Probabilistic and thermodynamic aspects of dynamical systems.

Area of Science:

• Statistical mechanics
• Theoretical physics
• Dynamical systems theory

Background:

• Irreversible behavior is a key feature of macroscopic, mesoscopic, and microscopic systems.
• Understanding the underlying dynamics is crucial for describing irreversibility.

Purpose of the Study:

• To outline a probabilistic approach to irreversible behavior in dynamical systems.
• To identify signatures of complex dynamics in spectral properties.
• To explore the connection between entropy in nonequilibrium steady states and phase space dynamics.

Main Methods:

• Analysis of spectral properties of Liouville, Frobenius-Perron, and Fokker-Planck operators.
• Introduction of entropy and entropy production-like quantities.
• Exploration of nonequilibrium steady states.

Main Results:

• Signatures of complexity in underlying dynamics were identified on spectral properties.
• Connections between entropy properties and phase space dynamics were explored.
• A framework for understanding irreversibility through a probabilistic approach was established.

Conclusions:

• The probabilistic approach provides insights into irreversible behavior across different scales.
• Spectral properties serve as indicators of dynamical complexity.
• Entropy-based quantities offer a link between nonequilibrium thermodynamics and phase space characteristics.