The Boltzmann-Grad limit of the periodic Lorentz gas

Pages 225-298 from Volume 174 (2011), Issue 1 by Jens Marklof, Andreas Strömbergsson

Abstract

We study the dynamics of a point particle in a periodic array of spherical scatterers and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (Boltzmann-Grad limit). A generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a Markov process with memory two.

  • [Balint07] Go to document P. Bálint and I. P. Tóth, "Exponential decay of correlations in multi-dimensional dispersing billiards," Ann. Henri Poincaré, vol. 9, iss. 7, pp. 1309-1369, 2008.
    @article {Balint07, MRKEY = {2453251},
      AUTHOR = {B{á}lint, P{é}ter and T{ó}th, Imre P{é}ter},
      TITLE = {Exponential decay of correlations in multi-dimensional dispersing billiards},
      JOURNAL = {Ann. Henri Poincaré},
      FJOURNAL = {Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics},
      VOLUME = {9},
      YEAR = {2008},
      NUMBER = {7},
      PAGES = {1309--1369},
      ISSN = {1424-0637},
      MRCLASS = {37D50 (37A25 37A60)},
      MRNUMBER = {2453251},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1007/s00023-008-0389-1},
      ZBLNUMBER = {1162.37018},
      }
  • [Bleher92] Go to document P. M. Bleher, "Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon," J. Statist. Phys., vol. 66, iss. 1-2, pp. 315-373, 1992.
    @article {Bleher92, MRKEY = {1149489},
      AUTHOR = {Bleher, P. M.},
      TITLE = {Statistical properties of two-dimensional periodic {L}orentz gas with infinite horizon},
      JOURNAL = {J. Statist. Phys.},
      FJOURNAL = {Journal of Statistical Physics},
      VOLUME = {66},
      YEAR = {1992},
      NUMBER = {1-2},
      PAGES = {315--373},
      ISSN = {0022-4715},
      CODEN = {JSTPSB},
      MRCLASS = {82C21 (82C40)},
      MRNUMBER = {1149489},
      MRREVIEWER = {Min Ping Qian},
      ZBLNUMBER = {0925.82147},
      DOI = {10.1007/BF01060071},
     }
  • [Boca07] Go to document F. P. Boca and A. Zaharescu, "The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit," Comm. Math. Phys., vol. 269, iss. 2, pp. 425-471, 2007.
    @article {Boca07, MRKEY = {2274553},
      AUTHOR = {Boca, Florin P. and Zaharescu, Alexandru},
      TITLE = {The distribution of the free path lengths in the periodic two-dimensional {L}orentz gas in the small-scatterer limit},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {269},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {425--471},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {37A25 (11B57 37A60 37D50 82C05 82C40)},
      MRNUMBER = {2274553},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1007/s00220-006-0137-7},
      ZBLNUMBER = {1143.37002},
      }
  • [Boldrighini83] Go to document C. Boldrighini, L. A. Bunimovich, and Y. G. Sinai, "On the Boltzmann equation for the Lorentz gas," J. Statist. Phys., vol. 32, iss. 3, pp. 477-501, 1983.
    @article {Boldrighini83, MRKEY = {0725107},
      AUTHOR = {Boldrighini, C. and Bunimovich, L. A. and Sinai, Ya. G.},
      TITLE = {On the {B}oltzmann equation for the {L}orentz gas},
      JOURNAL = {J. Statist. Phys.},
      FJOURNAL = {Journal of Statistical Physics},
      VOLUME = {32},
      YEAR = {1983},
      NUMBER = {3},
      PAGES = {477--501},
      ISSN = {0022-4715},
      CODEN = {JSTPSB},
      MRCLASS = {82A40 (60K35 82A05)},
      MRNUMBER = {0725107},
      MRREVIEWER = {Carlo Cercignani},
      ZBLNUMBER = {0583.76092},
      DOI = {10.1007/BF01008951},
     }
  • [Bourgain98] Go to document J. Bourgain, F. Golse, and B. Wennberg, "On the distribution of free path lengths for the periodic Lorentz gas," Comm. Math. Phys., vol. 190, iss. 3, pp. 491-508, 1998.
    @article {Bourgain98, MRKEY = {1600299},
      AUTHOR = {Bourgain, Jean and Golse, Fran{ç}ois and Wennberg, Bernt},
