Dear all,
We’ll have an online statphys seminar given by Dr Sergio Andraus from 10am, Monday, June 14.
Those who wish to join the seminar please send an e-mail to me at
hatano@iis.u-tokyo.ac.jp <mailto:hatano@iis.u-tokyo.ac.jp>
I’ll send the zoom link.
best regards,
Naomichi Hatano
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Speaker: Dr. Sergio Andraus (Dept. Phys., U. Tokyo)
Date: 10am, Monday, June 14
Title: Fractal dimension of collision times in a log-potential Brownian particle system
Abstract:
Starting in the 1960s, the study of one-dimensional Brownian particle systems conditioned never to collide has been an important research topic due to its relationship to multiple polymer chains in two dimensions [1], and wetting and melting transitions [2] among many others. The non-colliding constraint manifests itself as a logarithmic repulsion potential between particles with a couplingconstant beta equal to two. These systems, more generally known as the Dyson model [3], can be defined for any real positive value of beta, but the non-colliding property disappears when beta lies between zero and one [4], suggesting the existence of some sort of phase transition. While it is known that in this “high temperature regime” particles collide almost surely, the properties of the collision times have been unclear until now. We focus on the set of collision times, and we find that it has a fractal structure. Moreover, we show that the fractal dimension of the set is non-zero when beta lies between zero and one, and that it vanishes when beta is greater than one, which is a behavior that resembles that of an order parameter.
This is joint work with N. Hufnagel, TU Dortmund.
[1] P. G. de Gennes, J. Chem. Phys. 48 (1968) 2257 – 2259 <doi.org/10.1063/1.1669420>
[2] M. E. Fisher, J. Stat. Phys. 34 (1984) 667 – 729 <doi.org/10.1007/BF01009436>
[3] F. J. Dyson, J. Math. Phys. 3 (1962) 1191 – 1198 <doi.org/10.1063/1.1703862>
[4] N. Demni, C. R. Acad. Sci. Paris, Ser. I 347 (2009) 1125 – 1128 <dx.doi.org/10.1016/j.crma.2009.08.003>
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Naomichi Hatano
Institute of Industrial Science, University of Tokyo
Kashiwanoha 5-1-5, Kashiwa, Chiba 277-8574, JAPAN
Phone: +81-4-7136-6961
Fax: +81-4-7136-6978
hatano-lab.iis.u-tokyo.ac.jp/hatano/
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