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February  2021, 41(2): 747-775. doi: 10.3934/dcds.2020299

Large time behavior of exchange-driven growth

1. 

WMG, University of Warwick, UK, Mathematics Institute, University of Warwick, UK

2. 

Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60 (room 2.023), D-53115 Bonn, Germany

*Corresponding author: E.esenturk.1@warwick.ac.uk

Received  April 2019 Revised  April 2020 Published  August 2020

Exchange-driven growth (EDG) is a model in which pairs of clusters interact by exchanging single unit with a rate given by a kernel $ K(j, k) $. Despite EDG model's common use in the applied sciences, its rigorous mathematical treatment is very recent. In this article we study the large time behaviour of EDG equations. We show two sets of results depending on the properties of the kernel $ (i) $ $ K(j, k) = b_{j}a_{k} $ and $ (ii) $ $ K(j, k) = ja_{k}+b_{j}+\varepsilon\beta_{j}\alpha_{k} $. For type I kernels, under the detailed balance assumption, we show that the system admits unique equilibrium up to a critical mass $ \rho_{s} $ above which there is no equilibrium. We prove that if the system has an initial mass below $ \rho_{s} $ then the solutions converge to a unique equilibrium distribution strongly where if the initial mass is above $ \rho_{s} $ then the solutions converge to cricital equilibrium distribution in a weak sense. For type II kernels, we do not make any assumption of detailed balance and equilibrium is shown as a consequence of contraction properties of solutions. We provide two separate results depending on the monotonicity of the kernel or smallness of the total mass. For the first case we prove exponential convergence in the number of clusters norm and for the second we prove exponential convergence in the total mass norm.

Citation: Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299
References:
[1]

J. BeltránM. Jara and C. Landim, A martingale problem for an absorbed diffusion: The nucleation phase of condensing zero range processes, Prob. Theo. Rel. Fi., 169 (2017), 1169-1220.  doi: 10.1007/s00440-016-0749-6.  Google Scholar

[2]

J. M. BallJ. Carr and O. Penrose, Becker-Doring Cluster equations: Basic properties ad asymptotic behavior of solutions, Comm. Math. Phys., 104 (1986), 657-692.  doi: 10.1007/BF01211070.  Google Scholar

[3]

J. Cañizo, Convergence to equilibrium for the discrete coagulation-fragmentation equations with detailed balance, J. Stat. Phys., 129 (2007), 1-26.  doi: 10.1007/s10955-007-9373-2.  Google Scholar

[4]

J. CaoP. Chleboun and S. Grosskinsky, Dynamics of condensation in the totally asymmetric inclusion process, J. Stat. Phys., 155 (2014), 523-543.  doi: 10.1007/s10955-014-0966-2.  Google Scholar

[5]

J. Carr and F. P. da Costa, Asymptotic behavior of solutions to the coagulation-fragmentation equations II: weak fragmentation, J. Stat. Phys., 77 (1994), 89-123.  doi: 10.1007/BF02186834.  Google Scholar

[6]

Y. X. Chau, C. Connaughton and S. Grosskinsky, Explosive condensation in symmetric mass transport models, J. Stat. Mech.: Theo. Exp., 2015 (2015), P11031. Google Scholar

[7]

E. Esenturk, Mathematical theory of exchange-driven growth, Nonlinearity, 31 (2018), 3460-3483.  doi: 10.1088/1361-6544/aaba8d.  Google Scholar

[8]

E. Esenturk and C. Connaughton, Role of zero clusters in exchange-driven growth with and without input, Phys. Rev. E, 101 (2020), 052134-052146.   Google Scholar

[9]

N. Fournier and S. Mischler, Exponential trend to equilibrium for discrete coagulation equations with strong fragmentation and without balance condition, Proc. R. Soc. Lond. Ser. A, 460 (2004), 2477-2486.  doi: 10.1098/rspa.2004.1294.  Google Scholar

[10]

C. Godréche, Dynamics of condensation in zero-range processes, J. Phys. A: Math. Gen., 36 (2003), 6313-6328.  doi: 10.1088/0305-4470/36/23/303.  Google Scholar

[11]

S. GrosskinskyG. M. Schütz and H. Spohn, Condensation in the zero range process: Stationary and dynamical properties, J. Stat. Phys., 113 (2003), 389-410.  doi: 10.1023/A:1026008532442.  Google Scholar

[12]

C. Godréche and J. M. Drouffe, Coarsening dynamics of zero-range processes, J. Phys. A: Math. Theo., 50 (2016), 015005, 24. doi: 10.1088/1751-8113/50/1/015005.  Google Scholar

[13]

S. Ispalatov, P. L. Krapivsky and S. Redner, Wealth distributions in models of capital exchange, Euro. J. Phys. B., 2 (1998), 267. Google Scholar

