The Annals of Applied Probability

Phase transitions in the one-dimensional Coulomb gas ensembles

Tatyana S. Turova

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider the system of particles on a finite interval with pairwise nearest neighbours interaction and external force. This model was introduced by Malyshev [Probl. Inf. Transm. 51 (2015) 31–36] 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 recent results on the zero temperature case (ground states). We derive the asymptotics for the mean and for the variances of the distances between the neighbouring charges. We prove 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 proofs rely on a conditional central limit theorem for nonidentical random variables, which has an interest on its own.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 2 (2018), 1249-1291.

Dates
Received: December 2016
Revised: July 2017
First available in Project Euclid: 11 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1523433635

Digital Object Identifier
doi:10.1214/17-AAP1329

Mathematical Reviews number (MathSciNet)
MR3784499

Zentralblatt MATH identifier
06897954

Subjects
Primary: 82B21: Continuum models (systems of particles, etc.) 82B26: Phase transitions (general) 60F05: Central limit and other weak theorems

Keywords
Coulomb gas phase transitions Gibbs ensemble

Citation

Turova, Tatyana S. Phase transitions in the one-dimensional Coulomb gas ensembles. Ann. Appl. Probab. 28 (2018), no. 2, 1249--1291. doi:10.1214/17-AAP1329. https://projecteuclid.org/euclid.aoap/1523433635


Export citation

References

  • [1] Aizenman, M. and Martin, P. A. (1980). Structure of Gibbs states of one dimensional Coulomb systems. Comm. Math. Phys. 78 99–116.
  • [2] Ameur, Y., Hedenmalm, H. and Makarov, N. (2015). Random normal matrices and Ward identities. Ann. Probab. 43 1157–1201.
  • [3] Ameur, Y., Kang, N.-G. and Makarov, N. Rescaling Ward identities in the random normal matrix model. arXiv:1410.4132.
  • [4] Bauerschmidt, R., Bourgade, P., Nikula, M. and Yau, H.-T. Local density for two-dimensional one-component plasma. arXiv:1510.02074.
  • [5] Bauerschmidt, R., Bourgade, P., Nikula, M. and Yau, H.-T. The two-dimensional Coulomb plasma: Quasi-free approximation and central limit theorem. arXiv:1609.08582.
  • [6] Cunden, F. D., Facchi, P. and Vivo, P. (2016). A shortcut through the Coulomb gas method for spectral linear statistics on random matrices. J. Phys. A: Math. Theor. 49 35202.
  • [7] Diaconis, P. and Freedman, D. A. (1988). Conditional limit theorems for exponential families and finite versions of de Finetti’s theorem. J. Theoret. Probab. 1 381–410.
  • [8] Edwards, S. F. and Lenard, A. (1962). Exact statistical mechanics of a one-dimensional system with Coulomb forces. II. The method of functional integration. J. Math. Phys. 3 778–792.
  • [9] Fedoryuk, M. V. (1987). Asimptotika: Integraly i Ryady. Nauka, Moscow. (Russian) [Asymptotics: Integrals and Series].
  • [10] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley.
  • [11] Johansson, K. (1991). Separation of phases at low temperatures in a one-dimensional continuous gas. Comm. Math. Phys. 141 259–278.
  • [12] Johansson, K. (1995). On separation of phases in one-dimensional gases. Comm. Math. Phys. 169 521–561.
  • [13] Leblé, T., Serfaty, S. and Zeitouni, O. (2017). Large deviations for the two-dimensional two-component plasma. Comm. Math. Phys. 350 301–360.
  • [14] Lenard, A. (1961). Exact statistical mechanics of a one-dimensional system with Coulomb forces. J. Math. Phys. 2 682.
  • [15] Lenard, A. (1963). Exact statistical mechanics of a one-dimensional system with Coulomb forces. III. Statistics of the electric field. J. Math. Phys. 4 533.
  • [16] Malyshev, V. A. (2015). Phase transitions in the one-dimensional Coulomb medium. Probl. Inf. Transm. 51 31–36.
  • [17] Malyshev, V. A. and Zamyatin, A. A. (2015). One-dimensional Coulomb multiparticle systems. Adv. Math. Phys.
  • [18] Petrov, V. V. (1975). Sums of Independent Random Variables. Springer.
  • [19] Serfaty, S. (2015). Coulomb Gases and Ginzburg–Landau Vortices. European Mathematical Society.