IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v217y1995i1p1-21.html
   My bibliography  Save this article

Transport properties on a random comb

Author

Listed:
  • Balakrishnan, V.
  • Van den Broeck, C.

Abstract

We study the random walk of a particle in a random comb structure, both in the presence of a biasing field and an the field-free case. We show that the mean-field treatment of the quenched disorder can be exactly mapped on to a continuous time random walk (CTRW) on the backbone of the comb, with a definite waiting time density. We find an exact expression for this central quantity. The Green function for the CTRW is then obtained. Its first and second moments determine the drift and diffusion at all times. We show that the drift velocity v vanishes asymptotically for power-law and stretched-exponential distributions of branch lengths on the comb, whatever be the biasing field strength. For an exponential branch-length distribution, v is a nonmonotonic function of the bias, increasing initially to a maximum and then decreasing to zero at a critical value. In the field-free case, anomalous diffusion occurs for a range of power-law distributions of the branch length. The corresponding exponent for the mean square displacement is obtained, as is the asymptotic form of the positional probability distribution for the random walk. We show that normal diffusion occurs whenever the mean branch length is finite, and present a simple formula for the effective diffusion constant; these results are extended to regular (nonrandom) combs as well. The physical reason for anomalous drift or diffusion is traced to the properties of the distribution of a first passage time (on a finite chain) that controls the effective waiting time density of the CTRW.

Suggested Citation

  • Balakrishnan, V. & Van den Broeck, C., 1995. "Transport properties on a random comb," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 217(1), pages 1-21.
  • Handle: RePEc:eee:phsmap:v:217:y:1995:i:1:p:1-21
    DOI: 10.1016/0378-4371(95)00083-J
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/037843719500083J
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000

    File URL: https://libkey.io/10.1016/0378-4371(95)00083-J?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Weiss, George H. & Havlin, Shlomo, 1986. "Some properties of a random walk on a comb structure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 134(2), pages 474-482.
    2. Kehr, K.W. & Kutner, R., 1982. "Random walk on a random walk," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 110(3), pages 535-549.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kotak, Jesal D. & Barma, Mustansir, 2022. "Bias induced drift and trapping on random combs and the Bethe lattice: Fluctuation regime and first order phase transitions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 597(C).
    2. Valerii M Sukhorukov & Jürgen Bereiter-Hahn, 2009. "Anomalous Diffusion Induced by Cristae Geometry in the Inner Mitochondrial Membrane," PLOS ONE, Public Library of Science, vol. 4(2), pages 1-14, February.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Pottier, N., 1994. "Analytic study of a model of biased diffusion on a random comblike structure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 208(1), pages 91-123.
    2. Arkhincheev, V.E., 2020. "The capture of particles on absorbing traps in the medium with anomalous diffusion: The effective fractional order diffusion equation and the slow temporal asymptotic of survival probability," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 550(C).
    3. Endre Csáki & Antónia Földes, 2022. "Strong Approximation of the Anisotropic Random Walk Revisited," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2879-2895, December.
    4. Spišák, Daniel, 1994. "Two-dimensional diffusion of particles with dipolar-like interaction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 209(1), pages 42-50.
    5. Baskin, Emmanuel & Iomin, Alexander, 2011. "Electrostatics in fractal geometry: Fractional calculus approach," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 335-341.
    6. Pece Trajanovski & Petar Jolakoski & Ljupco Kocarev & Trifce Sandev, 2023. "Ornstein–Uhlenbeck Process on Three-Dimensional Comb under Stochastic Resetting," Mathematics, MDPI, vol. 11(16), pages 1-28, August.
    7. Dzhanoev, A.R. & Sokolov, I.M., 2018. "The effect of the junction model on the anomalous diffusion in the 3D comb structure," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 330-336.
    8. Muszkieta, Monika & Janczura, Joanna & Weron, Aleksander, 2021. "Simulation and tracking of fractional particles motion. From microscopy video to statistical analysis. A Brownian bridge approach," Applied Mathematics and Computation, Elsevier, vol. 396(C).
    9. Iomin, A. & Zaburdaev, V. & Pfohl, T., 2016. "Reaction front propagation of actin polymerization in a comb-reaction system," Chaos, Solitons & Fractals, Elsevier, vol. 92(C), pages 115-122.
    10. Huckaby, Dale A. & Hubbard, Joseph B., 1983. "A random walk on a random channel with absorbing barriers," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 122(3), pages 602-610.
    11. Valerii M Sukhorukov & Jürgen Bereiter-Hahn, 2009. "Anomalous Diffusion Induced by Cristae Geometry in the Inner Mitochondrial Membrane," PLOS ONE, Public Library of Science, vol. 4(2), pages 1-14, February.
    12. Kutner, Ryszard & Świtała, Filip, 2004. "Remarks on the possible universal mechanism of the non-linear long-term autocorrelations in financial time-series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(1), pages 244-251.
    13. Sandev, Trifce & Schulz, Alexander & Kantz, Holger & Iomin, Alexander, 2018. "Heterogeneous diffusion in comb and fractal grid structures," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 551-555.
    14. Csáki, Endre & Csörgo, Miklós & Földes, Antónia & Révész, Pál, 2011. "On the local time of random walk on the 2-dimensional comb," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1290-1314, June.
    15. Aslangul, C. & Pottier, N. & Chvosta, P., 1994. "Analytic study of a model of diffusion on a random comblike structure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 203(3), pages 533-565.
    16. Endre Csáki & Antónia Földes, 2022. "On the Local Time of the Half-Plane Half-Comb Walk," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1247-1261, June.
    17. Endre Csáki & Antónia Földes, 2020. "Random Walks on Comb-Type Subsets of $$\mathbb {Z}^2$$ Z 2," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2233-2257, December.
    18. Méndez, Vicenç & Iomin, Alexander, 2013. "Comb-like models for transport along spiny dendrites," Chaos, Solitons & Fractals, Elsevier, vol. 53(C), pages 46-51.
    19. Iomin, Alexander, 2011. "Fractional-time Schrödinger equation: Fractional dynamics on a comb," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 348-352.
    20. Kotak, Jesal D. & Barma, Mustansir, 2022. "Bias induced drift and trapping on random combs and the Bethe lattice: Fluctuation regime and first order phase transitions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 597(C).

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:217:y:1995:i:1:p:1-21. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.