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SUMMARY:Clay Public Lecture: Diffusion in the random Lorentz gas - Balint 
 Toth (University of Bristol)
DTSTART:20241120T160000Z
DTEND:20241120T170000Z
UID:TALK220171@talks.cam.ac.uk
DESCRIPTION:Since the pioneering works of Hendrik Lorentz (1905) and Paul 
 and Tatiana Ehrenfest (1912) the deterministic (Hamiltonian) motion of a p
 oint-like particle exposed to the action of a collection of fixed\, random
 ly located short range scatterers has been a much studied model of physica
 l diffusion under fully deterministic (Hamiltonian) dynamics\, with random
  initial conditions. This model of physical diffusion is known under the n
 ame of "random Lorentz gas" or "random wind-tree model". Celebrated milest
 ones on the route to better mathematical understanding of this model of tr
 ue physical diffusion are the Kinetic Limits for the tagged particle traje
 ctory under the so-called Boltzmann-Grad (a.k.a. low density)\, or weak co
 upling approximations [Gallavotti (1970)\, Spohn (1978)\, Boldrighini-Buni
 movich-Sinai (1982)\, respectively\, Kesten-Papanicolaou (1980)]. Under a 
 second diffusive space-time scaling limit - done as a second step\, after 
 the kinetic approximations- the central limit theorem (CLT) and invariance
  principle (IP) for the tagged particle motion follow. However\, the CLT/I
 P under bare diffusive space-time scaling (without first applying the kine
 tic approximations) remains a Holy Grail. In recent work we have obtained 
 some intermediate results\, partially interpolating between the two-steps-
 limit (first kinetic\, then diffusive - as described above) and the bare-d
 iffusive-limit (Holy Grail). We establish the Invariance Principle for the
  tagged particle trajectories under a joint kinetic+diffusive limiting pro
 cedure\, performed simultaneously rather than successively\, reaching sign
 ificantly longer time scales than in earlier works. The Holy Grail (i.e.\,
  CLT under bare diffusive scaling) remains\, however\, beyond reach. &nbsp
 \;I will prersent a survey of the main problems and (historic and more rec
 ent) results\, accessible for a broad range of mathematicians.
LOCATION:Seminar Room 1\, Newton Institute
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