  Thermodynamics can be formulated in either of two approaches, the
phenomenological approach, which refers to the macroscopic properties of
systems, and the statistical approach, which describes systems in terms of
their microscopic constituents. We establish a connection between these two
approaches by means of a new axiomatic framework that can take errors and
imprecisions into account. This link extends to systems of arbitrary sizes
including microscopic systems, for which the treatment of imprecisions is
pertinent to any realistic situation. Based on this, we identify the quantities
that characterise whether certain thermodynamic processes are possible with
entropy measures from information theory. In the error-tolerant case, these
entropies are so-called smooth min and max entropies. Our considerations
further show that in an appropriate macroscopic limit there is a single entropy
measure that characterises which state transformations are possible. In the
case of many independent copies of a system (the so-called i.i.d. regime), the
relevant quantity is the von Neumann entropy.
