Dynamic fluctuation-dissipation theory for generalized Langevin equations: Constructive constraints, stability, and...

Area of Science:

• Statistical Mechanics
• Theoretical Physics
• Non-equilibrium Thermodynamics

Background:

• Generalized Langevin equations (GLEs) describe complex systems dynamics.
• Understanding equilibrium and non-equilibrium properties is crucial.
• The connection between system dynamics and thermodynamic behavior requires clarification.

Purpose of the Study:

• Develop a dynamic theory for systems governed by generalized Langevin equations.
• Establish conditions for the existence of equilibrium behavior and its fluctuation-dissipation implications.
• Investigate the role of physical constraints in thermodynamic equilibrium and linear response theory.

Main Methods:

• Utilizing an initial-value formulation for the dynamic theory.
• Deriving physical constraints: dissipative stability and stochastic realizability.
• Analyzing systems with the property of local realizability, equivalent to Markovian embedding.

Main Results:

• Conditions for equilibrium behavior and fluctuation-dissipation relations are provided.
• For systems with local realizability, dissipative stability and stochastic realizability ensure thermodynamic equilibrium.
• Violation of these constraints leads to the absence of thermodynamic equilibrium, even for dissipative systems.

Conclusions:

• The developed theory reconstructs Kubo theory when physical constraints are met.
• The absence of thermodynamic equilibrium is demonstrated when dissipative stability or stochastic realizability is violated.
• Implications for linear response theory are significant, highlighting the necessity of these constraints for predictable system behavior.