      TITLE = {On the distribution of free path lengths for the periodic {L}orentz gas},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {190},
      YEAR = {1998},
      NUMBER = {3},
      PAGES = {491--508},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {82C40 (58F40)},
      MRNUMBER = {1600299},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1007/s002200050249},
      ZBLNUMBER = {0910.60082},
      }
  • [Bunimovich80] Go to document L. A. Bunimovich and Y. G. Sinai, "Statistical properties of Lorentz gas with periodic configuration of scatterers," Comm. Math. Phys., vol. 78, iss. 4, pp. 479-497, 1980/81.
    @article {Bunimovich80, MRKEY = {0606459},
      AUTHOR = {Bunimovich, L. A. and Sinai, Ya. G.},
      TITLE = {Statistical properties of {L}orentz gas with periodic configuration of scatterers},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {78},
      YEAR = {1980/81},
      NUMBER = {4},
      PAGES = {479--497},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {82A05 (58F11 58F13 60F17 60K35)},
      MRNUMBER = {0606459},
      MRREVIEWER = {Sheldon Goldstein},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0459.60099},
      DOI = {10.1007/BF02046760},
     }
  • [Caglioti03] Go to document E. Caglioti and F. Golse, "On the distribution of free path lengths for the periodic Lorentz gas. III," Comm. Math. Phys., vol. 236, iss. 2, pp. 199-221, 2003.
    @article {Caglioti03, MRKEY = {1981109},
      AUTHOR = {Caglioti, Emanuele and Golse, Fran{ç}ois},
      TITLE = {On the distribution of free path lengths for the periodic {L}orentz gas. {III}},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {236},
      YEAR = {2003},
      NUMBER = {2},
      PAGES = {199--221},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {82C05 (37A60 37D50 82C40)},
      MRNUMBER = {1981109},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1007/s00220-003-0825-5},
      ZBLNUMBER = {1041.82016},
      }
  • [Caglioti07] Go to document E. Caglioti and F. Golse, "The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions," C. R. Math. Acad. Sci. Paris, vol. 346, iss. 7-8, pp. 477-482, 2008.
    @article {Caglioti07, MRKEY = {2417573},
      AUTHOR = {Caglioti, Emanuele and Golse, Fran{ç}ois},
      TITLE = {The {B}oltzmann-{G}rad limit of the periodic {L}orentz gas in two space dimensions},
      JOURNAL = {C. R. Math. Acad. Sci. Paris},
      FJOURNAL = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
      VOLUME = {346},
      YEAR = {2008},
      NUMBER = {7-8},
      PAGES = {477--482},
      ISSN = {1631-073X},
      MRCLASS = {82C40 (37D50)},
      MRNUMBER = {2417573},
      DOI = {10.1016/j.crma.2008.01.016},
      ZBLNUMBER = {1145.82019},
      }
  • [Chernov94] Go to document N. I. Chernov, "Statistical properties of the periodic Lorentz gas. Multidimensional case," J. Statist. Phys., vol. 74, iss. 1-2, pp. 11-53, 1994.
    @article {Chernov94, MRKEY = {1257815},
      AUTHOR = {Chernov, N. I.},
      TITLE = {Statistical properties of the periodic {L}orentz gas. {M}ultidimensional case},
      JOURNAL = {J. Statist. Phys.},
      FJOURNAL = {Journal of Statistical Physics},
      VOLUME = {74},
      YEAR = {1994},
      NUMBER = {1-2},
      PAGES = {11--53},
      ISSN = {0022-4715},
      CODEN = {JSTPSB},
      MRCLASS = {58F11 (58F03 58F15 82C40)},
      MRNUMBER = {1257815},
      MRREVIEWER = {Andr{á}s Kr{á}mli},
      ZBLNUMBER = {0946.37500},
      DOI = {10.1007/BF02186805},
     }
  • [Dahlqvist97] Go to document P. Dahlqvist, "The Lyapunov exponent in the Sinai billiard in the small scatterer limit," Nonlinearity, vol. 10, iss. 1, pp. 159-173, 1997.
    @article {Dahlqvist97, MRKEY = {1430746},
      AUTHOR = {Dahlqvist, Per},
      TITLE = {The {L}yapunov exponent in the {S}inai billiard in the small scatterer limit},
