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Uspekhi Mat. Nauk, 2009, Volume 64, Issue 4(388), Pages 73–124 (Mi umn9305)  

This article is cited in 27 scientific papers (total in 27 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

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 $\mathbf{J}=\frac12\mathbf{D}\mathbf{E}|\log\|\mathbf{E}\||+\mathscr{O}(\|\mathbf{E}\|)$, 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.

DOI: https://doi.org/10.4213/rm9305

Full text: PDF file (1003 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2009, 64:4, 651–699

Bibliographic databases:

UDC: 517.53/.57
MSC: Primary 78A35, 82C05, 82C40; Secondary 37D50
Received: 22.05.2009

Citation: D. I. Dolgopyat, N. I. Chernov, “Anomalous current in periodic Lorentz gases with infinite horizon”, Uspekhi Mat. Nauk, 64:4(388) (2009), 73–124; Russian Math. Surveys, 64:4 (2009), 651–699

Citation in format AMSBIB
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\paper Anomalous current in periodic Lorentz gases with infinite horizon
\jour Uspekhi Mat. Nauk
\yr 2009
\vol 64
\issue 4(388)
\pages 73--124
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\vol 64
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\pages 651--699
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    Citing articles on Google Scholar: Russian citations, English citations
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    2. Chernov N., “The work of Dmitry Dolgopyat on physical models with moving particles”, J. Mod. Dyn., 4:2 (2010), 243–255  crossref  mathscinet  zmath  isi  scopus
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    4. Nándori P., “Recurrence properties of a special type of heavy-tailed random walk”, J. Stat. Phys., 142:2 (2011), 342–355  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. Bálint P., Chernov N., Dolgopyat D., “Limit theorems for dispersing billiards with cusps”, Comm. Math. Phys., 308:2 (2011), 479–510  crossref  mathscinet  zmath  adsnasa  isi  scopus
    6. Dettmann C.P., “New horizons in multidimensional diffusion: the Lorentz gas and the Riemann hypothesis”, J. Stat. Phys., 146:1 (2012), 181–204  crossref  zmath  adsnasa  isi  scopus
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    9. Karlis A., Diakonos F., Petri C., Schmelcher P., “Criticality and strong intermittency in the Lorentz channel”, Phys. Rev. Lett., 109:11 (2012), 110601, 5 pp.  crossref  adsnasa  isi  elib  scopus
    10. Zhang H.-K., “Free path of billiards with flat points”, Discret. Contin. Dyn. Syst., 32:12 (2012), 4445–4466  crossref  mathscinet  zmath  isi  scopus
    11. Bálint P., Borbély G., Varga A.N., “Statistical properties of the system of two falling balls”, Chaos, 22:2 (2012), 026104, 30 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus
    12. A. S. Kraemer, D. P. Sanders, “Embedding quasicrystals in a periodic cell: dynamics in quasiperiodic structures”, Phys. Rev. Lett., 111:12 (2013), 125501  crossref  adsnasa  isi  scopus
    13. M. F. Demers, Hong-Kun Zhang, “A functional analytic approach to perturbations of the Lorentz gas”, Commun. Math. Phys., 324:3 (2013), 767–830  crossref  mathscinet  zmath  adsnasa  isi  scopus
    14. Chernov N., Zhang H.-K., Zhang P., “Electrical current in Sinai billiards under general small forces”, J. Stat. Phys., 153:6 (2013), 1065–1083  crossref  mathscinet  zmath  adsnasa  isi  scopus
    15. N. Chernov, A. Korepanov, “Spatial structure of Sinai–Ruelle–Bowen measures”, Phys. D, 285 (2014), 1–7  crossref  mathscinet  zmath  isi  scopus
    16. G. Cristadoro, Th. Gilbert, M. Lenci, D. P. Sanders, “Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards”, Phys. Rev. E, 90:2 (2014), 022106  crossref  isi  scopus
    17. C. P. Dettmann, “Diffusion in the Lorentz gas”, Commun. Theor. Phys., 62:4 (2014), 521  crossref  mathscinet  zmath  isi  scopus
    18. G. Cristadoro, Th. Gilbert, M. Lenci, D. P. Sanders, “Machta-Zwanzig regime of anomalous diffusion in infinite-horizon billiards”, Phys. Rev. E, 90:5 (2014), 050102(R)  crossref  isi  scopus
    19. Nandori P., Szasz D., Varju T., “Tail Asymptotics of Free Path Lengths For the Periodic Lorentz Process: on Dettmann's Geometric Conjectures”, Commun. Math. Phys., 331:1 (2014), 111–137  crossref  mathscinet  zmath  isi  scopus
    20. 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
    21. Dolgopyat D., Nandori P., “Nonequilibrium Density Profiles in Lorentz Tubes With Thermostated Boundaries”, 69, no. 4, 2016, 649–692  mathscinet  zmath  isi
    22. Marklof J., Toth B., “Superdiffusion in the Periodic Lorentz Gas”, Commun. Math. Phys., 347:3 (2016), 933–981  crossref  mathscinet  zmath  isi  elib  scopus
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