### Abstract

We consider two related problems: the first is the minimization of the "Coulomb renormalized energy" of Sandier-Serfaty, which corresponds to the total Coulomb interaction of point charges in a uniform neutralizing background (or rather variants of it). The second corresponds to the minimization of the Hamiltonian of a 2D "Coulomb gas" or "one-component plasma", a system of n point charges with Coulomb pair interaction, in a confining potential (minimizers of this energy also correspond to "weighted Fekete sets"). In both cases, we investigate the microscopic structure of minimizers, that is, at the scale corresponding to the interparticle distance. We show that in any large enough microscopic set, the value of the energy and the number of points are "rigid" and completely determined by the macroscopic density of points. In other words, points and energy are "equidistributed" in space (modulo appropriate scalings). The number of points in a ball is in particular known up to an error proportional to the radius of the ball. We also prove a result on the maximal and minimal distances between points. Our approach involves fully exploiting the minimality by reducing to minimization problems with fixed boundary conditions posed on smaller subsets.

Original language | English (US) |
---|---|

Pages (from-to) | 3035-3093 |

Number of pages | 59 |

Journal | International Mathematics Research Notices |

Volume | 2015 |

Issue number | 11 |

DOIs | |

State | Published - 2015 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*International Mathematics Research Notices*,

*2015*(11), 3035-3093. https://doi.org/10.1093/imrn/rnu031

**Renormalized energy equidistribution and local charge balance in 2D coulomb systems.** / Nodari, Simona Rota; Serfaty, Sylvia.

Research output: Contribution to journal › Article

*International Mathematics Research Notices*, vol. 2015, no. 11, pp. 3035-3093. https://doi.org/10.1093/imrn/rnu031

}

TY - JOUR

T1 - Renormalized energy equidistribution and local charge balance in 2D coulomb systems

AU - Nodari, Simona Rota

AU - Serfaty, Sylvia

PY - 2015

Y1 - 2015

N2 - We consider two related problems: the first is the minimization of the "Coulomb renormalized energy" of Sandier-Serfaty, which corresponds to the total Coulomb interaction of point charges in a uniform neutralizing background (or rather variants of it). The second corresponds to the minimization of the Hamiltonian of a 2D "Coulomb gas" or "one-component plasma", a system of n point charges with Coulomb pair interaction, in a confining potential (minimizers of this energy also correspond to "weighted Fekete sets"). In both cases, we investigate the microscopic structure of minimizers, that is, at the scale corresponding to the interparticle distance. We show that in any large enough microscopic set, the value of the energy and the number of points are "rigid" and completely determined by the macroscopic density of points. In other words, points and energy are "equidistributed" in space (modulo appropriate scalings). The number of points in a ball is in particular known up to an error proportional to the radius of the ball. We also prove a result on the maximal and minimal distances between points. Our approach involves fully exploiting the minimality by reducing to minimization problems with fixed boundary conditions posed on smaller subsets.

AB - We consider two related problems: the first is the minimization of the "Coulomb renormalized energy" of Sandier-Serfaty, which corresponds to the total Coulomb interaction of point charges in a uniform neutralizing background (or rather variants of it). The second corresponds to the minimization of the Hamiltonian of a 2D "Coulomb gas" or "one-component plasma", a system of n point charges with Coulomb pair interaction, in a confining potential (minimizers of this energy also correspond to "weighted Fekete sets"). In both cases, we investigate the microscopic structure of minimizers, that is, at the scale corresponding to the interparticle distance. We show that in any large enough microscopic set, the value of the energy and the number of points are "rigid" and completely determined by the macroscopic density of points. In other words, points and energy are "equidistributed" in space (modulo appropriate scalings). The number of points in a ball is in particular known up to an error proportional to the radius of the ball. We also prove a result on the maximal and minimal distances between points. Our approach involves fully exploiting the minimality by reducing to minimization problems with fixed boundary conditions posed on smaller subsets.

UR - http://www.scopus.com/inward/record.url?scp=84941896207&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84941896207&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnu031

DO - 10.1093/imrn/rnu031

M3 - Article

VL - 2015

SP - 3035

EP - 3093

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 11

ER -