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 C Q T Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Outline Outline Motivation 1 Theory 2 Monte Carlo (MC) Simulations 3 Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model C Q T Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Outline Outline Motivation 1 Theory 2 Monte Carlo (MC) Simulations 3 Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model C Q T Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Outline Outline Motivation 1 Theory 2 Monte Carlo (MC) Simulations 3 Square lattice NN Ising model Triangular lattice Ising model Square lattice NNN Ising model C Q T Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Theory MC Simulations Evaporated Condensed Balancing fluctuations vs interface free energy, i.e., entropy vs energy C Q T Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Theory MC Simulations Droplet formation: nucleation of “wrong” phase fluid droplet in gas phase, or “ − ” Ising droplet in “ + ” phase distribution log P ( m ) -1 -0.5 0 0.5 1 magnetisation m Fisher; Binder & Kalos; Furukawa & Binder; Pleimling & Selke; Neuhaus & Hager; . . . Biskup, L. Chayes & Kotecký, Europhys. Lett. 60 (2002) 21; Comm. Math. Phys. 242 C Q T (2003) 137 Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Theory MC Simulations Theory: Equilibrium Droplet Formation 2D Ising model formulation lattice gas: spin up = black = vacancy spin down = white = particle M = − m 0 v L + m 0 ( V − v L ) − m 0 v L ���� � �� � droplet background ⇒ δ M ≡ M − M 0 = − 2 v L m 0 m 0 ( V − v L ) C Q T Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Theory MC Simulations Gaussian fluctuations around peak: � − ( δ M ) 2 � � − ( 2 m 0 v L ) 2 � exp = exp 2 V χ 2 V χ � � m 2 � − � m � 2 � χ = χ ( β ) = β V = susceptibility Interface free energy of droplet: √ v L � � exp − τ W τ W = τ W ( β ) = interfacial free energy per unit volume of optimal Wulff shaped droplet C Q T Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Theory MC Simulations 2D Ising model Wulff shapes at various temperatures: T = T c T = 2 . 000 T = 1 . 500 T = 1 . 000 T = 0 . 300 T = 0 . 050 T = 0 . 005 ⇒ for 1 . 0 � T ≤ T c ≈ 2 . 27 the Wulff shape is almost isotropic C Q T Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Theory MC Simulations Balancing the exponents of the two limiting cases: = 2 m 2 v 3 / 2 ∆ = ( 2 m 0 v L ) 2 / ( 2 V χ ) L 0 √ v L V τ W χτ W Terms are equally important for ∆ ! = 1: � 2 χτ W � 2 / 3 ⇒ v L ⇒ − δ M = θ V 2 / 3 with θ = √ 2 m 0 − δ M ≫ θ V 2 / 3 : droplet dominates − δ M ≪ θ V 2 / 3 : fluctuations dominate “Isoperimetric reasoning” (Biskup et al. ) shows that either of these two cases dominate – but no droplets of intermediate C Q T size can exist. Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Theory MC Simulations In general, a single large droplet of size v d coexists with small fluctuations taking v L − v d of the total excess. √ v d , absorbs fraction large droplet costs e − τ W δ M d = − 2 v d m 0 of δ M fluctuations cost e − ( δ M − δ M d ) 2 / ( 2 V χ ) For large systems, probability for magnetization excess: � √ √ v d − ( δ M − δ M d ) 2 − δ M q λ + ∆( 1 − λ ) 2 � 2 m 0 Φ ∆ ( λ ) , − τ W e − τ W = e 2 V χ Φ ∆ ( λ ) = where λ = δ M d /δ M is the fraction taken up by the droplet. ⇒ Optimize Φ ∆ ( λ ) in λ for given ∆ C Q T Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Theory MC Simulations m 2 v 3 / 2 Recall: λ = δ M d /δ M , ∆ = 2 L 0 V χτ W ∆ < ∆ c : λ min = 0 ∆ = ∆ c = ( 1 / 2 )( 3 / 2 ) 3 / 2 ≈ 0 . 92: λ min = λ c = 2 / 3 ∆ > ∆ c : λ min > 2 / 3 2 1.5 Φ ∆ ( λ ) 1 0.5 Φ 0 . 5 Φ ∆ c Φ 2 . 