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Cluster-cluster aggregation with evaporation and deposition, University of Manchester, 02 June 2010

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Presentation given at University of Manchester Theoretical Physics seminar, June 02, 2010

Presentation given at University of Manchester Theoretical Physics seminar, June 02, 2010


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  • 1. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Stationary State of CCA with Evaporation: p = 0 Summary and Conclusions Cluster-cluster aggregation with evaporation and deposition Colm Connaughton Mathematics Institute and Centre for Complexity Science, University of Warwick Collaborators: R. Rajesh (Chennai), Oleg Zaboronski (Warwick) UoM Theoretical Physics Seminars 02 June 2010 Colm Connaughton CCA
  • 2. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Stationary State of CCA with Evaporation: p = 0 Summary and Conclusions Outline 1 Cluster–Cluster Aggregation (CCA) Cluster Aggregation: Applications The Takayasu Model: A Mathematical Model of CCA Takayasu Model with Evaporation 2 Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Correlations and the Breakdown of Self-Similarity in CCA 3 Stationary State of CCA with Evaporation: p = 0 Growing Phase Exponential Phase Critical Phase 4 Summary and Conclusions Colm Connaughton CCA
  • 3. Cluster–Cluster Aggregation (CCA) Cluster Aggregation: Applications Stationary State of CCA without Evaporation: p = 0 The Takayasu Model: A Mathematical Model of CCA Stationary State of CCA with Evaporation: p = 0 Takayasu Model with Evaporation Summary and Conclusions Outline 1 Cluster–Cluster Aggregation (CCA) Cluster Aggregation: Applications The Takayasu Model: A Mathematical Model of CCA Takayasu Model with Evaporation 2 Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Correlations and the Breakdown of Self-Similarity in CCA 3 Stationary State of CCA with Evaporation: p = 0 Growing Phase Exponential Phase Critical Phase 4 Summary and Conclusions Colm Connaughton CCA
  • 4. Cluster–Cluster Aggregation (CCA) Cluster Aggregation: Applications Stationary State of CCA without Evaporation: p = 0 The Takayasu Model: A Mathematical Model of CCA Stationary State of CCA with Evaporation: p = 0 Takayasu Model with Evaporation Summary and Conclusions Cluster–Cluster Aggregation: Physical Examples Particles of one material dis- persed in another. Transport is diffusive or advective. Interac- tions between particles. clustering / sedimentation flocculation gelation phase separation Not to be confused with Diffusion–Limited Aggregation. Colm Connaughton CCA
  • 5. Cluster–Cluster Aggregation (CCA) Cluster Aggregation: Applications Stationary State of CCA without Evaporation: p = 0 The Takayasu Model: A Mathematical Model of CCA Stationary State of CCA with Evaporation: p = 0 Takayasu Model with Evaporation Summary and Conclusions Geomorphology: A model of river networks Scheidegger (1967) Rivulets flow downhill southeast or southwest randomly (diffusion). New rivulets appear randomly (injection). When rivulets intersect they combine to produce streams (aggregation). Interested in distribution of river sizes. Colm Connaughton CCA
