Relative Entropy, Gaussian Concentration and Uniqueness of Equilibrium States

Area of Science:

• Statistical Mechanics
• Mathematical Physics
• Quantum Information Theory

Background:

• Lattice spin systems are fundamental models in statistical mechanics.
• Gibbs measures describe the equilibrium states of physical systems.
• Relative entropy is a key measure of distinguishability between probability distributions.

Purpose of the Study:

• To establish a connection between Gaussian concentration bounds and the positivity of relative entropy density.
• To demonstrate the utility of Gaussian concentration bounds for proving the uniqueness of Gibbs measures.
• To extend existing results on Gibbs measure uniqueness with a novel and concise proof.

Main Methods:

• Utilizing an abstract Gaussian concentration bound.
• Applying techniques from mathematical physics and probability theory.
• Developing a novel proof strategy for relative entropy density positivity.

Main Results:

• Proved that an abstract Gaussian concentration bound implies positivity of the lower relative entropy density for a general class of lattice spin systems.
• Established the uniqueness of translation-invariant Gibbs measures directly from the Gaussian concentration bound.
• Achieved these results through a significantly shorter and more general proof compared to prior work.

Conclusions:

• Gaussian concentration bounds provide a powerful tool for analyzing lattice spin systems.
• The positivity of relative entropy density is a crucial consequence of these bounds.
• This work offers a streamlined approach to proving the uniqueness of Gibbs measures, advancing the field of statistical mechanics.