Sannolikhets- och statistikseminarium: Coulomb Gas Ensembles with Local Interactions

  • Datum:
  • Plats: Ångströmlaboratoriet 64119
  • Föreläsare: Tatyana Turova, Lund
  • Kontaktperson: Fiona Skerman
  • Seminarium

Abstract: We consider the system of particles on a finite interval with pair-wise interaction and external force. We take into account interactions between nearest, second nearest, and so on, up till K-th nearest neighbours, where K ≥ 1 is fixed arbitrarily. This model was introduced by Malyshev [1] to study the flow of charged particles on a rigorous mathematical level. It is a simplified version of a 3-dimensional classical Coulomb gas model. We study Gibbs distribution at finite positive temperature extending results [2] on the zero temperature case (ground states). We derived in [3] the asymptotics for the mean and for the variances of the distances between the neighbouring charges when K = 1. It is proved that depending on the strength of the external force there are several phase transitions in the local structure of the configuration of the particles in the limit when the number of particles goes to infinity. We identify 5 different phases for any positive temperature. The proof relies on a conditional central limit theorem for non-identical independent random variables. To study case K > 1 we derive first the exponential decay of the involved random variables. This allows to extend some results of [3] for a model with an arbitrary range of interactions.

[1] Malyshev, V. A.: Phase transitions in the one-dimensional Coulomb medium. Problems of Information Transmission, v. 51, no. 1, 31-36 (2015).
[2] Malyshev, V. A., and Zamyatin A. A.: One-Dimensional Coulomb Multiparticle Systems. Advances in Mathematical Physics, Article ID 857846 (2015).
[3] Turova, T. S. Phase transitions in the one-dimensional Coulomb gas ensembles. Ann. Appl. Probab. 28 (2018), no. 2, 1249-1291.