Analytical techniques for linear response formula of equilibrium states

Area of Science:

• Dynamical systems theory
• Ergodic theory
• Mathematical physics

Background:

• Equilibrium states are fundamental in statistical mechanics and dynamical systems.
• Understanding the differentiability of their probabilities is crucial for analyzing system behavior.
• Existing research has explored various aspects of equilibrium states, necessitating refined analytical tools.

Purpose of the Study:

• To present a novel analytical framework for proving the differentiability of equilibrium state probabilities.
• To investigate the relationship between Birkhoff metrics in cones and anisotropic spaces.
• To apply this framework to re-examine existing examples in equilibrium state research.

Main Methods:

• Development of an analytical framework based on differentiability.
• Exploration of the interplay between Birkhoff metrics and anisotropic spaces.
• Application of established theorems and introduction of new ones ('folklore theorems').

Main Results:

• The paper successfully establishes an analytical framework for differentiability of equilibrium state probabilities.
• Key theorems relating Birkhoff metrics, cones, and anisotropic spaces are proven.
• The framework is validated by revisiting and analyzing previously studied examples.

Conclusions:

• The presented framework offers a robust method for analyzing the differentiability of equilibrium state probabilities.
• The proven theorems provide valuable tools for the study of dynamical systems.
• This work contributes to a deeper understanding of equilibrium states in complex systems.