  • 6. Cluster–Cluster Aggregation (CCA) Cluster Aggregation: Applications Stationary State of CCA without Evaporation: p = 0 The Takayasu Model: A Mathematical Model of CCA Stationary State of CCA with Evaporation: p = 0 Takayasu Model with Evaporation Summary and Conclusions Self-Organised Criticality: Directed Sandpiles Grains added at top. If O(xi ) = 2 then it topples and its grains are given to it’s two neighbours 1 level down producing an "avalaunche" . Simplest model of SOC. Avalaunche size distribution: P(s) ∼ s−4/3 (Dhar and Ramaswamy Can be mapped to river 1989) network flowing "uphill". Colm Connaughton CCA
  • 7. Cluster–Cluster Aggregation (CCA) Cluster Aggregation: Applications Stationary State of CCA without Evaporation: p = 0 The Takayasu Model: A Mathematical Model of CCA Stationary State of CCA with Evaporation: p = 0 Takayasu Model with Evaporation Summary and Conclusions Cluster–Cluster Aggregation: Takayasu Model Lattice Rd with particles of Racz (1985), Takayasu et al. integer mass. (1988) Nt (x, m)=number of mass Diffusion rate: DNt (x, m)/2d m on site x at time t. Nt (x, m) → Nt (x, m) − 1 Nt (x + n, m) → Nt (x + n, m) + 1 Aggregation rate: gK (m1 , m2 )Nt (x, m1 )Nt (x, m2 ) Nt (x, m1 ) → Nt (x, m1 ) − 1 Nt (x, m2 ) → Nt (x, m2 ) − 1 Nt (x, m1 + m2 ) → Nt (x, m1 + m2 ) + 1 red-(1-10), green-(10-50), Injection rate: q blue-(50-500) Nt (x, m) → Nt (x, m) + 1 Colm Connaughton CCA
  • 8. Cluster–Cluster Aggregation (CCA) Cluster Aggregation: Applications Stationary State of CCA without Evaporation: p = 0 The Takayasu Model: A Mathematical Model of CCA Stationary State of CCA with Evaporation: p = 0 Takayasu Model with Evaporation Summary and Conclusions Cluster–Cluster Aggregation: Takayasu Model Model parameters: D - diffusion constant ∆M j ∆M k ∆M j+ ∆M k q - mass injection rate m mj mk mi g - reaction rate Physical details are in the kernel: K (m1 , m2 ) ∼ mλ . Definition Cn (m1 , . . . , mn )(∆V )n i dmi = probability of having particles of masses, mi , in the intervals [mi , mi + dmi ] in a volume ∆V . ∞ ∂ Nm (t) J = δ(m − m0 ) + dm1 dm2 C2 (m1 , m2 ) δ(m−m1 −m2 ) ∂t m0 0 ∞ − 2 dm1 dm2 C2 (m, m1 ) δ(m2 −m−m1 ) 0 Colm Connaughton CCA
  • 9. Cluster–Cluster Aggregation (CCA) Cluster Aggregation: Applications Stationary State of CCA without Evaporation: p = 0 The Takayasu Model: A Mathematical Model of CCA Stationary State of CCA with Evaporation: p = 0 Takayasu Model with Evaporation Summary and Conclusions Mean-field Theory: Smoluchowski Dynamics Mean Field Approximation: C2 (m1 , m2 , t) ≈ Nm1 (t)Nm2 (t) Well-mixed. No spatial correlations. Then Nm (t) satisfies the Smoluchowski (1917) kinetic equation : ∞ ∂Nm (t) = dm1 dm2 K (m1 , m2 )Nm1 Nm2 δ(m − m1 − m2 ) ∂t 0 ∞ − dm1 dm2 K (m, m1 )Nm Nm1 δ(m2 − m − m1 ) 0 ∞ − dm1 dm2 K (m2 , m)Nm Nm2 δ(m1 − m2 − m) 0 + (q/m0 ) δ(m − m0 ) − DM [Nm ] Colm Connaughton CCA
  • 10. Cluster–Cluster Aggregation (CCA) Cluster Aggregation: Applications Stationary State of CCA without Evaporation: p = 0 The Takayasu Model: A Mathematical Model of CCA Stationary State of CCA with Evaporation: p = 0 Takayasu Model with Evaporation Summary and Conclusions Takayasu Model with Evaporation 2.5 Upper bound Mean field Numerics (MF) Evaporation rate: p Nt (x, m) 2 Numerics (1D) Nt (x, m) → Nt (x, m) − 1 Deposition rate, q 1.5 0 J> Mass balance is non-trivial in e, as ph a “closed” system : Krapivsky g 1 in w ro G 0.5 se, J=0 & Redner (1995) pha ntial Exp one Similar behaviour in open 0 0 0.5 1 1.5 2 2.5 3 system with injection: Evaporation rate, p Majumdar et al (2000) Colm Connaughton CCA
