Loading [MathJax]/jax/output/SVG/config.js
Russian Mathematical Surveys
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Uspekhi Mat. Nauk:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Russian Mathematical Surveys, 2009, Volume 64, Issue 4, Pages 651–699
DOI: https://doi.org/10.1070/RM2009v064n04ABEH004630
(Mi rm9305)
 

This article is cited in 38 scientific papers (total in 38 papers)

Anomalous current in periodic Lorentz gases with infinite horizon

D. I. Dolgopyata, N. I. Chernovb

a University of Maryland, College Park
b University of Alabama at Birmingham
References:
Abstract: Electric current is studied in a two-dimensional periodic Lorentz gas in the presence of a weak homogeneous electric field. When the horizon is finite, that is, free flights between collisions are bounded, the resulting current $\mathbf{J}$ is proportional to the voltage difference $\mathbf{E}$, that is, $\mathbf{J}=\frac12\mathbf{D}^*\mathbf{E}+o(\|\mathbf{E}\|)$, where $\mathbf{D}^*$ is the diffusion matrix of a Lorentz particle moving freely without an electric field (see a mathematical proof in [1]). This formula agrees with Ohm's classical law and the Einstein relation. Here the more difficult model with an infinite horizon is investigated. It is found that infinite corridors between scatterers allow the particles (electrons) to move faster, resulting in an abnormal current (causing ‘superconductivity’). More precisely, the current is now given by $\mathbf{J}=\frac12\mathbf{D}\mathbf{E}\bigl|\log\|\mathbf{E}\|\bigr|+\mathscr{O}(\|\mathbf{E}\|)$, where $\mathbf{D}$ is the ‘superdiffusion’ matrix of a Lorentz particle moving freely without an electric field. This means that Ohm's law fails in this regime, but the Einstein relation (suitably interpreted) still holds. New results are also obtained for the infinite-horizon Lorentz gas without external fields, complementing recent studies by Szász and Varjú [2].
Bibliography: 31 titles.
Keywords: Lorentz gas, billiards, diffusion, electric current, Ohm's law.
Received: 22.05.2009
Bibliographic databases:
Document Type: Article
UDC: 517.53/.57
MSC: Primary 78A35, 82C05, 82C40; Secondary 37D50
Language: English
Original paper language: Russian
Citation: D. I. Dolgopyat, N. I. Chernov, “Anomalous current in periodic Lorentz gases with infinite horizon”, Russian Math. Surveys, 64:4 (2009), 651–699
Citation in format AMSBIB
\Bibitem{DolChe09}
\by D.~I.~Dolgopyat, N.~I.~Chernov
\paper Anomalous current in periodic Lorentz gases with infinite horizon
\jour Russian Math. Surveys
\yr 2009
\vol 64
\issue 4
\pages 651--699
\mathnet{http://mi.mathnet.ru//eng/rm9305}
\crossref{https://doi.org/10.1070/RM2009v064n04ABEH004630}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2583573}
\zmath{https://zbmath.org/?q=an:1179.78021}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2009RuMaS..64..651D}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000275492400003}
\elib{https://elibrary.ru/item.asp?id=20425303}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77951526945}
Linking options:
  • https://www.mathnet.ru/eng/rm9305
  • https://doi.org/10.1070/RM2009v064n04ABEH004630
  • https://www.mathnet.ru/eng/rm/v64/i4/p73
  • This publication is cited in the following 38 articles:
    1. Jens Marklof, Andreas Strömbergsson, “Kinetic Theory for the Low-Density Lorentz Gas”, Memoirs of the AMS, 294:1464 (2024)  crossref
    2. Françoise Pène, Dalia Terhesiu, “Strong mixing for the periodic Lorentz gas flow with infinite horizon”, Trans. Amer. Math. Soc., 2024  crossref
