Analytic form of a two-dimensional critical distribution.

Area of Science:

• Statistical physics
• Condensed matter theory
• Mathematical physics

Background:

• Understanding critical phenomena in two-dimensional systems is crucial.
• The Edwards-Wilkinson interface model and spin-wave models exhibit complex fluctuation dynamics.

Purpose of the Study:

• To derive an analytic form for the distribution of order parameter fluctuations.
• To investigate the nature of width fluctuations in 2D critical systems.
• To explore the applicability of mathematical functions to describe these distributions.

Main Methods:

• Analysis of the characteristic function of the distribution.
• Exact expression using a gamma function quotient.
• Approximation using a Charlier series with a Gumbel distribution kernel.

Main Results:

• The characteristic function is precisely a quotient of gamma functions.
• A Charlier series converges to the exact distribution over a limited range.
• The derived methods allow calculation of temperature dependence.
• Insights into Gumbel-like distributions in non-extreme equilibrium quantities are provided.

Conclusions:

• An exact analytical solution for order parameter fluctuation distributions in 2D critical models is established.
• The findings connect statistical mechanics with special functions.
• The study offers a framework for understanding equilibrium fluctuations beyond extreme value statistics.