      JOURNAL = {Nonlinearity},
      FJOURNAL = {Nonlinearity},
      VOLUME = {10},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {159--173},
      ISSN = {0951-7715},
      CODEN = {NONLE5},
      MRCLASS = {58F11 (82C05)},
      MRNUMBER = {1430746},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1088/0951-7715/10/1/011},
      ZBLNUMBER = {0907.58038},
      }
  • [davies] E. B. Davies, One-parameter Semigroups, London: Academic Press [Harcourt Brace Jovanovich Publishers], 1980, vol. 15.
    @book {davies, MRKEY = {0591851},
      AUTHOR = {Davies, Edward Brian},
      TITLE = {One-parameter Semigroups},
      SERIES = {London Math. Soc. Monogr.},
      VOLUME = {15},
      PUBLISHER = {Academic Press [Harcourt Brace Jovanovich Publishers]},
      ADDRESS = {London},
      YEAR = {1980},
      PAGES = {viii+230},
      ISBN = {0-12-206280-9},
      MRCLASS = {47D05},
      MRNUMBER = {0591851},
      MRREVIEWER = {S. Kurepa},
      ZBLNUMBER = {0457.47030},
      }
  • [dolgopyat] Go to document D. Dolgopyat, D. Szász, and T. Varjú, "Recurrence properties of planar Lorentz process," Duke Math. J., vol. 142, iss. 2, pp. 241-281, 2008.
    @article {dolgopyat, MRKEY = {2401621},
      AUTHOR = {Dolgopyat, Dmitry and Sz{á}sz, Domokos and Varj{ú},
      Tam{á}s},
      TITLE = {Recurrence properties of planar {L}orentz process},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {142},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {241--281},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {37D50 (37A50 37C30 60F05 60K40)},
      MRNUMBER = {2401621},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1215/00127094-2008-006},
      ZBLNUMBER = {1136.37022},
      }
  • [Gallavotti69] Go to document G. Gallavotti, "Divergences and approach to equilibrium in the Lorentz and the wind-tree-models," Phys. Rev., vol. 185, pp. 308-322, 1969.
    @article{Gallavotti69,
      author={Gallavotti, G.},
      TITLE={Divergences and approach to equilibrium in the {L}orentz and the wind-tree-models},
      JOURNAL={Phys. Rev.},
      VOLUME={185},
      YEAR={1969},
      PAGES={308--322},
      DOI = {10.1103/PhysRev.185.308},
     }
  • [Golse06] F. Golse, "The periodic Lorentz gas in the Boltzmann-Grad limit," in International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, pp. 183-201.
    @incollection {Golse06, MRKEY = {2275676},
      AUTHOR = {Golse, Fran{ç}ois},
      TITLE = {The periodic {L}orentz gas in the {B}oltzmann-{G}rad limit},
      BOOKTITLE = {International {C}ongress of {M}athematicians. {V}ol. {\rm III}},
      PAGES = {183--201},
      PUBLISHER = {Eur. Math. Soc., Zürich},
      YEAR = {2006},
      MRCLASS = {82C40 (37A60 37D50)},
      MRNUMBER = {2275676},
      MRREVIEWER = {Silvia Caprino},
      ZBLNUMBER = {1109.82020},
      }
  • [Golse07] F. Golse, "On the periodic Lorentz gas and the Lorentz kinetic equation," Ann. Fac. Sci. Toulouse Math., vol. 17, iss. 4, pp. 735-749, 2008.
    @article {Golse07, MRKEY = {2499853},
      AUTHOR = {Golse, Fran{ç}ois},
      TITLE = {On the periodic {L}orentz gas and the {L}orentz kinetic equation},
      JOURNAL = {Ann. Fac. Sci. Toulouse Math.},
      FJOURNAL = {Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6},
      VOLUME = {17},
      YEAR = {2008},
      NUMBER = {4},
      PAGES = {735--749},
      ISSN = {0240-2963},
      MRCLASS = {82C40},
      MRNUMBER = {2499853},
      ZBLNUMBER = {1166.82304},
      }
  • [Golse00] Go to document F. Golse and B. Wennberg, "On the distribution of free path lengths for the periodic Lorentz gas. II," M2AN Math. Model. Numer. Anal., vol. 34, iss. 6, pp. 1151-1163, 2000.
    @article {Golse00, MRKEY = {1812731},
      AUTHOR = {Golse, Fran{ç}ois and Wennberg, Bernt},
