  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.
