Computer Simulations of Self-Repelling Fractional Brownian Motion

  • Jinky Bornales
  • Cresente Cabahug
  • Roel Baybayon
  • Sim Bantayan
  • Beverly Gemao
Conference paper
First Online:
Part of the Trends in Mathematics book series (TM)

Abstract

Self-repelling fractional Brownian motion (fBm) has been constructed, generalizing the Edwards model for the conformations of chain polymers. In this context of particular interest is the predicted scaling behaviour of their end-to-end length, i.e. the anomalous diffusion of self-repelling fBm. We briefly present the model and a heuristic formula of the scaling behaviour for general dimension and Hurst index, and then our computer simulations of self-repelling fBm paths, their method and first results.

Keyword

Fractional Brownian motion 

Mathematics Subject Classification (2010)

60G22 
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Notes

Acknowledgements

This work was financed by Portuguese national funds through FCT – Fundaão para a Ciência e Tecnologia within the project PTDC/MAT-STA/1284/2012. We are grateful to W. Bock and S. Eleuterio for sharing their simulation program and for generous advice. Roel Baybayon and Sim Bantayan are also grateful to the Department of Science and Technology – ASTHRD for the scholarship grant.

References

  1. 1.
    Besold, G., Guo, H., Zuckermann, M.J.: Off-lattice Monte Carlo simulation of the discrete Edwards model. J. Polym. Sci. B Polym. Phys. 38, 10531068 (2000)CrossRefGoogle Scholar
  2. 2.
    Biagini, F., Hu, Y., Oksendal, B., Zhang, T.: Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, Berlin (2007)MATHGoogle Scholar
  3. 3.
    Bock, W., Bornales, J., Eleuterio, S.: preprint, in preparationGoogle Scholar
  4. 4.
    Bock, W., Bornales, J., Streit, L.: Self-repelling fractional Brownian motion – the case of higher order self-intersections. arXiv:1311.4375 [math-ph]Google Scholar
  5. 5.
    Bornales, J., Oliveira, M.J., Streit, L.: Self-repelling fractional Brownian motion – a generalized Edwards model for chain polymers. In: Accardi, L., Freudenberg, W., Ohya, M. (eds.) Quantum Bio-Informatics V. Proceedings of Quantum Bio-Informatics 2011. Quantum Probability and White Noise Analysis, vol. 30, pp. 389–401. World Scientific, Singapore/New Jersey/London (2013)Google Scholar
  6. 6.
    Dekeyzer, R., Maritan, A., Stella, A.: Excluded-volume effects in linear polymers: universality of generalized self-avoiding walks. Phys. Rev. B 31, 4659–4662 (1985)CrossRefGoogle Scholar
  7. 7.
    Dekeyzer, R., Maritan, A., Stella, A.: Random walks with intersections: static and dynamic fractal properties Phys. Rev. A 36, 2338–2351 (1987)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Edwards, S.F.: The statistical mechanics of polymers with excluded volume. Proc. R. Soc. 85, 613–624 (1965)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Flory, P.J.: Principles of Polymer Chemistry. Cornell University Press, Ithaca (1953)Google Scholar
  10. 10.
    Grothaus, M., Oliveira, M.J., Silva, J.-L., Streit, L.: Self-avoiding fBm – the Edwards model. J. Stat. Phys. 145, 1513–1523 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    van der Hofstad, R., König, W.: A survey of one-dimensional random polymers. J. Stat. Phys. 103, 915–944 (2001)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hu, Y., Nualart, D.: Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab. 33, 948–983 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kosmas, M.K., Freed, K.F.: On scaling theories of polymer solutions. J. Chem. Phys. 69, 3647–3659 (1978)CrossRefGoogle Scholar
  14. 14.
    Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., Teller, E.: Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953)CrossRefGoogle Scholar
  15. 15.
    Mishura, Y.: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics, vol. 1929. Springer, Berlin/Heidelberg (2008)Google Scholar
  16. 16.
    Nienhuis, B.: Exact critical point and critical exponents of O(n) models in two dimensions. Phys. Rev. Lett. 49, 1062 (1982)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Pelissetto, A., Vicari, E.: Critical phenomena and renormalization-group theory. Phys. Rep. 368, 549–727 (2002)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Talagrand, M.: Multiple points of trajectories of multiparameter fractional Brownian motion. Prob. Theory Relat. Fields 122, 545–563 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Varadhan, S.R.S.: Appendix to Euclidean quantum field theory by K. Symanzik. In: Jost, R. (ed.) Local Quantum Theory, p. 285. Academic Press, New York (1970)Google Scholar
  20. 20.
    Westwater, J.: On Edwards’ model for polymer chains. Commun. Math. Phys. 72, 131–174 (1980)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Jinky Bornales
    • 1
  • Cresente Cabahug
    • 1
  • Roel Baybayon
    • 1
  • Sim Bantayan
    • 1
  • Beverly Gemao
    • 1
  1. 1.Physics Department, Mindanao State University – Iligan Institute of TechnologyIligan CityPhilippines

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