IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v466y2017icp69-78.html
   My bibliography  Save this article

Equivalence of deterministic walks on regular lattices on the plane

Author

Listed:
  • Rechtman, Ana
  • Rechtman, Raúl

Abstract

We consider deterministic walks on square, triangular and hexagonal two dimensional lattices. In each case, there is a scatterer at every lattice site that can be in one of two states that forces the walker to turn either to his/her immediate right or left. After the walker is scattered, the scatterer changes state. A lattice with an arrangement of scatterers is an environment. We show that there are only two environments for which the scattering rules are injective, mirrors or rotators, on the three lattices. On hexagonal lattices Webb and Cohen (2014), proved that if a walker with a given initial position and velocity moves through an environment of mirrors (rotators) then there is an environment of rotators (mirrors) through which the walker would move with the same trajectory. We refer to these trajectories on mirror and rotator environments as equivalent walks. We prove the equivalence of walks on square and triangular lattices and include a proof of the equivalence of walks on hexagonal lattices. The proofs are based both on the geometry of the lattice and the structure of the scattering rule.

Suggested Citation

  • Rechtman, Ana & Rechtman, Raúl, 2017. "Equivalence of deterministic walks on regular lattices on the plane," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 466(C), pages 69-78.
  • Handle: RePEc:eee:phsmap:v:466:y:2017:i:c:p:69-78
    DOI: 10.1016/j.physa.2016.08.077
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437116306045
    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/j.physa.2016.08.077?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. Cohen, E.G.D. & Wang, F., 1995. "Novel phenomena in Lorentz lattice gases," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 219(1), pages 56-87.
    Full references (including those not matched with items on IDEAS)

    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.

      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:466:y:2017:i:c:p:69-78. 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.