Minimal work principle: proof and counterexamples

TL;DR

This paper investigates the validity of the minimal work principle in quantum systems, proving it holds without level crossing and providing counterexamples where it fails, supported by an exactly solvable model.

Contribution

It proves the minimal work principle for quantum systems without level crossing and presents counterexamples demonstrating its violation when crossings occur.

Findings

Minimal work principle holds if adiabatic energy levels do not cross.

Counterexamples show violations when level crossing occurs.

Exactly solvable model confirms theoretical results.

Abstract

The minimal work principle states that work done on a thermally isolated equilibrium system is minimal for adiabatically slow (reversible) realization of a given process. This principle, one of the formulations of the second law, is studied here for finite (possibly large) quantum systems interacting with macroscopic sources of work. It is shown to be valid as long as the adiabatic energy levels do not cross. If level crossing does occur, counter examples are discussed, showing that the minimal work principle can be violated and that optimal processes are neither adiabatically slow nor reversible. The results are corroborated by an exactly solvable model.