Instantons in Aggregation Kinetics Colm Connaughton a , Roger Tribe b - PowerPoint PPT Presentation

Instantons in Aggregation Kinetics Colm Connaughton a , Roger Tribe b and Oleg Zaboronski c a Complexity Centre, University of Warwick b Department of Mathematics, University of Warwick c Department of Mathematics, University of Warwick Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 1/20

Plan The model Mean field theory The formalism Rate functions via DZO path integrals Instanton energy and mass conservation Fast and slow gelation probabilities: constant kernel Large deviations principle Solution of instanton equations The statistics of mass flux Fast gelation and the non-gelling probability: multiplicative kernel Fast gelation: LDP and instanton equations Non-gelling near gelation time: LDP , results Conclusions Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 2/20

Markus-Luzhnikov model Microstate: N = N 1 , N 2 , . . . N m = # of particles of mass m ∈ { 1 , 2 , 3 , . . . } Coagulation: N m 1 → N m 1 − 1 Classical kernels : N m 2 → N m 2 − 1 N m 1 + m 2 → N m 1 + m 2 + 1 λ ( k, l ) = 1 (Constant) λ ( k, l ) = kl (Multiplicative) Rate: λ ( m 1 , m 2 ) N m 1 N m 2 λ ( k, l ) = ( k + l ) / 2 (Sum) Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 3/20

Problem statement Monomer initial condition: N m (0) = Mδ ( m, 1) Mass conservation � m mN m ( t ) = M Complete gelation event: N m ( t ) = δ ( m, M ) def = � Equivalently, N ( t ) m N m ( t ) = 1 Gelation time: T G = E ( τ | N τ = 1) Find: Prob ( N t = 1) for t << T G Find: Prob ( N t >> 1) for t > T G Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 4/20

Smoluchowski (mean field) theory m 1 ˙ � λ ( m ′ , m − m ′ ) N ( m − m ′ ) N ( m ′ ) N m = 2 m ′ =1 ∞ � λ ( m, m ′ ) N ( m ′ ) − N ( m ) m ′ =1 - Smoluchowski equation (SE) Can be rigorously related to ML model in the scaling limit N t → ∞ for certain kernels Cannot be used to describe complete gelation ( N t ∼ 1 ) Suffers from finite time singularities for some kernels (e. g. the multiplicative kernel) Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 5/20

The formalism Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 6/20

Path integral expression for P ( N t = 1) � � D µ ( z ( τ ′ ) , ¯ z ( τ ′ )) exp[ − S eff ] P ( N t = 1) = τ ′ � t �� � z M � � S eff = dτ z m ¯ ˙ z m + h ( z , ¯ z ) − log z M ( t )¯ 1 (0) 0 m z ) = − 1 � h ( z , ¯ λ m 1 ,m 2 (¯ z m 1 + m 2 − ¯ z m 1 ¯ z m 2 ) z m 1 z m 2 2 m 1 , m 2 Method: Doi-Zeldovich-Ovchinnikov. Note the presence of boundary terms. Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 7/20

Path integral expression for P ( N t = fM ) � � 1 D µ ( z ( τ ′ ) , ¯ z ( τ ′ )) exp[ − S eff ] , P ( N t = fM ) = ( fM )! τ ′ f ∈ (0 , 1) . � M � fM   � t �� � � z M S eff = dτ z m ¯ ˙ z m + h ( z , ¯ z ) − log z k ( t ) ¯ 1 (0)   0 m k =1 Note the difference in the boundary terms. Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 8/20

Laplace approximation for the path integral Laplace formula: P ( N t = 1) ∼ exp {− S eff [ z c , ¯ z c ] } m ( τ )) solve δS eff = 0 subject to: Here ( z c z c m ( τ ) , ¯ z m ( t ) = δ m,M (Fast gelation) z m (0)¯ z m (0) = Mδ m, 1 , z m ( t )¯ M � z m ( t ) = fM (Non-gelation) z m (0)¯ z m (0) = Mδ m, 1 , z k ( t )¯ k =1 General applicability condition: the PI is dominated by trajectories close to the instanton trajectory Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 9/20

Euler-Lagrange (instanton) equations z m = 1 � ˙ λ m 1 ,m 2 ( δ m,m 1 + m 2 − ¯ z m 1 δ m,m 2 − ¯ z m 2 δ m,m 1 ) z m 1 z m 2 2 m 1 ,m 2 z m = − 1 � ˙ ¯ λ m 1 ,m 2 (¯ z m 1 + m 2 − ¯ z m 1 ¯ z m 2 )( z m 1 δ m,m 2 + z m 2 δ m,m 1 ) 2 m 1 , m 2 Integrals of motion: E = h ( z c , ¯ z c ) (’Instanton energy’) m mz c z c M = � m ¯ m (Mass) Special solution: ¯ z ≡ 1 ; z solves Smoluchowski equation, E = 0 z m ( t ) - the symbol of the occupation N m ( t ) = z m ( t )¯ number operator Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 10/20

