Work extraction from microcanonical bath

TL;DR

This paper investigates the maximum work extractable from a large microcanonical spin bath at positive temperature, revealing that it can be significantly larger than a single spin's energy and differs from canonical systems.

Contribution

It provides a novel analysis of work extraction from microcanonical states, introducing a microcanonical free energy concept involving linear entropy, and highlights differences from canonical thermodynamics.

Findings

Maximum extractable work scales as O(√N ln N) for large N.

Work extraction can surpass single-spin energy but remains below total bath energy.

Microcanonical free energy involves linear entropy instead of von Neumann entropy.

Abstract

We determine the maximal work extractable via a cyclic Hamiltonian process from a positive-temperature ($T>0$) microcanonical state of a $N\gg 1$ spin bath. The work is much smaller than the total energy of the bath, but can be still much larger than the energy of a single bath spin, e.g. it can scale as ${\cal O}(\sqrt{N\ln N})$. Qualitatively same results are obtained for those cases, where the canonical state is unstable (e.g., due to a negative specific heat) and the microcanonical state is the only description of equilibrium. For a system coupled to a microcanonical bath the concept of free energy does {\it not generally} apply, since such a system|starting from the canonical equilibrium density matrix $\rho_T$ at the bath temperature $T$|can enhance the work extracted from the microcanonical bath without changing its state $\rho_T$. This is impossible for any system coupled to a canonical thermal bath due to the relation between the maximal work and free energy. But the concept of free energy still applies for a sufficiently large $T$. Here we find a compact expression for the {\it microcanonical free-energy} and show that in contrast to the canonical case it contains a {\it linear entropy} instead of the von Neumann entropy.