner, A. Nußbaumer, M. Weigel & WJ, Eur. Phys. J. B 67 (2009)

459

Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation