Exchange-driven growth with a source and sink of particles LML - - PowerPoint PPT Presentation

Exchange-driven growth with a source and sink of particles

LML Summer School 2017 Francis Aznaran supervised by Dr Colm Connaughton

Warwick University

August 18th 2017

Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 1 / 14

Motivation

Droplet growth via evaporation and recondensation Exchange of capital between economically interacting individuals Formation of smog Formation of planets from interstellar dust

Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 2 / 14

Physical principles

Exchange kernel is symmetric, K(i, j) = K(j, i), and homogeneous, K(ai, aj) = a2λK(i, j) Spatial homogeneity Law of mass action Modelling choices: Clusters of size 0 Conservation of clusters Kernel homogeneity index: K(i, j) = (ij)λ

Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 3 / 14

Rate equation derivation

dck dt = K(k + 1, j)ck+1cj − 2K(k, j)ckcj + K(j, k − 1)ck−1cj

Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 4 / 14

Rate equation derivation

dck dt =

∞

• j=1

K(k + 1, j)ck+1cj − 2K(k, j)ckcj + K(j, k − 1)ck−1cj

Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 4 / 14

Rate equation derivation

dck dt =

∞

• j=1

K(k + 1, j)ck+1cj − 2K(k, j)ckcj + K(j, k − 1)ck−1cj + K(k + 1, 0)ck+1c0 − K(k, 0)ckc0

Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 4 / 14

ODE system

For clusters of size k = 0: dc0 dt = 0 and for k = 1, ..., M − 1 : dck dt =

M

• j=1

K(k + 1, j)ck+1cj − 2K(k, j)ckcj + K(j, k − 1)ck−1cj and for clusters of maximal size k = M : dcM dt =

M−1

• j=1

−2K(M, j)cMcj + K(j, M − 1)cM−1cj − 2K(M, M)c2

M + K(M, M − 1)cM−1cM

Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 5 / 14

Scaling behaviour

Conservation of mass: ∞ kckdk = 1 Typical size: s(t) := M2(t) M1(t) where Mλ(t) := ∞

0 kλck(t)dk is the λ-th moment.

Ben-Naim and Krapivsky, 2003, proposed a self-similar ansatz ck(t) = s(t)αF(z) where z =

k s(t), α = (3 − 2λ)−1 for λ < 3 2, and showed that for λ = 0,

s(t) ∼ t

1 3 . Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 6 / 14

Scaling behaviour with source turned on

New rate equations: dck dt =

• j

(...) + Jδk,1 so that M1 ∼ t. Krapivsky, 2015, showed s(t) ∼ t2 for K(i, j) = ij. We generalised this to s(t) ∼ t

2 3−2λ

for K(i, j) = (ij)λ.

Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 7 / 14

Imposing a sink

Desorption: dck dt =

• j

(...) + Jδk,1 − Γ(k)ck Evaporation: dck dt =

• j

(...) + Jδk,1 +

• Γ(k + 1)ck+1 − Γ(k)ck,

0 ≤ k ≤ M − 1 −Γ(M)cM, k = M where Γ(k) = γ0kγ.

Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 8 / 14

Solver

dc dt = f (c) where c(t) =    c0(t) . . . cM(t)    and f (c) =      −γ00γ

• j(K(2, j)c2cj − ...) + J − γ01γ

. . .

• j(−2K(M, j)cMcj + ...) − γ0Mγ

     and a ‘monodisperse’ initial condition: ck(0) = δ1,k. Predictor-corrector method with fixed timestep (3rd order local error in time). Python, within Jupyter notebooks.

Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 9 / 14

Convergence and code validation

Scaling analysis: Computation time ∼

• T

simulated time M2 numerical cutoff

Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 10 / 14

Experimentation: Scaling regimes and possibly phase transitions

‘Waiting long enough’ to observe scaling Don’t confuse numerical regularisation with a physical process Is there a non-equilibrium steady state?

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Conclusions

Existence of steady states Settling time for the steady state increased with the ratio

J γ0

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The steady state had an approximately exponential tail Larger systems may exhibit a phase transition (cf. Connaughton, Rajesh, Zaboronski, 2010).

Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 13 / 14

References

• Ben-Naim, E. and Krapivsky, P.L. “Exchange Driven Growth”. Physical

Review E, 68(3):031104, 2003.

• Connaughton, C. “Instantaneous gelation and explosive condensation in

non-equilibrium cluster growth”. Applied Mathematics seminar, University

• f Exeter, 2016.
• Connaughton, C. “Oscillatory kinetics in cluster-cluster aggregation”.

Lecture at WCPM MIRaW Day, University of Warwick, 2017.

• Connaughton, C., Rajesh, R., and Zaboronski, O. “On the

non-equilibrium phase transition in evaporation-deposition models”. J.

• Stat. Mech. Theor. E., P09016, 2010.
• Krapivsky, P. L. “Mass Exchange Processes with Input”. Journal of

Physics A: Mathematical and Theoretical 48.20: 205003, 2015.

• Krapivsky, P. L. A Kinetic View of Statistical Physics. Cambridge

University Press, Cambridge, 2010.

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