Minimum relative entropy distributions with a large mean are Gaussian.

Area of Science:

• Statistical physics
• Information theory
• Evolutionary dynamics

Background:

• Entropy optimization is a powerful tool across diverse scientific fields.
• Relative entropy (Kullback-Leibler divergence) measures differences between probability distributions.
• Constrained optimization problems are common in physical and biological systems.

Purpose of the Study:

• To find a posterior distribution minimizing relative entropy given a prior and a fixed, large mean constraint.
• To explore the mathematical properties of such distributions.
• To demonstrate practical applications in physics and evolutionary biology.

Main Methods:

• Formulating a constrained optimization problem.
• Applying mathematical analysis to derive properties of the solution distribution.
• Investigating implications for dissipative dynamics and evolutionary natural selection.

Main Results:

• Solutions to the constrained problem are approximately Gaussian distributions.
• The equilibrium distribution of a confined Brownian particle is independent of the potential's shape.
• An H-type theorem is derived, showing increasing entropy for standardized fitness distributions in evolving populations.

Conclusions:

• The Gaussian approximation provides a simplified model for complex systems.
• Findings offer insights into equilibrium states of physical systems and the dynamics of natural selection.
• Entropy optimization principles offer a unified framework for understanding diverse phenomena.