  We investigate the dynamics of ergotropy in open systems under Markovian and
non-Markovian evolutions. In this scenario, we begin by formulating the
ergotropy of an arbitrary qubit state in terms of energy and coherence. Thus,
we determine the conditions for ergotropy freezing and ergotropy sudden death
as a consequence of the system-bath interaction. In order to use ergotropy as a
resource for energy extraction in the form of work in an open-system scenario,
we adopt the entropy-based formulation of quantum thermodynamics. In this
approach, the work gains an additional environment-induced component, which may
be present even for constant Hamiltonians. We then establish an analytical
relationship between the environment-induced work and ergotropy, providing an
interpretation of environment-induced work in terms of variation of ergotropy.
In particular, energy transfer by environment-induced work can be performed up
to a limit, which is governed by the energy cost to transit between the initial
and final passive states of the quantum dynamics. We illustrate these results
for qubit states evolving under non-dissipative and dissipative quantum
processes.
