Skip to main content
SpringerLink
Log in

Lattice Models Solvable Through the Full Interval Method on Links

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

Abstract

A two state model on a one dimensional lattice is considered, where the evolution of the state of each site is determined by the states of that site and its neighboring sites. Corresponding to this original lattice, a derived lattice is introduced the sites of which are the links of the original lattice. It is shown that there is only one reaction on the original lattice, which results in the derived lattice being solvable through the full interval method. And that reaction corresponds to the one dimensional stochastic non-consensus opinion model. A one dimensional non-consensus opinion model with deterministic evolution has already been introduced. Here this is extended to be a model which has a stochastic evolution. Discrete time evolution of such a model is investigated, including the two limiting cases of small probabilities for evolution (resulting to an effectively continuous-time evolution), and deterministic evolution. The formal solution to the evolution equation is obtained and the large time behavior of the system is investigated. Some special cases are studied in more detail.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Lee, B.P.: Renormalization group calculation for the reaction kA to OE. J. Phys. A27, 2633–2652 (1994)

    ADS  Google Scholar 

  2. Cardy, J.L.: Renormalization group approach to reaction-diffusion problems. In: J.M. Drouffe, J.B. Zuber (eds.) The Mathematical Beauty of Physics, Adv. Ser. Math. Phys., vol. 24. World Scientific, Singapore (1997)

  3. Alcaraz, F.C., Droz, M., Henkel, M., Rittenberg, V.: Reaction–diffusion processes, critical dynamics, and quantum chains. Ann. Phys. NY 230, 250–302 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. Krebs, K., Pfannmuller, M.P., Wehefritz, B., Hinrichsen, H.: Finite-size scaling studies of one-dimensional reaction-diffusion systems. Part I. Analytical results. J. Stat. Phys. 78, 1429–1470 (1995)

    Article  ADS  MATH  Google Scholar 

  5. Simon, H.: Concentration for one and two-species one-dimensional reaction-diffusion systems. J. Phys. A28, 6585–6604 (1995)

    ADS  Google Scholar 

  6. Privman, V., Cadilhe, A.M.R., Glasser, M.L.: Exact solutions of anisotropic diffusion-limited reactions with coagulation and annihilation. J. Stat. Phys. 81, 881–899 (1995)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Henkel, M., Orlandini, E., Schütz, G.M.: Equivalences between stochastic systems. J. Phys. A28, 6335–6344 (1995)

    ADS  Google Scholar 

  8. Henkel, M., Orlandini, E., Santos, J.: Reaction–diffusion processes from equivalent integrable quantum chains. Ann. Phys. NY 259, 163–231 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Lushnikov, A.A.: Binary reaction \(1+1\rightarrow 0\) in one dimension. Sov. Phys. JETP 64, 811–815 (1986)

    Google Scholar 

  10. Ben-Avraham, D.: The method of inter-particle distribution functions for diffusion-reaction systems in one dimension. Mod. Phys. Lett. B09, 895–919 (1995)

    Article  ADS  Google Scholar 

  11. Castellano, C., Fortunato, S., Loreto, V.: Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591–646 (2009)

    Article  ADS  Google Scholar 

  12. Shao, J., Havlin, S., Stanley, H.E.: Dynamic opinion model and invasion percolation. Phys. Rev. Lett. 103, 018701 (2009)

    Article  ADS  Google Scholar 

  13. Ben-Avraham, D.: Exact solution of the nonconsensus opinion model on the line. Phys. Rev. E83, 050101 (2011)

    ADS  Google Scholar 

  14. Li, Q., Braunstein, L.A., Wang, H., Shao, J., Stanley, H.E., Havlin, S.: Non-consensus opinion models on complex networks. J. Stat. Phys. 151, 92–112 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. Helbing, D.: Traffic and related Self-driven many-particle systems. Rev. Mod. Phys. 73, 1067–1141 (2001)

    Article  ADS  Google Scholar 

  16. Rajewsky, N., Santen, L., Schadschneider, A., Schreckenberg, M.: The asymmetric exclusion process: comparison of update procedures. J. Stat. Phys. 92, 151–194 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Henkel, M., Hinrichsen, H., Lübeck, S.: Non-equilibrium phase transitions. Absorbing phase transitions, vol. 1. Springer, Berlin (2008)

    MATH  Google Scholar 

  18. Aghamohammadi, A., Khorrami, M.: Models solvable through the empty-interval method. Eur. Phys. J. B47, 583–586 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  19. Aghamohammadi, A., Khorrami, M.: The spectrum and the phase transition of models solvable through the full interval method. J. Stat. Mech. 2012, P07023 (2012)

Download references

Acknowledgments

This work was supported by the research council of the Alzahra University.

Author information

Authors and Affiliations

  1. Department of Physics, Alzahra University, Tehran, 19938-93973, Iran

    Amir Aghamohammadi & Mohammad Khorrami

Authors
  1. Amir Aghamohammadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohammad Khorrami
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amir Aghamohammadi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aghamohammadi, A., Khorrami, M. Lattice Models Solvable Through the Full Interval Method on Links. J Stat Phys 157, 1320–1330 (2014). https://doi.org/10.1007/s10955-014-1097-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-014-1097-5

Keywords

Search

Navigation