  • 11. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Stationary State of CCA with Evaporation: p = 0 Correlations and the Breakdown of Self-Similarity in CCA Summary and Conclusions Outline 1 Cluster–Cluster Aggregation (CCA) Cluster Aggregation: Applications The Takayasu Model: A Mathematical Model of CCA Takayasu Model with Evaporation 2 Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Correlations and the Breakdown of Self-Similarity in CCA 3 Stationary State of CCA with Evaporation: p = 0 Growing Phase Exponential Phase Critical Phase 4 Summary and Conclusions Colm Connaughton CCA
  • 12. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Stationary State of CCA with Evaporation: p = 0 Correlations and the Breakdown of Self-Similarity in CCA Summary and Conclusions Kolmogorov 1941 Theory of Turbulence LU Reynolds number R = ν . Energy injected into large eddies. Energy removed from small eddies at viscous scale. Transfer by interaction between eddies. Concept of inertial range K41 : In the limit of ∞ R, all small scale statistical properties depend only on the local scale, k, and the energy dissipation rate, ǫ. Dimensional analysis : 2 5 E (k) = cǫ 3 k − 3 Kolmogorov spectrum Colm Connaughton CCA
  • 13. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Stationary State of CCA with Evaporation: p = 0 Correlations and the Breakdown of Self-Similarity in CCA Summary and Conclusions 4 Structure Functions and the 5 -Law Structure functions : Sn (r ) = (u(x + r ) − u(x))n . Scaling form in stationary state: lim lim lim Sn (r ) = Cn (ǫr )ζn . r →0 ν→0 t→∞ K41 theory gives ζn = n . 3 4 5 Law : S3 (r ) = − 4 ǫr . Thus ζ3 = 1 (exact). 5 Colm Connaughton CCA
  • 14. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Stationary State of CCA with Evaporation: p = 0 Correlations and the Breakdown of Self-Similarity in CCA Summary and Conclusions Stationary State of CCA Suppose particles having 1 Spectrum profiles : lambda=1.5 nu=0.5 P=0 Nd=10 IC=N t=1.079349e-02 m > M are removed. t=4.844532e-01 Stationary state is obtained for t=1.384435e+00 t=1.999137e+00 t=2.331577e+00 1e-05 t=2.474496e+00 1e-10 large t when J = 0. Stationary state is a balance N(omega) 1e-15 between injection and 1e-20 dissipation. Constant mass flux 1e-25 in range [m0 , M] 1e-30 1 10 100 1000 omega 10000 100000 1e+06 Essentially non-equilibrium: no detailed balance. Colm Connaughton CCA
  • 15. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Stationary State of CCA with Evaporation: p = 0 Correlations and the Breakdown of Self-Similarity in CCA Summary and Conclusions Kolmogorov Theory of CCA (Constant Kernel) Dimensional analysis λ = 0: (3−2x )d (d +2)x −2d −2 Nm = c1 J x−1 D d −2 g d −2 m−x Two possible values for Kolmogorov exponent: 3 2d + 2 xg = xD = . 2 d +2 Self-similarity of higher order correlation functions: (3n−2γn )d (d +2)γn +(2d +2)n γn Cn (m1 , . . . , mn ) = cn J γn −n D d −2 g d −2 (m1 . . . mn )− n , g 3 D 2d + 2 γn = n γn = n. 2 d +2 Colm Connaughton CCA
  • 16. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Stationary State of CCA with Evaporation: p = 0 Correlations and the Breakdown of Self-Similarity in CCA Summary and Conclusions Kolmogorov Solution of Smoluchowski Equation Zakharov Transformation: Nm = Cm−x ) 1 m2 −m −m ′ mm1 m2 δ(m 2 m (m1 , m2 ) → ( ′ , ′ ) ) m2 m2 δ( −m 2 m −m −m ′ m2 mm2 1 −m 1 δ(m (m1 , m2 ) → ( , ′ ). 2 ) m m1 ′ m1 m1 ∞ C2 0= dm1 dm2 K (m1 , m2 ) (m1 m2 )−x m2−λ−2x 2 0 2x−λ−2 2x−λ−2 2x−ζ−2 m − m1 − m2 δ(m − m1 − m2 ) x = (λ + 3)/2. C depends on K . If K (m1 , m2 )=(m1 m2 )λ/2 : J λ+3 Nm = m− 2 . 2πg Colm Connaughton CCA
