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: Z Z d + ; IC: M t =0 = 0 Diffusion-aggregation: 1 / 2 d ( M t ( x ) , M t ( y )) − − − − − − → (0 , M t ( x ) + M t ( y )) , x ∼ y q Deposition: M t ( x ) − → M t ( x ) + 1 pχ ( M t ( x ) > 0) Evaporation: M t ( x ) − − − − − − − − − − − − − → M t ( x ) − 1 SM seminar. Warwick, 10.11.2011 – p. 3/18

Phase transition t →∞ → ∞ (Growing phase) q > q c ( p ) ⇒ E ( M t (0)) − − − − t →∞ q < q c ( p ) ⇒ E ( M t (0)) − − − − → ρ M < ∞ (Stationary phase) Discovered using numerical simulations and MF analysis Majumdar, Krishnamurthy, Barma, PRL 81 3691 (1998) P ( m ) ∼ e − m/m ∗ , q < q c ( p ) P ( m ) ∼ m − τ , q > q c ( p ) Conjecture : τ = 2 d +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. q c ( p ) = inf { q ≥ 0 : lim t →∞ E ( M t (0)) = ∞} Claim. E ( M t (0)) is increasing in q, t , decreasing in p Therefore : lim t →∞ E ( M t (0)) = ∞ for q > q c ( p ) lim t →∞ E ( M t (0)) < ∞ for q < q c ( p ) SM seminar. Warwick, 10.11.2011 – p. 6/18

Existence of growing phase Notations: P n ( { ( m k , x k ) } n k =1 ; t ) =Prob. of finding particles with masses m 1 , m 2 , . . . m n at lattice sites x 1 , x 2 , . . . x n at time t Moment equation: d dt E ( M t (0)) = q − ps ( t ) s ( t ) = � ∞ m =1 P ( m ; t ) ≤ 1 - occupation probability Flux J ( t ) ≡ q − ps ( t ) ≥ q − p q > p ⇒ lim t →∞ E ( M t (0)) = ∞ ⇒ q c ( p ) ≤ p Differential inequality estimate of P ( m = 0; t ) : q c ( p ) ≤ 1 � � p 2 + 2 � p − 2 + 2 SM seminar. Warwick, 10.11.2011 – p. 7/18

Existence of stationary phase Moment equation: d dt E ( M t (0) 2 ) = 2 E ( M t (0) M t (1))+ q + ps ( t ) − 2( p − q ) E ( M t (1)) Monotonicity lemma [Ligget]: d dt E ( M t (0) 2 ) ≥ 0 ; d dt E ( M t (0)) ≥ 0 ⇒ q ≥ ps ( t ) Anti-correlation lemma [van den Berg-Kesten-Reimers]: E ( M t (0) M t (1)) ≤ E ( M t (0)) 2 M 1 ( t ) 2 − ( p − q ) M 1 ( t ) + q ≥ 0 for all t ≥ 0 If ( p − q ) 2 − 4 q > 0 , M 1 ( t ) ≡ E ( M t (0)) ≤ C < ∞ ∀ t ⇒ q c ( 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 3.5 Upper bound d=1 d=2 d=3 3 Long range hops Mean field bound 2.5 Deposition rate, q 2 Growing phase, J>0 1.5 1 0.5 Stationary phase, J=0 0 0.5 1 1.5 2 2.5 3 3.5 4 Evaporation rate, p lim d →∞ q c ( p ) = q MF ( p ) ( Open problem ) c 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 0.5 q=1.00 0.4 0.3 q=0.75 J agg 0.2 q=0.50 0.1 q ≈ q c 0 10 0 10 1 10 2 10 3 10 4 10 5 10 6 m Simulation: t = ∞ , p = 1 . 0 , q c ≈ 0 . 31 Observe: for q > q c , lim m →∞ J agg ( m ) = J > 0 SM seminar. Warwick, 10.11.2011 – p. 11/18

Balance equation I� m� J� (m)� ev� J� (1)� agg� J� (m)� agg� 0� 1� m� Mass� I m + J ( m ) agg = J (1) agg + J ( m ) ev J (1) agg = qP (0) J ( m ) ( ev ) = pmP ( m + 1) I m = 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 m →∞ J ( m ) = pmP ( m + 1) − − − − → 0 ev m →∞ I 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 ( m 1 , m 2 ) ⇒ 1 � m 1 � P ( m 1 , m 2 ) = ( m 1 m 2 ) 3 / 2 Φ , m 1 , m 2 >> 1 m 2 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 1 d EDM (0 , J ) : P n ([ m, 0]; [ m, 1]; . . . ; [ m, n ]; t = ∞ ) ∼ m − γ n 3 + n ( n − 1) γ n = 4 n (PRL 94 194503 (2005), RG) 6 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 0 k=1 -10 k=2 -20 -30 k=3 -40 ln[P k (m)] -50 5 Simulation Theory 4 -60 3 -70 γ n 2 -80 1 -1.33 0 -90 -3.00 0 0.5 1 1.5 2 2.5 3 -5.04 n -100 0 2 4 6 8 10 12 14 ln(m) Nonlinear scaling in 1 d : γ 1 = 4 / 3 , γ 2 = 3 , γ 3 = 5 . Is J > 0 phase equivalent to p = 0 aggregation model? ( Open k=1 problem ) SM seminar. Warwick, 10.11.2011 – p. 15/18

On the nature of J = 0 phase for m >> 1 J ( m ) ≈ pmP ( m ) ev µ =1 P ( µ ) ≈ ( q − p ) � ∞ J (1) agg − I m ≈ pP (0) − ( p − q ) � m m P ( µ ) m 3 / 2 e − m/m ∗ A Mean field distribution: P ( m ) ∼ | J (1) agg − I m | << 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 1 d 10 0 10 0 J agg 10 -2 pmP(m+1) 10 -2 10 -4 10 -6 10 -4 10 -8 10 -10 0 10 20 30 40 50 10 -6 m 10 -8 10 -10 J agg P(m) 10 -12 0 5 10 15 20 25 30 35 40 45 m J ( m ) The relation lim m →∞ agg = 1 seems to hold in one J ( m ) ev 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? SM seminar. Warwick, 10.11.2011 – p. 18/18