      TITLE = {On the distribution of free path lengths for the periodic {L}orentz gas. {II}},
      JOURNAL = {M2AN Math. Model. Numer. Anal.},
      FJOURNAL = {M2AN. Mathematical Modelling and Numerical Analysis},
      VOLUME = {34},
      YEAR = {2000},
      NUMBER = {6},
      PAGES = {1151--1163},
      ISSN = {0764-583X},
      MRCLASS = {82C40 (35F15 37D50)},
      MRNUMBER = {1812731},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1051/m2an:2000121},
      ZBLNUMBER = {1006.82025},
      }
  • [hirsch] M. W. Hirsch, Differential Topology, New York: Springer-Verlag, 1994, vol. 33.
    @book {hirsch, MRKEY = {1336822},
      AUTHOR = {Hirsch, Morris W.},
      TITLE = {Differential Topology},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {33},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1994},
      PAGES = {x+222},
      ISBN = {0-387-90148-5},
      MRCLASS = {57-01 (58-01)},
      MRNUMBER = {1336822},
      ZBLNUMBER = {0356.57001},
      }
  • [Lorentz05] H. Lorentz, "Le mouvement des électrons dans les métaux," Arch. Néerl., vol. 10, pp. 336-371, 1905.
    @article{Lorentz05,
      author={Lorentz, H.},
      TITLE={Le mouvement des électrons dans les métaux},
      JOURNAL={Arch. Néerl.},
      VOLUME={10},
      YEAR={1905},
      PAGES={336--371},
      JFMNUMBER = {36.0922.02},
     }
  • [partI] Go to document J. Marklof and A. Strömbergsson, "The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems," Ann. of Math., vol. 172, iss. 3, pp. 1949-2033, 2010.
    @article {partI, MRKEY = {2726104},
      AUTHOR = {Marklof, Jens and Str{ö}mbergsson, Andreas},
      TITLE = {The distribution of free path lengths in the periodic {L}orentz gas and related lattice point problems},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {172},
      YEAR = {2010},
      NUMBER = {3},
      PAGES = {1949--2033},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {82C05 (37Axx 60F17)},
      MRNUMBER = {2726104},
      DOI = {10.4007/annals.2010.172.1949},
      ZBLNUMBER = {05850191},
      }
  • [partIII] Go to document J. Marklof and A. Strömbergsson, "Kinetic transport in the two-dimensional periodic Lorentz gas," Nonlinearity, vol. 21, iss. 7, pp. 1413-1422, 2008.
    @article {partIII, MRKEY = {2425325},
      AUTHOR = {Marklof, Jens and Str{ö}mbergsson, Andreas},
      TITLE = {Kinetic transport in the two-dimensional periodic {L}orentz gas},
      JOURNAL = {Nonlinearity},
      FJOURNAL = {Nonlinearity},
      VOLUME = {21},
      YEAR = {2008},
      NUMBER = {7},
      PAGES = {1413--1422},
      ISSN = {0951-7715},
      CODEN = {NONLE5},
      MRCLASS = {82B40 (37D50 82C05 82C40)},
      MRNUMBER = {2425325},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1088/0951-7715/21/7/001},
      ZBLNUMBER = {1153.82020},
      }
  • [partIV] Go to document J. Marklof and A. Strömbergsson, The periodic Lorentz gas in the Boltzmann-Grad limit: Asymptotic estimates.
    @misc{partIV,
      author = {Marklof, Jens and Str{ö}mbergsson, Andreas},
      TITLE = {The periodic {L}orentz gas in the {B}oltzmann-{G}rad limit: {A}symptotic estimates},
      NOTE={{\it GAFA online first} (2011)},
      SORTYEAR={2015},
      DOI={10.1007/s00039-011-0116-9},
      }
  • [Melbourne05] Go to document I. Melbourne and M. Nicol, "Almost sure invariance principle for nonuniformly hyperbolic systems," Comm. Math. Phys., vol. 260, iss. 1, pp. 131-146, 2005.
    @article {Melbourne05, MRKEY = {2175992},
      AUTHOR = {Melbourne, Ian and Nicol, Matthew},
      TITLE = {Almost sure invariance principle for nonuniformly hyperbolic systems},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {260},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {131--146},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {37D25 (37A99 37C40 82B05)},