  • 17. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Stationary State of CCA with Evaporation: p = 0 Correlations and the Breakdown of Self-Similarity in CCA Summary and Conclusions The Takayasu Model in Low Dimensions 1 In d ≤ 2. Mean field Spatially extended Mean field -4/3 m-3/2 scaling exponents are not 0.1 m correct. 0.01 3 In 1-d x = 2 becomes P(m) x = 4 (exact). 0.001 3 0.0001 Reason is development of 1e-05 spatial correlations 1e-06 1 10 100 1000 generated by recurrence m property of random walks. Colm Connaughton CCA
  • 18. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Stationary State of CCA with Evaporation: p = 0 Correlations and the Breakdown of Self-Similarity in CCA Summary and Conclusions Spatial Correlations Visualising spatial correlations: 5 x 10-3 Regular Diffusion Definition β=2.0 Levy diffusion 4 x 10-3 β=1.6 Levy diffusion Pm (x) = Probability of finding a β=1.0 Levy diffusion particle of mass greater than Density auto-correlation Random hopping 3 x 10-3 m at a distance x from a 2 x 10-3 particle of mass m. 1 x 10-3 Heavy particles develop zones 0 x 100 of exclusion. 0 20 40 x 60 80 100 Aggregation of heavy particles is suppressed relative to MF estimates. Colm Connaughton CCA
  • 19. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Stationary State of CCA with Evaporation: p = 0 Correlations and the Breakdown of Self-Similarity in CCA Summary and Conclusions A Theoretical Approach m∈Z + 1 A set {ni,m }x ∈Rd determines a configuration. i 2 Write a Master equation for time evolution of P({ni,m }). 3 Convert master equation into a Schrodinger equation : d |ψ(t) = −H[ai,m , a† ] |ψ(t) i,m dt using Doi’s formalism. Path integral representation gives a continuous field theory having critical dimension 2: ∗ ,t,D,g,J] Nm (t) = DφDφ∗ φ(m, t) e−Seff [φ,φ 4 Use standard techniques to compute correlation functions. Colm Connaughton CCA
  • 20. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Stationary State of CCA with Evaporation: p = 0 Correlations and the Breakdown of Self-Similarity in CCA Summary and Conclusions Renormalisation of reaction rate Mean-field answer obtained from summing tree diagrams but in d ≤ 2, loops are divergent as t → ∞. The only loop diagrams which correct the average density are those which renormalise the reaction rate : Resumming: g → gR (m) d 2d +2 Nm ∼ φm = (J/D) d +2 m− d +2 xD is renormalised mean field exponent (Rajesh and Majumdar (2000) by other means). Colm Connaughton CCA
  • 21. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Stationary State of CCA with Evaporation: p = 0 Correlations and the Breakdown of Self-Similarity in CCA Summary and Conclusions Renormalisation of correlation functions in d < 2 ½ ½ First diagram gives MF answer : + Rµ1 Rµ2 = Rµ1 Rµ2 . ¾ ¾ Singularities in third and fourth ½ ½ diagram are removed by λ → λR . ½ + + Singularity in second is not. ¾ Higher correlations also require ¾ ¾ multiplicative renormalisation. Final result : Cn (m1 , . . . , mn ) ∼ m−γ(n) 2d + 2 ǫ n(n − 1) γ(n) = n+ + O(ǫ2 ). d +2 d +2 2 where ǫ = 2 − d . Colm Connaughton CCA
