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SUMMARY:Invariance principle for the random Lorentz gas beyond the [Boltzm
 ann-Grad / Gallavotti-Spohn] limit - Balint Toth (Bristol)
DTSTART:20181009T130000Z
DTEND:20181009T140000Z
UID:TALK111589@talks.cam.ac.uk
CONTACT:Perla Sousi
DESCRIPTION:Let hard ball scatterers of radius $r$ be placed in $\\mathbb 
 R^d$\, centred at the points of a Poisson point process of intensity $\\rh
 o$. The volume fraction $r^d \\rho$ is assumed to be sufficiently low so t
 hat with positive probability the origin is not covered by a scatterer or 
 trapped in a finite domain fully surrounded by scatterers. The Lorentz pro
 cess is the trajectory of a point-like particle starting from the origin w
 ith randomly oriented unit velocity subject to elastic collisions with the
  fixed (infinite mass) scatterers. The question of diffusive scaling limit
  of this process is a major open problem in classical statistical physics.
  \n \nGallavotti (1969) and Spohn (1978) proved that under the so-called B
 oltzmann-Grad limit\, when $r \\to 0$\, $\\rho \\to \\infty$ so that $r^{d
 -1}\\rho \\to 1$ and the time scale is fixed\, the Lorentz process (descri
 bed informally above) converges to a Markovian random flight process\, wit
 h independent exponentially distributed free flight times and Markovian sc
 atterings. It is essentially straightforward to see that taking a second d
 iffusive scaling limit (after the Gallavotti-Spohn limit) yields invarianc
 e principle. \n \nI will present new results going beyond the [Boltzmann-G
 rad / Gallavotti-Spohn] limit\, in $d=3$: Letting $r \\to 0$\, $\\rho \\to
  \\infty$ so that $r^{d-1}\\rho \\to 1$ (as in B-G) and *simultaneously* r
 escaling time by $T \\sim r^{-2+\\epsilon}$ we prove invariance principle 
 (under diffusive scaling) for the Lorentz trajectory. Note that the B-G li
 mit and diffusive scaling are done simultaneously and not in sequel. The p
 roof is essentially based on control of the effect of re-collisions by pro
 babilistic coupling arguments. The main arguments are valid in $d=3$ but n
 ot in $d=2$. \n \nJoint work with Chris Lutsko (Bristol)
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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