[14]

P.-E. Jabin and B. Niethammer, On the rate of convergence to equilibrium in the Becker–Döring equations, J. Diff. Equ., 191 (2003), 518-543.  doi: 10.1016/S0022-0396(03)00021-4.  Google Scholar

[15]

W. Jatuviriyapornchai and S. Grosskinsky, Coarsening dynamics in condensing zero-range processes and size-biased birth death chains, J. Phys A: Math. Theo., 49 (2016), 185005, 19pp. doi: 10.1088/1751-8113/49/18/185005.  Google Scholar

[16]

W. Jatuviriyapornchai and S. Grosskinsky, Derivation of mean-field equations for stochastic particle systems, Stoch. Proc. Appl., 129 (2019), 1455-1475.  doi: 10.1016/j.spa.2018.05.006.  Google Scholar

[17] P. L. KrapivskyS. Redner and E. Ben-Naim, A Kinetic View of Statistical Physics, Cambridge University Press, 2010.  doi: 10.1017/CBO9780511780516.  Google Scholar
[18]

J. Ke and Z. Lin, Kinetics of migration-driven aggregation processes with birth and death, Phys. Rev. E., 67 (2002), 031103. Google Scholar

[19]

P. Laurencot and S. Mischler, From the Becker–Döring to the Lifshitz–Slyozov–Wagner equations, J. Stat. Phys., 106 2002,957–991. doi: 10.1023/A:1014081619064.  Google Scholar

[20]

F. Leyvraz and S. Redner, Scaling theory for migration-driven aggregate growth, Phys. Rev. Lett., 88 (2002), 068301. Google Scholar

[21]

P. R. Murray Ryan, Algebraic decay to equilibrium for the Becker–Döring equations, SIAM J. Math. Anal., 48 (2016), 2819-2842.  doi: 10.1137/15M1038578.  Google Scholar

[22]

E. B. Naim and P. L. Krapivsky, Exchange-driven growth, Phys. Rev. E, 68 (2003), 031104. Google Scholar

[23]

B. Niethammer, Self-similarity in Smoluchowski's coagulation equation, Jahresber. Dtsch. Math. Ver., 116 (2014), 43-65.  doi: 10.1365/s13291-014-0085-7.  Google Scholar

[24]

A. Schlichting, The exchange-driven growth model: Basic properties and long-time behaviour, J. Nonlin. Sci, 30 (2019), 793-830.  doi: 10.1007/s00332-019-09592-x.  Google Scholar

[25]

M. Slemrod, Trend to equilibrium in the Becker–Doring cluster equations, Nonlinearity, 2 (1989), 429-443.  doi: 10.1088/0951-7715/2/3/004.  Google Scholar

[26]

J. Wattis, An introduction to mathematical models of coagulation-fragmentation processes: a discrete deterministic mean-field approach, Phys. D, 222 (2006), 1-20.  doi: 10.1016/j.physd.2006.07.024.  Google Scholar

[27]

B. Waclaw and M. R. Evans, Explosive condensation in a mass transport model, Physical Rev. Lett., 108 (2012), 070601. Google Scholar

show all references

References:
[1]

J. BeltránM. Jara and C. Landim, A martingale problem for an absorbed diffusion: The nucleation phase of condensing zero range processes, Prob. Theo. Rel. Fi., 169 (2017), 1169-1220.  doi: 10.1007/s00440-016-0749-6.  Google Scholar

[2]

J. M. BallJ. Carr and O. Penrose, Becker-Doring Cluster equations: Basic properties ad asymptotic behavior of solutions, Comm. Math. Phys., 104 (1986), 657-692.  doi: 10.1007/BF01211070.  Google Scholar

[3]

J. Cañizo, Convergence to equilibrium for the discrete coagulation-fragmentation equations with detailed balance, J. Stat. Phys., 129 (2007), 1-26.  doi: 10.1007/s10955-007-9373-2.  Google Scholar

[4]

J. CaoP. Chleboun and S. Grosskinsky, Dynamics of condensation in the totally asymmetric inclusion process, J. Stat. Phys., 155 (2014), 523-543.  doi: 10.1007/s10955-014-0966-2.  Google Scholar

[5]

J. Carr and F. P. da Costa, Asymptotic behavior of solutions to the coagulation-fragmentation equations II: weak fragmentation, J. Stat. Phys., 77 (1994), 89-123.  doi: 10.1007/BF02186834.  Google Scholar

[6]

Y. X. Chau, C. Connaughton and S. Grosskinsky, Explosive condensation in symmetric mass transport models, J. Stat. Mech.: Theo. Exp., 2015 (2015), P11031. Google Scholar