    3. Songzi Li, “Rates of convergence for the superdiffusion in the Boltzmann–Grad limit of the periodic Lorentz gas”, Stochastic Processes and their Applications, 154 (2022), 26  crossref
    4. Dmitry Dolgopyat, Péter Nándori, “Infinite measure mixing for some mechanical systems”, Advances in Mathematics, 410 (2022), 108757  crossref
    5. Dolgopyat D., Nandori P., “on Mixing and the Local Central Limit Theorem For Hyperbolic Flows”, Ergod. Theory Dyn. Syst., 40:1 (2020), PII S0143385718000299, 142–174  crossref  mathscinet  isi
    6. Lutsko Ch., Toth B., “Invariance Principle For the Random Lorentz Gas-Beyond the Boltzmann-Grad Limit”, Commun. Math. Phys., 379:2 (2020), 589–632  crossref  mathscinet  isi  scopus
    7. Borgers Ch., Greengard C., “On the Mean Square Displacement in Levy Walks”, SIAM J. Appl. Math., 80:3 (2020), 1175–1196  crossref  mathscinet  isi
    8. Grigo A., “a Rigorous Derivation of Haff'S Law For a Periodic Two-Disk Fluid”, J. Stat. Phys., 176:4 (2019), 806–835  crossref  mathscinet  isi
    9. Zarfat L., Peletskyi A., Barkai E., Denisov S., “Infinite Horizon Billiards: Transport At the Border Between Gauss and Levy Universality Classes”, Phys. Rev. E, 100:4 (2019), 042140  crossref  mathscinet  isi
    10. Domokos Szász, The Abel Prize, The Abel Prize 2013-2017, 2019, 299  crossref
    11. Balint P., Nandori P., Szasz D., Toth I.P., “Equidistribution For Standard Pairs in Planar Dispersing Billiard Flows”, Ann. Henri Poincare, 19:4 (2018), 979–1042  crossref  mathscinet  zmath  isi  scopus
    12. Zarfaty L. Peletskyi A. Fouxon I. Denisov S. Barkai E., “Dispersion of Particles in An Infinite-Horizon Lorentz Gas”, Phys. Rev. E, 98:1 (2018), 010101  crossref  mathscinet  isi  scopus
    13. Dettmann C.P., Marklof J., Strombergsson A., “Universal Hitting Time Statistics for Integrable Flows”, J. Stat. Phys., 166:3-4, SI (2017), 714–749  crossref  mathscinet  zmath  isi  scopus
    14. Bunimovich L.A., Grigo A., “Transport Processes from Mechanics: Minimal and Simplest Models”, J. Stat. Phys., 166:3-4, SI (2017), 750–764  crossref  mathscinet  zmath  isi  scopus
    15. Feliczaki R.M., Vicentini E., Gonzalez-Borrero P.P., “Dynamical Transition on the Periodic Lorentz Gas: Stochastic and Deterministic Approaches”, Phys. Rev. E, 96:5 (2017), 052117  crossref  mathscinet  isi  scopus
    16. Dolgopyat D., Nandori P., “Nonequilibrium Density Profiles in Lorentz Tubes With Thermostated Boundaries”, 69, no. 4, 2016, 649–692  mathscinet  zmath  isi
    17. Marklof J., Toth B., “Superdiffusion in the Periodic Lorentz Gas”, Commun. Math. Phys., 347:3 (2016), 933–981  crossref  mathscinet  zmath  isi  elib  scopus
    18. Dmitry Dolgopyat, Péter Nándori, “Nonequilibrium Density Profiles in Lorentz Tubes with Thermostated Boundaries”, Comm Pure Appl Math, 69:4 (2016), 649  crossref
    19. Kraemer A.S., Schmiedeberg M., Sanders D.P., “Horizons and Free-Path Distributions in Quasiperiodic Lorentz Gases”, 92, no. 5, 2015, 052131  crossref  isi  scopus
    20. N. Chernov, A. Korepanov, “Spatial structure of Sinai–Ruelle–Bowen measures”, Phys. D, 285 (2014), 1–7  crossref  mathscinet  zmath  isi  scopus
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Успехи математических наук Russian Mathematical Surveys
    Statistics & downloads:
    Abstract page:792
    Russian version PDF:222
    English version PDF:26
    References:98
    First page:9
     
      Contact us:
    math-net2025_01@mi-ras.ru
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2025