      MRNUMBER = {2175992},
      MRREVIEWER = {Slobodan N. Simi{ć}},
      DOI = {10.1007/s00220-005-1407-5},
      ZBLNUMBER = {1084.37024},
      }
  • [Melbourne07] Go to document I. Melbourne and M. Nicol, "A vector-valued almost sure invariance principle for hyperbolic dynamical systems," Ann. Probab., vol. 37, iss. 2, pp. 478-505, 2009.
    @article {Melbourne07, MRKEY = {2510014},
      AUTHOR = {Melbourne, Ian and Nicol, Matthew},
      TITLE = {A vector-valued almost sure invariance principle for hyperbolic dynamical systems},
      JOURNAL = {Ann. Probab.},
      FJOURNAL = {The Annals of Probability},
      VOLUME = {37},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {478--505},
      ISSN = {0091-1798},
      CODEN = {APBYAE},
      MRCLASS = {37A50 (37D20 37D25 37D50 60F17 82C40)},
      MRNUMBER = {2510014},
      MRREVIEWER = {Stefano Galatolo},
      DOI = {10.1214/08-AOP410},
      ZBLNUMBER = {1176.37006},
      }
  • [Newton82] R. G. Newton, Scattering Theory of Waves and Particles, Second ed., New York: Springer-Verlag, 1982.
    @book {Newton82, MRKEY = {0666397},
      AUTHOR = {Newton, Roger G.},
      TITLE = {Scattering Theory of Waves and Particles},
      SERIES = {Texts and Monographs in Physics},
      EDITION = {{S}econd},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1982},
      PAGES = {xx+743},
      ISBN = {0-387-10950-1},
      MRCLASS = {81-02 (78-02 78A45 81Fxx)},
      MRNUMBER = {0666397},
      MRREVIEWER = {Pierre C. Sabatier},
      ZBLNUMBER = {0496.47011},
      }
  • [Rudin] W. Rudin, Real and Complex Analysis, Third ed., New York: McGraw-Hill Book Co., 1987.
    @book {Rudin, MRKEY = {0924157},
      AUTHOR = {Rudin, Walter},
      TITLE = {Real and Complex Analysis},
      EDITION = {{T}hird},
      PUBLISHER = {McGraw-Hill Book Co.},
      ADDRESS = {New York},
      YEAR = {1987},
      PAGES = {xiv+416},
      ISBN = {0-07-054234-1},
      MRCLASS = {00A05 (26-01 30-01 46-01)},
      MRNUMBER = {0924157},
      ZBLNUMBER = {0925.00005},
      }
  • [Szasz07] Go to document D. Szász and T. Varjú, "Limit laws and recurrence for the planar Lorentz process with infinite horizon," J. Stat. Phys., vol. 129, iss. 1, pp. 59-80, 2007.
    @article {Szasz07, MRKEY = {2349520},
      AUTHOR = {Sz{á}sz, Domokos and Varj{ú},
      Tam{á}s},
      TITLE = {Limit laws and recurrence for the planar {L}orentz process with infinite horizon},
      JOURNAL = {J. Stat. Phys.},
      FJOURNAL = {Journal of Statistical Physics},
      VOLUME = {129},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {59--80},
      ISSN = {0022-4715},
      MRCLASS = {37D50 (37N20 60F05 82C40)},
      MRNUMBER = {2349520},
      MRREVIEWER = {Nikolai Chernov},
      DOI = {10.1007/s10955-007-9367-0},
      ZBLNUMBER = {1128.82011},
      }
  • [Spohn78] Go to document H. Spohn, "The Lorentz process converges to a random flight process," Comm. Math. Phys., vol. 60, iss. 3, pp. 277-290, 1978.
    @article {Spohn78, MRKEY = {0496299},
      AUTHOR = {Spohn, Herbert},
      TITLE = {The {L}orentz process converges to a random flight process},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {60},
      YEAR = {1978},
      NUMBER = {3},
      PAGES = {277--290},
      ISSN = {0010-3616},
      MRCLASS = {82.60},
      MRNUMBER = {0496299},
      MRREVIEWER = {Burton H. Voorhees},
      ZBLNUMBER = {0381.60099},
      DOI = {10.1007/BF01612893},
     }

Keywords
, , ,
DOI
http://doi.org/10.4007/annals.2011.174.1.7
MR
2811599
Zbl
05960701
Milestones
Received: 4 January 2008
Accepted: 6 May 2010
Published online: 1 July 2011

Authors

Jens Marklof

School of Mathematics
University of Bristol
Bristol BS8 1TW
United Kingdom

Andreas Strömbergsson

Department of Mathematics
Box 480
Uppsala University
SE-75106 Uppsala
Sweden