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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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