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.
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@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},
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MRNUMBER = {2453251},
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AUTHOR = {Bleher, P. M.},
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@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},
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CODEN = {CMPHAY},
MRCLASS = {37A25 (11B57 37A60 37D50 82C05 82C40)},
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} -
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@article {Boldrighini83, MRKEY = {0725107},
AUTHOR = {Boldrighini, C. and Bunimovich, L. A. and Sinai, Ya. G.},
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@article {Bourgain98, MRKEY = {1600299},
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@article {Bunimovich80, MRKEY = {0606459},
AUTHOR = {Bunimovich, L. A. and Sinai, Ya. G.},
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@article {Caglioti07, MRKEY = {2417573},
AUTHOR = {Caglioti, Emanuele and Golse, Fran{ç}ois},
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ZBLNUMBER = {1145.82019},
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ISSN = {0951-7715},
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@book {davies, MRKEY = {0591851},
AUTHOR = {Davies, Edward Brian},
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PUBLISHER = {Academic Press [Harcourt Brace Jovanovich Publishers]},
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PAGES = {viii+230},
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Tam{á}s},
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@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}},
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@article {Golse07, MRKEY = {2499853},
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AUTHOR = {Marklof, Jens and Str{ö}mbergsson, Andreas},
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AUTHOR = {Marklof, Jens and Str{ö}mbergsson, Andreas},
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YEAR = {2008},
NUMBER = {7},
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@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)},
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} -
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AUTHOR = {Newton, Roger G.},
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} -
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@book {Rudin, MRKEY = {0924157},
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ZBLNUMBER = {0381.60099},
DOI = {10.1007/BF01612893},
}