  • 22. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Stationary State of CCA with Evaporation: p = 0 Correlations and the Breakdown of Self-Similarity in CCA Summary and Conclusions Multi-scaling of higher order correlation functions RG calculation shows 10 presence of multi-scaling γkolm(n) 8 one loop in the particle distribution 6 for high masses. γ (n) 4 Compare exponents 2 obtained from ǫ-expansion 0 with measurements from 0 1 2 3 4 5 n Monte-Carlo simulations Montecarlo measure- in d = 1. ments of multiscaling Why is agreement so exponents in Takayasu good? model. Note special property of n = 2... Colm Connaughton CCA
  • 23. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Stationary State of CCA with Evaporation: p = 0 Correlations and the Breakdown of Self-Similarity in CCA Summary and Conclusions 4 Analogue of the 5 -Law γ(2) = 3 is an exact - a counterpart of the 4/5 law. Confirms multiscaling in this model without using ǫ-expansion. Stationary state: ∞ 0 = dm1 dm2 C(m1 , m2 ) δ(m−m1 −m2 ) 0 ∞ − dm1 dm2 C(m, m1 ) δ(m2 −m−m1 ) 0 ∞ − dm1 dm2 C(m, m2 ), δ(m1 −m2 −m) 0 Scaling form : C(m1 , m2 ) = (m1 m2 )−h ψ(m1 /m2 ). Zakharov transformation and constant flux give h = 3. Colm Connaughton CCA
  • 24. Cluster–Cluster Aggregation (CCA) Growing Phase Stationary State of CCA without Evaporation: p = 0 Exponential Phase Stationary State of CCA with Evaporation: p = 0 Critical Phase Summary and Conclusions Outline 1 Cluster–Cluster Aggregation (CCA) Cluster Aggregation: Applications The Takayasu Model: A Mathematical Model of CCA Takayasu Model with Evaporation 2 Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Correlations and the Breakdown of Self-Similarity in CCA 3 Stationary State of CCA with Evaporation: p = 0 Growing Phase Exponential Phase Critical Phase 4 Summary and Conclusions Colm Connaughton CCA
  • 25. Cluster–Cluster Aggregation (CCA) Growing Phase Stationary State of CCA without Evaporation: p = 0 Exponential Phase Stationary State of CCA with Evaporation: p = 0 Critical Phase Summary and Conclusions Nonequilibrium Phase Transition 2.5 Upper bound Low evaporation: growing 2 Mean field Numerics (MF) phase - M(t) ∼ t. Numerics (1D) High evaporation: Deposition rate, q 1.5 J >0 exponential phase - e, as ph g 1 M(t) ∼ constant. in w ro G 0 , J= 0.5 tia l ph ase Critical line q = qc (p) onen Exp 0 separates the two 0 0.5 1 1.5 2 2.5 3 Evaporation rate, p regimes. MF Mean field: qc (p) = p + 2 − 2 p+1 1 Upper bound: qc (p) ≤ p−2+ p2 + 4 2 Colm Connaughton CCA
  • 26. Cluster–Cluster Aggregation (CCA) Growing Phase Stationary State of CCA without Evaporation: p = 0 Exponential Phase Stationary State of CCA with Evaporation: p = 0 Critical Phase Summary and Conclusions Growing phase in mean field Most aspects of system are amenable to analytic analysis at mean field level. lattice: 1000, λ=10.0, q=1.0 lattice: 1000, λ=10.0, q=1.0 101 101 p=0.0 100 p=0.815 100 p=0.415 -1 p=0.315 10 10-1 p=0.215 Aggregation Flux Mass distribution 10-2 m-3/2 m-5/2 10-2 10-3 10-4 10-3 p=0.0 p=0.815 10-5 10-4 p=0.415 p=0.315 10-6 p=0.215 10-5 10-7 0 0 200 400 600 800 1000 10 101 102 103 m m Mass flux. Mass distribution. Colm Connaughton CCA
  • 27. Cluster–Cluster Aggregation (CCA) Growing Phase Stationary State of CCA without Evaporation: p = 0 Exponential Phase Stationary State of CCA with Evaporation: p = 0 Critical Phase Summary and Conclusions Growing Phase in 1D Aside from C2 (m) (known from constant flux) we have no analytic results for the 1-D case yet. Numerically observe the same multiscaling exponents. 0.5 0 q=1.00 k=1 -10 0.4 -20 k=2 -30 k=3 0.3 q=0.75 -40 ln[Pk(m)] Jagg -50 5 Simulation 0.2 4 Theory -60 3 γn -70 q=0.50 2 0.1 -80 1 0 -1.33 q≈qc -90 -3.00 0 0.5 1 1.5 2 2.5 3 0 n -5.04 0 1 2 3 4 5 6 -100 10 10 10 10 10 10 10 0 2 4 6 8 10 12 14 m ln(m) Conjecture that growing phase is in same universality class as the original Takayasu model (mass flux is modified). k=1 Colm Connaughton CCA
