  In one of its versions, the Second Law states: "It is impossible to construct
an engine which will work in a complete cycle, and produces no effect except
the raising of a weight and cooling of a heat reservoir." While the Second Law
is considered as one of the most robust laws of Nature, it is still challenging
how to interpret it in a fully quantum domain. Here we unpack the true meaning
of the "cyclicity" and formulate the Second Law for a generic quantum battery
via its asymptotic properties of a charging process rather than in terms of a
single cycle. As a paradigm, we propose a machine consisting of a battery that
repeatedly interacts with identically prepared systems. We then propose the
Second Law in the form: The ergotropy of the battery may increase indefinitely
if and only if systems are in a non-passive state. One of the most interesting
features of this new formulation is the appearance of the passive states that
naturally generalize the notion of the heat bath. In this paper, we provide a
handful of results that supports this formulation for diagonal systems.
Interestingly, our methodology meets a well-known theory of Markov chains,
according to which we classify the general charging processes based on the
passivity/non-passivity of charging systems. In particular, the adopted
mathematics allows us to distinguish a subtle asymptotic difference between the
indefinite increase of the battery's energy (induced by the maximally mixed
states) and of ergotropy (induced by the non-passive states) in terms of the
so-called null-recurrent versus transient Markov chains.
