Journal of Statistical Physics

, Volume 167, Issue 5, pp 1233–1243 | Cite as

Information Dimension of Stochastic Processes on Networks: Relating Entropy Production to Spectral Properties

  • Oliver Mülken
  • Sarah Heinzelmann
  • Maxim Dolgushev
Article

Abstract

We consider discrete stochastic processes, modeled by classical master equations, on networks. The temporal growth of the lack of information about the system is captured by its non-equilibrium entropy, defined via the transition probabilities between different nodes of the network. We derive a relation between the entropy and the spectrum of the master equation’s transfer matrix. Our findings indicate that the temporal growth of the entropy is proportional to the logarithm of time if the spectral density shows scaling. In analogy to chaos theory, the proportionality factor is called (stochastic) information dimension and gives a global characterization of the dynamics on the network. These general results are corroborated by examples of regular and of fractal networks.

Keywords

Networks Fractals Entropy Stochastic thermodynamics 

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, vol. 55. Courier Corporation, Washington (1964)MATHGoogle Scholar
  2. 2.
    Agliari, E., Blumen, A., Cassi, D.: Slow encounters of particle pairs in branched structures. Phys. Rev. E 89(5), 052147 (2014)ADSCrossRefGoogle Scholar
  3. 3.
    Agliari, E., Contucci, P., Giardina, C.: A Random Walk in Diffusion Phenomena and Statistical Mechanics. Advances in Disordered Systems, Random Processes and Some Applications, p. 43. Cambridge university press, Cambridge (2016)Google Scholar
  4. 4.
    Alexander, S., Orbach, R.: Density of states on fractals: fractons. J. Phys. Lett. 43, 625–631 (1982)CrossRefGoogle Scholar
  5. 5.
    Argyrakis, P.: Information dimension in random-walk processes. Phys. Rev. Lett. 59(15), 1729 (1987)ADSCrossRefGoogle Scholar
  6. 6.
    Ashcroft, N.W., Mermin, D.: Solid State Physics. Saunders, Philadelphia (1976)MATHGoogle Scholar
  7. 7.
    Blumen, A., Jurjiu, A., Koslowski, T., von Ferber, C.: Dynamics of Vicsek fractals, models for hyperbranched polymers. Phys. Rev. E 67(6), 061103 (2003)ADSCrossRefGoogle Scholar
  8. 8.
    Burioni, R., Cassi, D.: Fractals without anomalous diffusion. Phys. Rev. E 49(3), R1785 (1994)ADSCrossRefGoogle Scholar
  9. 9.
    Burioni, R., Cassi, D.: Spectral dimension of fractal trees. Phys. Rev. E 51(4), 2865 (1995)ADSCrossRefGoogle Scholar
  10. 10.
    Burioni, R., Cassi, D.: Random walks on graphs: ideas, techniques and results. J. Phys. A Math. Gen. 38(8), R45 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cai, C., Chen, Z.Y.: Rouse dynamics of a dendrimer model in the \(\theta \) condition. Macromolecules 30, 5104–5117 (1997)ADSCrossRefGoogle Scholar
  12. 12.
    Cherstvy, A.G., Chechkin, A.V., Metzler, R.: Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes. New J. Phys. 15(8), 083039 (2013)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Esposito, M.: Stochastic thermodynamics under coarse graining. Phys. Rev. E 85(4), 041125 (2012)ADSCrossRefGoogle Scholar
  14. 14.
    Fürstenberg, F., Dolgushev, M., Blumen, A.: Analytical model for the dynamics of semiflexible dendritic polymers. J. Chem. Phys. 136, 154904 (2012)ADSCrossRefGoogle Scholar
  15. 15.
    Gotlib, Y.Y., Markelov, D.A.: Theory of the relaxation spectrum of a dendrimer macromolecule. Polym. Sci. Ser. A 44(12), 1341–1350 (2002)Google Scholar
  16. 16.
    Goutsias, J., Jenkinson, G.: Markovian dynamics on complex reaction networks. Phys. Rep. 529(2), 199 (2013)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Gurtovenko, A.A., Markelov, D.A., Gotlib, Y.Y., Blumen, A.: Dynamics of dendrimer-based polymer networks. J. Chem. Phys. 119, 7579 (2003)ADSCrossRefGoogle Scholar
  18. 18.
    van Kampen, N.: Stochastic Processes in Physics and Chemistry. North Holland, Amsterdam (1990)MATHGoogle Scholar
  19. 19.
    Klafter, J., Zumofen, G., Blumen, A.: On the propagator of Sierpinski gaskets. J. Phys. A Math. Gen. 24, 4835–4842 (1991)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Metzler, R., Redner, S., Oshanin, G.: First-Passage Phenomena and Their Applications, vol. 35. World Scientific, Singapore (2014)CrossRefMATHGoogle Scholar
  21. 21.
    Mielke, J., Dolgushev, M.: Relaxation dynamics of semiflexible fractal macromolecules. Polymers 8(7), 263 (2016)CrossRefGoogle Scholar
  22. 22.
    Mülken, O., Blumen, A.: Efficiency of quantum and classical transport on graphs. Phys. Rev. E 73, 066117 (2006)ADSCrossRefGoogle Scholar
  23. 23.
    Mülken, O., Blumen, A.: Continuous-time quantum walks: models for coherent transport on complex networks. Phys. Rep. 502, 37–87 (2011)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (1993)MATHGoogle Scholar
  25. 25.
    Pitsianis, N., Bleris, G., Argyrakis, P.: Information dimension in fractal structures. Phys. Rev. B 39(10), 7097 (1989)ADSCrossRefGoogle Scholar
  26. 26.
    Reichl, L.E.: A Modern Course in Statistical Mechanics. University on Texas Press, Austin (1980)MATHGoogle Scholar
  27. 27.
    Schijven, P., Mülken, O.: Information Dimension of Dissipative Quantum Walks. arXiv preprint arXiv:1408.3037 (2014)
  28. 28.
    Seifert, U.: Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys. 75(12), 126001 (2012)ADSCrossRefGoogle Scholar
  29. 29.
    Van den Broeck, C., Esposito, M.: Ensemble and trajectory thermodynamics: a brief introduction. Phys. A 418, 6 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Physikalisches InstitutUniversität FreiburgFreiburgGermany
  2. 2.Institut Charles SadronUniversité de Strasbourg and CNRSStrasbourg CedexFrance

Personalised recommendations