[Paper Review] Absolute Irreversibility in Information Thermodynamics

Yûto Murashita|arXiv (Cornell University)|Jun 15, 2015

Advanced Thermodynamics and Statistical Mechanics4 references3 citations

TL;DR

This thesis introduces absolute irreversibility as a new class of irreversibility in nonequilibrium thermodynamics, defined via the singular part of a probability measure using Lebesgue’s decomposition theorem. It derives stronger nonequilibrium equalities that apply to systems where conventional equalities fail, such as free expansion and systems with traps, and provides a resolution to Gibbs’ paradox in classical mesoscopic regimes by identifying a missing $N!$ factor through absolute irreversibility.

ABSTRACT

Nonequilibrium equalities have attracted considerable interest in the context of statistical mechanics and information thermodynamics. What is remarkable about nonequilibrium equalities is that they apply to rather general nonequilibrium situations beyond the linear response regime. However, nonequilibrium equalities are known to be inapplicable to some important situations. In this thesis, we introduce a concept of absolute irreversibility as a new class of irreversibility that encompasses the entire range of those irreversible situations to which the conventional nonequilibrium equalities are inapplicable. In mathematical terms, absolute irreversibility corresponds to the singular part of probability measure and can be separated from the ordinary irreversible part by Lebesgue's decomposition theorem in measure theory. This theorem guarantees the uniqueness of the decomposition of probability measure into singular and nonsingular parts, which enables us to give a well-defined mathematical and physical meaning to absolute irreversibility. Consequently, we derive a new type of nonequilibrium equalities in the presence of absolute irreversibility. Inequalities derived from our nonequilibrium equalities give stronger restrictions on the entropy production during nonequilibrium processes than the conventional second-law like inequalities. Moreover, we present a new resolution of Gibbs' paradox from the viewpoint of absolute irreversibility. This resolution applies to a classical mesoscopic regime, where two prevailing resolutions of Gibbs' paradox break down.

Motivation & Objective

• To address the inapplicability of conventional nonequilibrium equalities in certain irreversible processes, such as free expansion and systems with traps.
• To formalize a new class of irreversibility—absolute irreversibility—using measure-theoretic decomposition.
• To derive stronger nonequilibrium equalities that constrain entropy production more tightly than the second law.
• To resolve Gibbs’ paradox in the classical mesoscopic regime, where quantum and extensivity-based resolutions fail.
• To unify information thermodynamics with absolute irreversibility in feedback-controlled processes.

Proposed method

• Uses Lebesgue’s decomposition theorem to separate a probability measure into absolutely continuous, singular continuous, and discrete parts.
• Defines absolute irreversibility as the singular part of the probability measure, which corresponds to events with zero probability density but nonzero measure.
• Applies the Radon-Nikodým derivative and singular decomposition to derive new integral nonequilibrium equalities valid even when the Jarzynski equality fails.
• Applies the framework to feedback-controlled systems, including the Szilard engine, to derive information-thermodynamic equalities under absolute irreversibility.
• Uses the uniqueness of measure decomposition to rigorously define the $N!$ factor in Gibbs’ paradox resolution via absolute irreversibility.
• Derives a new form of the second law that incorporates absolute irreversibility, yielding tighter bounds on entropy production.

Experimental results

Research questions

• RQ1 Can nonequilibrium equalities be extended to processes where the conventional Jarzynski equality fails due to singular probability measures?
• RQ2 How can absolute irreversibility, defined via the singular part of a measure, be used to derive stronger constraints on entropy production?
• RQ3 What is the role of absolute irreversibility in resolving Gibbs’ paradox in classical mesoscopic systems?
• RQ4 How do feedback-controlled processes, such as the Szilard engine, behave under absolute irreversibility?
• RQ5 Can the $N!$ factor in Gibbs’ paradox be derived from a physical principle of absolute irreversibility rather than quantum statistics or extensivity?

Key findings

• The paper establishes that absolute irreversibility corresponds to the singular part of a probability measure, which is uniquely identifiable via Lebesgue’s decomposition theorem.
• New nonequilibrium equalities are derived that apply to processes such as free expansion and systems with traps, where conventional equalities fail.
• These new equalities yield stronger bounds on entropy production than the second law, providing tighter thermodynamic constraints.
• The resolution of Gibbs’ paradox is achieved by identifying a missing $N!$ factor arising from absolute irreversibility in the classical mesoscopic regime.
• The framework applies to feedback-controlled systems, including multi-particle Szilard engines, and shows that unavailable information is linked to absolute irreversibility.
• The uniqueness of the measure decomposition ensures a well-defined physical interpretation of absolute irreversibility, distinct from ordinary irreversibility.