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CATEGORIES:Probability
SUMMARY:Invariance principle for the random Lorentz gas be
 yond the [Boltzmann-Grad / Gallavotti-Spohn] limit
  - Balint Toth (Bristol)
DTSTART;TZID=Europe/London:20181009T140000
DTEND;TZID=Europe/London:20181009T150000
UID:TALK111589AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/111589
DESCRIPTION:Let hard ball scatterers of radius $r$ be placed i
 n $\\mathbb R^d$\, centred at the points of a Pois
 son point process of intensity $\\rho$. The volume
  fraction $r^d \\rho$ is assumed to be sufficientl
 y low so that with positive probability the origin
  is not covered by a scatterer or trapped in a fin
 ite domain fully surrounded by scatterers. The Lor
 entz process is the trajectory of a point-like par
 ticle starting from the origin with randomly orien
 ted unit velocity subject to elastic collisions wi
 th the fixed (infinite mass) scatterers. The quest
 ion of diffusive scaling limit of this process is 
 a major open problem in classical statistical phys
 ics. \n \nGallavotti (1969) and Spohn (1978) prove
 d that under the so-called Boltzmann-Grad limit\, 
 when $r \\to 0$\, $\\rho \\to \\infty$ so that $r^
 {d-1}\\rho \\to 1$ and the time scale is fixed\, t
 he Lorentz process (described informally above) co
 nverges to a Markovian random flight process\, wit
 h independent exponentially distributed free fligh
 t times and Markovian scatterings. It is essential
 ly straightforward to see that taking a second dif
 fusive scaling limit (after the Gallavotti-Spohn l
 imit) yields invariance principle. \n \nI will pre
 sent new results going beyond the [Boltzmann-Grad 
 / Gallavotti-Spohn] limit\, in $d=3$: Letting $r \
 \to 0$\, $\\rho \\to \\infty$ so that $r^{d-1}\\rh
 o \\to 1$ (as in B-G) and *simultaneously* rescali
 ng time by $T \\sim r^{-2+\\epsilon}$ we prove inv
 ariance principle (under diffusive scaling) for th
 e Lorentz trajectory. Note that the B-G limit and 
 diffusive scaling are done simultaneously and not 
 in sequel. The proof is essentially based on contr
 ol of the effect of re-collisions by probabilistic
  coupling arguments. The main arguments are valid 
 in $d=3$ but not in $d=2$. \n \nJoint work with Ch
 ris Lutsko (Bristol)
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0W
 B
CONTACT:Perla Sousi
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