[7]

E. Esenturk, Mathematical theory of exchange-driven growth, Nonlinearity, 31 (2018), 3460-3483.  doi: 10.1088/1361-6544/aaba8d.  Google Scholar

[8]

E. Esenturk and C. Connaughton, Role of zero clusters in exchange-driven growth with and without input, Phys. Rev. E, 101 (2020), 052134-052146.   Google Scholar

[9]

N. Fournier and S. Mischler, Exponential trend to equilibrium for discrete coagulation equations with strong fragmentation and without balance condition, Proc. R. Soc. Lond. Ser. A, 460 (2004), 2477-2486.  doi: 10.1098/rspa.2004.1294.  Google Scholar

[10]

C. Godréche, Dynamics of condensation in zero-range processes, J. Phys. A: Math. Gen., 36 (2003), 6313-6328.  doi: 10.1088/0305-4470/36/23/303.  Google Scholar

[11]

S. GrosskinskyG. M. Schütz and H. Spohn, Condensation in the zero range process: Stationary and dynamical properties, J. Stat. Phys., 113 (2003), 389-410.  doi: 10.1023/A:1026008532442.  Google Scholar

[12]

C. Godréche and J. M. Drouffe, Coarsening dynamics of zero-range processes, J. Phys. A: Math. Theo., 50 (2016), 015005, 24. doi: 10.1088/1751-8113/50/1/015005.  Google Scholar

[13]

S. Ispalatov, P. L. Krapivsky and S. Redner, Wealth distributions in models of capital exchange, Euro. J. Phys. B., 2 (1998), 267. Google Scholar

[14]

P.-E. Jabin and B. Niethammer, On the rate of convergence to equilibrium in the Becker–Döring equations, J. Diff. Equ., 191 (2003), 518-543.  doi: 10.1016/S0022-0396(03)00021-4.  Google Scholar

[15]

W. Jatuviriyapornchai and S. Grosskinsky, Coarsening dynamics in condensing zero-range processes and size-biased birth death chains, J. Phys A: Math. Theo., 49 (2016), 185005, 19pp. doi: 10.1088/1751-8113/49/18/185005.  Google Scholar

[16]

W. Jatuviriyapornchai and S. Grosskinsky, Derivation of mean-field equations for stochastic particle systems, Stoch. Proc. Appl., 129 (2019), 1455-1475.  doi: 10.1016/j.spa.2018.05.006.  Google Scholar

[17] P. L. KrapivskyS. Redner and E. Ben-Naim, A Kinetic View of Statistical Physics, Cambridge University Press, 2010.  doi: 10.1017/CBO9780511780516.  Google Scholar
[18]

J. Ke and Z. Lin, Kinetics of migration-driven aggregation processes with birth and death, Phys. Rev. E., 67 (2002), 031103. Google Scholar

[19]

P. Laurencot and S. Mischler, From the Becker–Döring to the Lifshitz–Slyozov–Wagner equations, J. Stat. Phys., 106 2002,957–991. doi: 10.1023/A:1014081619064.  Google Scholar

[20]

F. Leyvraz and S. Redner, Scaling theory for migration-driven aggregate growth, Phys. Rev. Lett., 88 (2002), 068301. Google Scholar

[21]

P. R. Murray Ryan, Algebraic decay to equilibrium for the Becker–Döring equations, SIAM J. Math. Anal., 48 (2016), 2819-2842.  doi: 10.1137/15M1038578.  Google Scholar

[22]

E. B. Naim and P. L. Krapivsky, Exchange-driven growth, Phys. Rev. E, 68 (2003), 031104. Google Scholar

[23]

B. Niethammer, Self-similarity in Smoluchowski's coagulation equation, Jahresber. Dtsch. Math. Ver., 116 (2014), 43-65.  doi: 10.1365/s13291-014-0085-7.  Google Scholar

[24]

A. Schlichting, The exchange-driven growth model: Basic properties and long-time behaviour, J. Nonlin. Sci, 30 (2019), 793-830.  doi: 10.1007/s00332-019-09592-x.  Google Scholar

[25]

M. Slemrod, Trend to equilibrium in the Becker–Doring cluster equations, Nonlinearity, 2 (1989), 429-443.  doi: 10.1088/0951-7715/2/3/004.  Google Scholar

[26]

J. Wattis, An introduction to mathematical models of coagulation-fragmentation processes: a discrete deterministic mean-field approach, Phys. D, 222 (2006), 1-20.  doi: 10.1016/j.physd.2006.07.024.  Google Scholar

[27]

B. Waclaw and M. R. Evans, Explosive condensation in a mass transport model, Physical Rev. Lett., 108 (2012), 070601. Google Scholar

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