• Previous Article
    Attainability property for a probabilistic target in wasserstein spaces
  • DCDS Home
  • This Issue
  • Next Article
    Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case
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  February 2021 Early access  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 and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[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. 

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

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

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

[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.

[20]

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

[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.

[22]

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

[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.

[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.

[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.

[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.

[27]

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

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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[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. 

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

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

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

[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.

[20]

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

[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.

[22]

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

[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.

[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.

[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.

[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.

[27]

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

[1]

Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15

[2]

Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic and Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701

[3]

Marie Doumic, Miguel Escobedo. Time asymptotics for a critical case in fragmentation and growth-fragmentation equations. Kinetic and Related Models, 2016, 9 (2) : 251-297. doi: 10.3934/krm.2016.9.251

[4]

Christopher F. Novak. Discontinuity-growth of interval-exchange maps. Journal of Modern Dynamics, 2009, 3 (3) : 379-405. doi: 10.3934/jmd.2009.3.379

[5]

Claude-Michel Brauner, Michael L. Frankel, Josephus Hulshof, Alessandra Lunardi, G. Sivashinsky. On the κ - θ model of cellular flames: Existence in the large and asymptotics. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 27-39. doi: 10.3934/dcdss.2008.1.27

[6]

Bertrand Maury, Aude Roudneff-Chupin, Filippo Santambrogio. Congestion-driven dendritic growth. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1575-1604. doi: 10.3934/dcds.2014.34.1575

[7]

Shihu Li, Wei Liu, Yingchao Xie. Small time asymptotics for SPDEs with locally monotone coefficients. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4801-4822. doi: 10.3934/dcdsb.2020127

[8]

Yang Yang, Kam C. Yuen, Jun-Feng Liu. Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims. Journal of Industrial and Management Optimization, 2018, 14 (1) : 231-247. doi: 10.3934/jimo.2017044

[9]

Marcelo E. de Oliveira, Luiz M. G. Neto. Directional entropy based model for diffusivity-driven tumor growth. Mathematical Biosciences & Engineering, 2016, 13 (2) : 333-341. doi: 10.3934/mbe.2015005

[10]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3419-3440. doi: 10.3934/dcdss.2020426

[11]

Daniela Cárcamo-Díaz, Jesús F. Palacián, Claudio Vidal, Patricia Yanguas. Nonlinear stability of elliptic equilibria in hamiltonian systems with exponential time estimates. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5183-5208. doi: 10.3934/dcds.2021073

[12]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[13]

Bernard Ducomet. Asymptotics for 1D flows with time-dependent external fields. Conference Publications, 2007, 2007 (Special) : 323-333. doi: 10.3934/proc.2007.2007.323

[14]

Mustapha Mokhtar-Kharroubi, Quentin Richard. Time asymptotics of structured populations with diffusion and dynamic boundary conditions. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4087-4116. doi: 10.3934/dcdsb.2018127

[15]

Tomasz Komorowski. Long time asymptotics of a degenerate linear kinetic transport equation. Kinetic and Related Models, 2014, 7 (1) : 79-108. doi: 10.3934/krm.2014.7.79

[16]

Barry Simon. Zeros of OPUC and long time asymptotics of Schur and related flows. Inverse Problems and Imaging, 2007, 1 (1) : 189-215. doi: 10.3934/ipi.2007.1.189

[17]

Vladimir Varlamov. Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 675-702. doi: 10.3934/dcds.2001.7.675

[18]

David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135

[19]

Paul Deuring. Spatial asymptotics of mild solutions to the time-dependent Oseen system. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1833-1849. doi: 10.3934/cpaa.2021044

[20]

Mingming Chen, Xianguo Geng, Kedong Wang. Long-time asymptotics for the modified complex short pulse equation. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022060

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (200)
  • HTML views (210)
  • Cited by (0)

Other articles
by authors

[Back to Top]