From Non-Normalizable Boltzmann-Gibbs Statistics to Infinite-Ergodic Theory

Area of Science:

• Statistical Mechanics
• Theoretical Physics
• Complex Systems

Background:

• Particles in heat baths are fundamental to thermodynamics.
• External forces, like surface interactions, alter particle dynamics.
• Asymptotic states in driven systems are complex and often non-normalizable.

Purpose of the Study:

• To investigate the long-term behavior of a particle in a heat bath under a decaying external force.
• To analyze observables like energy in the emergent non-normalizable Boltzmann state.
• To develop a new non-equilibrium ensemble using a maximum entropy principle.

Main Methods:

• Analysis of a particle system with Lennard-Jones or logarithmic potentials.
• Calculation of time and ensemble averages for integrable observables.
• Derivation of a canonical-like ensemble from a maximum entropy principle with specific constraints.

Main Results:

• The system approaches a non-normalizable Boltzmann state over time.
• A novel out-of-equilibrium ensemble was derived, incorporating normalization, finite average energy, and linear mean-squared displacement.
• The study successfully merges infinite-ergodic theory with Boltzmann-Gibbs statistics.

Conclusions:

• The findings extend Boltzmann-Gibbs statistics to non-equilibrium systems.
• New perspectives on ergodicity are provided by integrating infinite-ergodic theory.
• The derived ensemble offers a valuable tool for studying driven systems approaching non-normalizable states.