# Second law of thermodynamics for batteries with vacuum state

## TL;DR

This paper explores how incorporating the vacuum state into battery models in stochastic thermodynamics alters fundamental identities and laws, leading to new inequalities and bounds on work fluctuations during thermodynamic processes.

## Contribution

It introduces the effects of the vacuum state in battery models, replacing the Jarzynski identity with inequalities and deriving bounds on work fluctuations.

## Key findings

- Jarzynski identity replaced by inequalities when vacuum state is included
- Lower bounds on work fluctuations set by vacuum occupation
- Batteries without vacuum allow fluctuation-free erasure

## Abstract

In stochastic thermodynamics work is a random variable whose average is bounded by the change in the free energy of the system. In most treatments, however, the work reservoir that absorbs this change is either tacitly assumed or modelled using unphysical systems with unbounded Hamiltonians (i.e. the ideal weight). In this work we describe the consequences of introducing the ground state of the battery and hence -- of breaking its translational symmetry. The most striking consequence of this shift is the fact that the Jarzynski identity is replaced by a family of inequalities. Using these inequalities we obtain corrections to the second law of thermodynamics which vanish exponentially with the distance of the initial state of the battery to the bottom of its spectrum. Finally, we study an exemplary thermal operation which realizes the approximate Landauer erasure and demonstrate the consequences which arise when the ground state of the battery is explicitly introduced. In particular, we show that occupation of the vacuum state of any physical battery sets a lower bound on fluctuations of work, while batteries without vacuum state allow for fluctuation-free erasure.