Skip to main content
SpringerLink
Log in

Convergence to Equilibrium for the Discrete Coagulation-Fragmentation Equations with Detailed Balance

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

Under the condition of detailed balance and some additional restrictions on the size of the coefficients, we identify the equilibrium distribution to which solutions of the discrete coagulation-fragmentation system of equations converge for large times, thus showing that there is a critical mass which marks a change in the behavior of the solutions. This was previously known only for particular cases as the generalized Becker–Döring equations. Our proof is based on an inequality between the entropy and the entropy production which also gives some information on the rate of convergence to equilibrium for solutions under the critical mass.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aizenman, M., Bak, T.A.: Convergence to equilibrium in a system of reacting polymers. Commun. Math. Phys. 65(3), 203–230 (1979)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Aldous, D.J.: Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernouilli 5(1), 3–48 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arnold, A., Carrillo, J.A., Desvillettes, L., Dolbeault, J., Jüngel, A., Lederman, C., Markowich, P.A., Toscani, G., Villani, C.: Entropies and equilibria of many-particle systems: an essay on recent research. Mon. Math. 142(1–2), 35–43 (2004)

    Article  MATH  Google Scholar 

  4. Ball, J.M., Carr, J.: Asymptotic behaviour of solutions to the Becker–Döring equations for arbitrary initial data. Proc. Roy. Soc. Edinb. Sect. A 108, 109–116 (1988)

    MATH  MathSciNet  Google Scholar 

  5. Ball, J.M., Carr, J.: The discrete coagulation-fragmentation equations: existence, uniqueness and density conservation. J. Stat. Phys. 61, 203–234 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  6. Ball, J.M., Carr, J., Penrose, O.: The Becker–Döring cluster equations: basic properties and asymptotic behaviour of solutions. Commun. Math. Phys. 104, 657–692 (1986)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Bonilla, L.L., Carpio, A., Neu, J.C.: Igniting homogeneous nucleation. In: F.J. Higuera, J. Jiménez, J.M. Vega (eds.) Simplicity, Rigor and Relevance in Fluid Mechanics. CIMNE, Barcelona (2004)

    Google Scholar 

  8. Cañizo Rincón, J.A.: Asymptotic behaviour of solutions to the generalized Becker–Döring equations for general initial data. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461(2064), 3731–3745 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Cañizo Rincón, J.A.: Some problems related to the study of interaction kernels: coagulation, fragmentation and diffusion in kinetic and quantum equations. PhD thesis. Universidad de Granada (2006)

  10. Carr, J.: Asymptotic behaviour of solutions to the coagulation-fragmentation equations, I: the strong fragmentation case. Proc. Roy. Soc. Edinb. Sect. A 121, 231–244 (1992)

    MATH  MathSciNet  Google Scholar 

  11. Carr, J., da Costa, F.P.: Asymptotic behavior of solutions to the coagulation-fragmentation equations, II: weak fragmentation. J. Stat. Phys. 77, 98–123 (1994)

    ADS  MathSciNet  Google Scholar 

  12. Carrillo, J., Desvillettes, L., Fellner, K.: Exponential decay towards equilibrium for the inhomogeneous Aizenman–Bak model. Preprint

  13. da Costa, F.P.: Asymptotic behaviour of low density solutions to the generalized Becker–Döring equations. Nonlinear Differ. Equ. Appl. 5, 23–37 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Drake, R.L.: A general mathematical survey of the coagulation equation. In: Hidy, G.M., Brock, J.R. (eds.) Topics in Current Aerosol Research (Part 2). International Reviews in Aerosol Physics and Chemistry, vol. 3, pp. 201–376. Pergamon, New York (1972)

    Google Scholar 

  15. Grabe, M., Neu, J., Oster, G., Nollert, P.: Protein interactions and membrane geometry. Biophys. J. 84, 654–868 (2003)

    Google Scholar 

  16. Jabin, P.-E., Niethammer, B.: On the rate of convergence to equilibrium in the Becker–Döring equations. J. Differ. Equ. 191, 518–543 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Laurençot, P., Mischler, S.: Convergence to equilibrium for the continuous coagulation-fragmentation equation. Bull. Sci. Math. 127(3), 179–190 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Laurençot, P., Mischler, S.: On coalescence equations and related models. In: Modeling and Computational Methods for Kinetic Equations. Model. Simul. Sci. Eng. Technol., pp. 321–356. Birkhäuser, Boston (2004)

    Google Scholar 

  19. Neu, J., Cañizo, J.A., Bonilla, L.L.: Three eras of micellization. Phys. Rev. E 66, 061406 (2002)

    Article  ADS  Google Scholar 

  20. Penrose, O.: The Becker–Döring equations at large times and their connection with the LSW theory of coarsening. J. Stat. Phys. 19, 243–267 (1997)

    Article  ADS  Google Scholar 

  21. Penrose, O., Lebowitz, J.L.: Towards a Rigorous Theory of Metastability. Studies in Statistical Mechanics, vol. VII. North-Holland, Amsterdam (1979)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CEREMADE, University of Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775, Paris, Cedex 16, France

    José A. Cañizo

Authors
  1. José A. Cañizo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to José A. Cañizo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cañizo, J.A. Convergence to Equilibrium for the Discrete Coagulation-Fragmentation Equations with Detailed Balance. J Stat Phys 129, 1–26 (2007). https://doi.org/10.1007/s10955-007-9373-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-007-9373-2

Keywords

Search

Navigation