Exact Extremal Statistics in the Classical 1D Coulomb Gas

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Mathematical Physics

Background:

• The one-dimensional Coulomb gas, or one-component plasma, is a fundamental model in statistical mechanics.
• Understanding the behavior of charged particles in confined systems is crucial for various physical phenomena.

Purpose of the Study:

• To analytically compute the probability distribution of the rightmost charge in a one-dimensional Coulomb gas with large N.
• To investigate the scaling functions and large deviation properties of this distribution.
• To compare the findings with existing models like Dyson's log gas.

Main Methods:

• Analytical computation of probability distributions.
• Analysis of scaling functions and asymmetric tail behavior.
• Calculation of large deviation functions.

Main Results:

• The probability distribution of the rightmost charge exhibits nontrivial scaling functions with asymmetric tails.
• This distribution is distinct from the Tracy-Widom distribution observed in Dyson's log gas.
• A third-order phase transition is identified, consistent with log gas behavior.

Conclusions:

• The one-dimensional Coulomb gas displays unique statistical properties for its extreme charge positions.
• The study provides a detailed analytical framework and numerical verification for these properties.
• This work contributes to a deeper understanding of interacting particle systems in one dimension.