Statistics of particles with long-ranged pairwise interactions
In this thesis, we investigate the equilibrium and dynamical properties of charged particles in one dimension interacting via long-range interactions. First, we focus on the equilibrium one-dimensional one-component plasma (1dOCP), also known as the jellium model. It consists of $N$ harmonically confined charges on a line interacting via pairwise repulsive Coulomb interactions. First, we introduce some well-known equilibrium properties of the model. Then we present the Coulomb gas method, which we later use to analyze the behavior of large deviations of different observables in the limit of a large number of particles. We find the large deviation form for the general full linear statistics, and we derive the exact formula for its variance. We continue by computing typical as well as large deviations of the gap between two middle particles. We also analyze typical and atypical fluctuations of the number of particles inside of a symmetric interval that is in the bulk of the system. The last equilibrium observable that we study is the truncated linear statistics, which describes the sum of the positions of the $N’$ rightmost particles. In the last chapter, we investigate the Langevin dynamics of the jellium model without the confining potential. We find that, for large times, there exists a correspondence between the dynamical system and the equilibrium 1dOCP. We use this connection to study fluctuations of the particle positions in the dynamical system.
Jury : Ada Altieri, Fabio Cunden (rapporteur), David Dean, Satya Majumdar (co-directeur de thèse), Sergei Nechaev (co-directeur de thèse), Grégory Schehr (invité), Gabriel Tellez (rapporteur), Christophe Texier.