On the calculation of inf [ S eff ] Claim. S c eff = − E · t + boundary terms Derivation: h ( z , ¯ z ) is homogeneous function of z of order 2 : � M � t � � dτ z m ¯ ˙ z m + h 0 m =1 � t M M � � ∂ h � � z m | t = z m ¯ 0 + dτ − z m + h ∂z m 0 m =0 m =1 M � z m | t = z m ¯ 0 − E ( t ) t m =0 N.B. E = 0 corresponds to mean field Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 11/20

Fast and slow gelation probabilities: the constant kernel Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 12/20

The large deviations principle for fast gelation. The limit : t << 1 , M = ∞ S c z M � z M (1)¯ � 1 (0) eff log P ( N τ ) ∼ − + log , τ τ where � 1 � S c eff = inf dτ [ z m ¯ ˙ z m + h ( z, ¯ z )] , { z ( t ) , ¯ z ( t ) } 0 m � z m (0+)¯ z m (0+) = ∞ · δ m, 1 , z m (1 − )¯ z m (1 − ) = 0 Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 13/20

Solving the instanton equations. Euler-Lagrange equation for N ( τ ) = � m z m ( τ )¯ z m ( τ ) : N ( τ ) = − 1 ˙ 2 N 2 ( τ ) + E, Boundary conditions: N (0) = ∞ , N (1) = 0 E = − p 2 2 < 0 � p � N ( τ ) = p tan 2 ( τ − τ 0 ) E = − π 2 2 Rate function: log P ( N t = 1) ∼ − π 2 2 t + O ( t 0 ) Really hard step: the estimate of the contribution from the boundary terms Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 14/20

Statistics of mass flux Non-equilibrium ’turbulent’ state: constant flux of mass through mass scales of the system. The average mass flux: J = M/τ (random quantity) Mean field flux: J mf = M/T G = M . � � − π 2 τ < M J + � J + →∞ � P ( J > J + ) = Pr = Pr N M/J + = 1 ∼ e 2 Jmf J + Left tail of flux distribution: � � τ > M Jmf � J − → 0 e − � P ( J < J − ) = Pr ∼ Pr N M/J − = 2 ∼ J − J − � L →∞ � � � Pr( J>J mf L ) 1 − π 2 Fluctuation relation: log ∼ L 2 Pr( J<J mf /L ) Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 15/20

Fast gelation and the non-gelling probability: the multiplicative kernel Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 16/20

Fast gelation event Typical gelation time: T G = Const M The scaling limit: M is fixed, t = θ/M , θ << 1 Small time LD principle still applies: log Pr ( N t = 1) ∼ − 1 t S eff + boundary terms Equations of motion: N ( τ ) = E − M 2 ˙ 2 , 0 < τ < t N (0) = M, N ( t ) = 1 Instanton energy: E = M 2 2 + 1 − M t Boundary terms dominate � 1 + O ( θ 0 ) ⇒ Algebraic decay � log P ( N t = 1) ∼ − M log θ of gelation probability Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 17/20

LDP for P ( N t = fM ) , f ∈ f (0 , 1) Scaling limit: M → ∞ , t = θ/M , θ ∼ 1 LD principle: 1 � � M log Pr N θ M = fM = − I ( θ ) + O (log( M ) /M ) 2 ( θ − θ mf ( f )) − θ mf ( f ) Rate function: I ( θ ) = 1 log( θ/θ mf ( f )) 2 θ mf ( f ) = 2(1 − f ) - mean field time to N = fM . Potential non-analyticity! Mean field evolution of density 1 Finite time singularity 0.5 N θ /M f 0 θ (f) −0.5 0 0.5 1 1.5 2 2.5 θ Rate function for f=1/2 2 1.5 I( θ ) 1 θ (f) 0.5 0 0 0.5 1 1.5 2 2.5 θ Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 18/20

A note for mathematicians. ML model can be restated as a stochastic differential equation driven by Poisson noise All scaling limits considered in the presentation correspond to the limit of weak noise All large deviation principles discussed in the talk follow from the standard Wentzel -Freidlin theory for SDE’s with Poisson noise. Lyon, Rare Events in Non-Equilibrium Systems, 11-15.06.2012 – p. 19/20

Conclusions Large deviations turned out to be an effective tool in the analysis of aggregation Rate function = Instanton energy × time + boundary terms Instanton energy = 0 corresponds to MF approximation Instanton equations: Mean field equation = Optimal noise fluctuation Solutions to instanton equations are globally well defined even for gelling kernels Reference : Colm Connaughton, Roger Tribe, Oleg Zaboronski On the statistics of rare events in Markus-Luzhnikov model , still in preparation Lyon, Rare Events in Non -Equilibrium Systems, 11-15.06.2012 – p. 20/20