0 0 0 0.5 1 1.5 2 λ C Q T � d + 1 � d + 1 General d : ∆ c = 1 d , λ c = 2 d d + 1 2 Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Theory MC Simulations Solution: λ = δ M d /δ M ≃ droplet size ∆ = 2 m 2 v 3 / 2 L ≃ scaled magnetization ( ∆ = 0: peak location) 0 V χτ W 1.2 1 0.8 2 / 3 excess λ 0.4 0.2 0 -0.2 0 0.5 ∆ c 1.5 2 2.5 3 scaling parameter ∆ C Q T Biskup et al. , Europhys. Lett. 60 (2002) 21; Comm. Math. Phys. 242 (2003) 137 Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Theory MC Simulations Solution: λ = δ M d /δ M ≃ droplet size ∆ = 2 m 2 v 3 / 2 L ≃ scaled magnetization ( ∆ = 0: peak location) 0 V χτ W 1.2 1 0.8 2 / 3 excess λ distribution log P ( m ) 0.4 0.2 -1 -0.5 0 0.5 1 magnetisation m 0 -0.2 ∆ c 0 0.5 1.5 2 2.5 3 scaling parameter ∆ C Q T Biskup et al. , Europhys. Lett. 60 (2002) 21; Comm. Math. Phys. 242 (2003) 137 Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Theory MC Simulations Numerical Studies Suppressed two-phase region: use multicanonical type of simulation in magnetisation 6000 3 10 P muca P muca 4000 P(M) P(M) -2 10 P can 2000 P can -7 0 10 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 M/V M/V Clear NON-random-walk behaviour observed ! C Q T Neuhaus & Hager, J. Stat. Phys. 116 (2003) 47 Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Theory MC Simulations Numerical Studies Suppressed two-phase region: use multicanonical type of simulation in magnetisation 6000 3 10 P muca P muca 4000 P(M) P(M) -2 10 P can 2000 P can -7 0 10 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 M/V M/V Clear NON-random-walk behaviour observed ! C Q T Neuhaus & Hager, J. Stat. Phys. 116 (2003) 47 Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Theory MC Simulations Reason: Two “hidden” barriers along transition trajectory -0.96 ∆ c ( L = ∞ ) -0.965 ∆ c ( L = 160) evaporation/ -0.97 magnetisation m condensation -0.975 -0.98 -0.985 -0.99 1 10 − 5 10 − 10 10 − 15 0 20000 40000 60000 80000 100000 distribution P ( m ) MC sweeps 0 − 1 /π -0.2 magnetisation m -0.4 droplet/strip -0.6 -0.8 -1 1 10 − 10 10 − 20 10 − 30 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 C Q T distribution P ( m ) MC sweeps Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Theory MC Simulations Reason: Two “hidden” barriers along transition trajectory -0.96 ∆ c ( L = ∞ ) -0.965 ∆ c ( L = 160) evaporation/ -0.97 magnetisation m condensation -0.975 -0.98 -0.985 -0.99 1 10 − 5 10 − 10 10 − 15 0 20000 40000 60000 80000 100000 distribution P ( m ) MC sweeps 0 − 1 /π -0.2 magnetisation m -0.4 droplet/strip -0.6 -0.8 -1 1 10 − 10 10 − 20 10 − 30 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 C Q T distribution P ( m ) MC sweeps Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Theory MC Simulations Droplet/strip barrier rather well understood in 2D: strip: interface length = 2 L circular droplet: Same length for radius R = L /π , area = L 2 /π # overturned spins = L 2 /π , hence (assuming isotropic interface tension) barrier located at about m = m 0 /π But 2 R = 2 π L < L : C Q T Leung & Zia, J. Phys. A 23 (1990) 4593 Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Square lattice NN Ising model Theory Triangular lattice Ising model MC Simulations 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 v L v L together with the “known constants” m 0 , χ , τ W yields ∆( m 0 , χ, τ W , v L ) . “micro-magnetical” simulation at: 1 − 2 v L � � M = − m 0 v L + m 0 ( V − v L ) M = m 0 V ⇒ V C Q T measure λ ( ≃ relative size of largest droplet) Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
Motivation Square lattice NN Ising model Theory Triangular lattice Ising model MC Simulations Square lattice NNN Ising model Algorithm: Kawasaki dynamics ( M = const.) measure λ = v d / v L , the largest droplet size v d (i.e., second largest cluster), by using the Hoshen-Kopelman algorithm Difficulty: v d is the area of the second largest cluster ⇒ What is inside and what is outside? C Q T Wolfhard Janke Transition Trajectory for Equilibrium Droplet Formation
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