Continuous thermomajorization and a complete set of laws for Markovian thermal processes

TL;DR

This paper introduces continuous thermomajorization to characterize Markovian thermal processes, providing necessary and sufficient conditions, generalized entropy inequalities, and explicit protocols for energy distribution transformations in quantum thermodynamics.

Contribution

It develops a comprehensive framework for Markovian thermal processes using continuous thermomajorization, including verifiable constraints and constructive protocols for state transformations.

Findings

Derived necessary and sufficient conditions for thermal state transformations.

Established a complete set of generalized entropy production inequalities.

Provided an efficient algorithm and Mathematica implementation for energy distribution analysis.

Abstract

The standard dynamical approach to quantum thermodynamics is based on Markovian master equations describing the thermalization of a system weakly coupled to a large environment, and on tools such as entropy production relations. Here we develop a new framework overcoming the limitations that the current dynamical and information theory approaches encounter when applied to this setting. More precisely, we introduce the notion of continuous thermomajorization, and employ it to obtain necessary and sufficient conditions for the existence of a Markovian thermal process transforming between given initial and final energy distributions of the system. These lead to a complete set of generalized entropy production inequalities including the standard one as a special case. Importantly, these conditions can be reduced to a finitely verifiable set of constraints governing non-equilibrium transformations under master equations. What is more, the framework is also constructive, i.e., it returns explicit protocols realizing any allowed transformation. These protocols use as building blocks elementary thermalizations, which we prove to be universal controls. Finally, we also present an algorithm constructing the full set of energy distributions achievable from a given initial state via Markovian thermal processes and provide a $\texttt{Mathematica}$ implementation solving $d=6$ on a laptop computer in minutes.