[Paper Review] Contributions to the Theory of Thermostated Systems

Ronald F. Fox|arXiv (Cornell University)|Sep 17, 2014

Advanced Thermodynamics and Statistical Mechanics7 references3 citations

TL;DR

This paper develops a theoretical framework for thermostated systems—systems in contact with a thermal reservoir—using Markovian dynamics and path integral formulations derived from the Smoluchowski equation. It establishes a rigorous foundation for nonequilibrium statistical mechanics, showing how the Crooks theorem and Jarzynski equality emerge naturally from this formalism, thereby unifying key results in stochastic thermodynamics for full-phase-space dynamics.

ABSTRACT

In this paper, a theory for systems in contact with a thermal reservoir is developed. We call such systems "thermostated systems." Interest in these systems has been vigorous for about the last 20 years and is illustrated by the Crooks theorem and the Jarzinski equality, two results for the description of systems in full phase space where all coordinates and momenta may be followed through time. These fundamental results follow from the behavior of Markovian systems in phase space for which path integral expressions can be derived from the Smoluchowski equation valid for Markovian systems.

Motivation & Objective

• To develop a comprehensive theoretical framework for systems in contact with a thermal reservoir, termed 'thermostated systems'.
• To unify fundamental nonequilibrium results such as the Crooks theorem and Jarzynski equality within a single dynamical formalism.
• To derive path integral expressions from the Smoluchowski equation for Markovian systems in full phase space.
• To provide a rigorous foundation for stochastic thermodynamics applicable to systems with time-evolving coordinates and momenta.
• To clarify the connection between Markovian stochastic dynamics and fluctuation theorems in statistical mechanics.

Proposed method

• Formulates thermostated systems using the Smoluchowski equation as the starting point for Markovian stochastic dynamics.
• Derives path integral representations for the probability evolution of trajectories in full phase space (coordinates and momenta).
• Applies the formalism to systems with time-dependent external protocols, enabling analysis of work and entropy production.
• Uses the Markovian assumption to ensure Markovian transition probabilities and time-local dynamics.
• Establishes equivalence between the path integral formulation and fluctuation theorems via functional integration.
• Demonstrates that the Crooks and Jarzynski theorems follow as direct consequences of the derived path integral structure.

Experimental results

Research questions

• RQ1 How can a consistent theoretical framework be constructed for thermostated systems in contact with a thermal reservoir?
• RQ2 What is the role of the Smoluchowski equation in deriving path integral expressions for full-phase-space dynamics?
• RQ3 How do the Crooks theorem and Jarzynski equality emerge from a unified Markovian stochastic dynamics framework?
• RQ4 What conditions ensure the validity of fluctuation theorems in systems with time-dependent protocols?
• RQ5 Can the path integral formulation derived from the Smoluchowski equation reproduce known results in nonequilibrium statistical mechanics?

Key findings

• The path integral formulation derived from the Smoluchowski equation provides a complete description of trajectory probabilities in full phase space for Markovian thermostated systems.
• The Crooks fluctuation theorem is shown to follow directly from the derived path integral structure under time-reversal symmetry.
• The Jarzynski equality is derived as a consequence of the same formalism, confirming its validity for Markovian systems with full phase-space observables.
• The framework unifies fluctuation theorems with stochastic dynamics, showing they are inherent properties of the underlying Markovian process.
• The results are consistent across different time-symmetric and time-asymmetric protocols, validating the robustness of the formalism.