[Paper Review] Fluctuations, Trajectory Entropy, and Ziegler's Maximum Entropy Production

В. Д. Селезнев, L. M. Martyushev|arXiv (Cornell University)|Dec 13, 2011

Advanced Thermodynamics and Statistical Mechanics35 references3 citations

TL;DR

This paper derives explicit expressions for trajectory entropy and entropy production in a small isolated system with two thermodynamic fluxes deviating slightly from equilibrium, using detailed balance and Onsager's fluctuation approximation. It demonstrates that Onsager's reciprocal relations emerge from both maximizing trajectory entropy and Ziegler's maximum entropy production principle, reconciling two interpretations of the latter as a physical law and an inference procedure.

ABSTRACT

We consider relaxation of an isolated system to the equilibrium using detailed balance condition and Onsager's fluctuation approximation. There is a small deviation from the equilibrium in two parameters. For this system, explicit expressions both for the dependence of trajectory entropy on random thermodynamic fluxes and for the dependence of entropy production on the most probable thermodynamic fluxes are obtained. Onsager's linear relations are obtained for the considered model using two methods (maximization of trajectory entropy and Ziegler's maximization of entropy production). Two existing interpretations of the maximum entropy production principle - as a physical principle and as an effective inference procedure - are discussed in the paper.

Motivation & Objective

• To analyze relaxation dynamics of an isolated system near equilibrium using detailed balance and Onsager's fluctuation approximation.
• To derive explicit expressions for trajectory entropy as a function of random thermodynamic fluxes.
• To express entropy production in terms of the most probable fluxes.
• To derive Onsager's linear reciprocal relations via two distinct optimization principles: trajectory entropy maximization and Ziegler's entropy production maximization.
• To clarify the conceptual status of the maximum entropy production principle as either a physical law or an inference procedure.

Proposed method

• Apply the detailed balance condition to model stochastic transitions between states in a small isolated system.
• Use Onsager's fluctuation approximation to describe small deviations from equilibrium in two thermodynamic fluxes.
• Construct the trajectory entropy as a function of fluctuating fluxes using statistical mechanics formalism.
• Maximize trajectory entropy with respect to fluxes to derive the most probable fluxes and their relations.
• Apply Ziegler's principle of maximum entropy production to the same system and derive the same linear flux-force relations.
• Compare the two derivation pathways to assess consistency and interpretative implications.

Experimental results

Research questions

• RQ1 How does trajectory entropy depend on random thermodynamic fluxes in a small system with two degrees of freedom near equilibrium?
• RQ2 Can Onsager's reciprocal relations be derived from maximizing trajectory entropy in this model?
• RQ3 Does Ziegler's principle of maximum entropy production yield the same linear relations as trajectory entropy maximization in this system?
• RQ4 What is the relationship between the maximum entropy production principle and statistical inference in non-equilibrium systems?
• RQ5 How do the two interpretations of the maximum entropy production principle—physical law versus inference procedure—coincide in this model?

Key findings

• Explicit analytical expressions are derived for trajectory entropy as a function of fluctuating thermodynamic fluxes in a two-flux system.
• The most probable fluxes, obtained by maximizing trajectory entropy, satisfy Onsager's reciprocal relations.
• Ziegler's principle of maximum entropy production leads to the same linear flux-force relations as trajectory entropy maximization.
• The two interpretations of the maximum entropy production principle—physical and inferential—yield identical results in this model, suggesting a deep connection between them.
• The derivation confirms the consistency of Onsager's linear response theory with both entropy maximization and entropy production maximization principles.