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 ) = a 2 λ K ( 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 dc k dt = K ( k + 1 , j ) c k +1 c j − 2 K ( k , j ) c k c j + K ( j , k − 1) c k − 1 c j Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 4 / 14
Rate equation derivation dc k dt = ∞ � K ( k + 1 , j ) c k +1 c j − 2 K ( k , j ) c k c j + K ( j , k − 1) c k − 1 c j j =1 Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 4 / 14
Rate equation derivation dc k dt = ∞ � K ( k + 1 , j ) c k +1 c j − 2 K ( k , j ) c k c j + K ( j , k − 1) c k − 1 c j j =1 + K ( k + 1 , 0) c k +1 c 0 − K ( k , 0) c k c 0 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: dc 0 dt = 0 and for k = 1 , ..., M − 1 : M dc k � dt = K ( k + 1 , j ) c k +1 c j − 2 K ( k , j ) c k c j + K ( j , k − 1) c k − 1 c j j =1 and for clusters of maximal size k = M : M − 1 dc M � = − 2 K ( M , j ) c M c j + K ( j , M − 1) c M − 1 c j dt j =1 − 2 K ( M , M ) c 2 M + K ( M , M − 1) c M − 1 c M Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 5 / 14
Scaling behaviour Conservation of mass: � ∞ kc k dk = 1 0 Typical size: s ( t ) := M 2 ( t ) M 1 ( t ) � ∞ 0 k λ c k ( t ) dk is the λ -th moment. where M λ ( t ) := Ben-Naim and Krapivsky, 2003, proposed a self-similar ansatz c k ( t ) = s ( t ) α F ( z ) s ( t ) , α = (3 − 2 λ ) − 1 for λ < 3 k where z = 2 , and showed that for λ = 0, 1 3 . s ( t ) ∼ t 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: dc k � dt = ( ... ) + J δ k , 1 j so that M 1 ∼ t . Krapivsky, 2015, showed s ( t ) ∼ t 2 for K ( i , j ) = ij . We generalised this to 2 s ( t ) ∼ t 3 − 2 λ 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: dc k � dt = ( ... ) + J δ k , 1 − Γ( k ) c k j Evaporation: � Γ( k + 1) c k +1 − Γ( k ) c k , 0 ≤ k ≤ M − 1 dc k � dt = ( ... ) + J δ k , 1 + − Γ( M ) c M , k = M j where Γ( k ) = γ 0 k γ . Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 8 / 14
Solver d c dt = f ( c ) c 0 ( t ) . . where c ( t ) = and . c M ( t ) − γ 0 0 γ j ( K (2 , j ) c 2 c j − ... ) + J − γ 0 1 γ � f ( c ) = . . . j ( − 2 K ( M , j ) c M c j + ... ) − γ 0 M γ � and a ‘monodisperse’ initial condition: c k (0) = δ 1 , 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: � T simulated time Computation time ∼ M 2 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? Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 11 / 14
Conclusions Existence of steady states J Settling time for the steady state increased with the ratio γ 0 Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 12 / 14
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 of 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. Francis Aznaran (Warwick University) Exchange-driven growth with a source and sink of particles August 18th 2017 14 / 14
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