  • 28. Cluster–Cluster Aggregation (CCA) Growing Phase Stationary State of CCA without Evaporation: p = 0 Exponential Phase Stationary State of CCA with Evaporation: p = 0 Critical Phase Summary and Conclusions Exponential Phase If p < pc the mass distribution decays Im Jev(m) exponentially and Jagg (1) Jagg (m) Jagg → 0 as m → ∞. Theory gives the mean 0 1 m Mass 10 0 field result: 100 Jagg 10-2 pmP(m+1) -2 10 10-4 10-6 1 (m) 10 -4 10-8 P(m + 1) ∼ Jagg 10 -6 10-10 0 10 20 30 40 50 pm m -8 10 -10 which numerics suggest is 10 -12 Jagg P(m) true for d < 2. 10 0 5 10 15 20 25 30 35 40 45 m Looks more like detailed balance. Colm Connaughton CCA
  • 29. Cluster–Cluster Aggregation (CCA) Growing Phase Stationary State of CCA without Evaporation: p = 0 Exponential Phase Stationary State of CCA with Evaporation: p = 0 Critical Phase Summary and Conclusions Critical Phase If p = pc the stationary lattice: 100000, λ=10.0, q=0.22 p=1.0 100 mass flux Jagg decays as a Nm C2(m) m-5/2 power law as m → ∞. 10-2 m-4 At mean field level: 10-4 Nm 5 10-6 Nm ∼ m − 2 10-8 Krapivsky & Redner 10-10 100 101 102 103 m (1995). Exponent is modified in N(m) and C2 (m) at the d = 1. Numerics gives critical point (mean field). 1.83. Colm Connaughton CCA
  • 30. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Stationary State of CCA with Evaporation: p = 0 Summary and Conclusions Outline 1 Cluster–Cluster Aggregation (CCA) Cluster Aggregation: Applications The Takayasu Model: A Mathematical Model of CCA Takayasu Model with Evaporation 2 Stationary State of CCA without Evaporation: p = 0 Kolmogorov Theory of Turbulence: An analogy Correlations and the Breakdown of Self-Similarity in CCA 3 Stationary State of CCA with Evaporation: p = 0 Growing Phase Exponential Phase Critical Phase 4 Summary and Conclusions Colm Connaughton CCA
  • 31. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Stationary State of CCA with Evaporation: p = 0 Summary and Conclusions Conclusions CCA is a broadly interesting and useful model in physics and elsewhere. There are useful analogies with turbulent systems. In d ≤ 2 diffusive fluctuations dominate the dynamics leading to a breakdown of mean-field theory and emergence of spatially correlated structures. Introduction of weak evaporation doesn’t change much. Stronger evaporation triggers transition from growing to exponential phase. Colm Connaughton CCA
  • 32. Cluster–Cluster Aggregation (CCA) Stationary State of CCA without Evaporation: p = 0 Stationary State of CCA with Evaporation: p = 0 Summary and Conclusions References Colm Connaughton, R. Rajesh and Oleg Zaboronski 1 "Phases of Evaporation–Deposition Models", To appear, (2010) 2 "Constant Flux Relation for Driven Dissipative Systems", Phys. Rev. Lett. 98, 080601 (2007) 3 "Cluster-Cluster Aggregation as an Analogue of a Turbulent Cascade", Physica D, Volume 222, 1-2 (2006) 4 "Breakdown of Kolmogorov Scaling in Models of Cluster Aggregation", Phys. Rev. Lett. 94, 194503 (2005) 5 "Stationary Kolmogorov solutions of the Smoluchowski aggregation equation with a source term", Phys. Rev. E 69, 061114 (2004) Colm Connaughton CCA

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