  Two approaches to describe the thermodynamics of a subsystem that interacts
with a thermal bath are considered. Within the first approach, the mean system
energy $E_{S}$ is identified with the expectation value of the system
Hamiltonian, which is evaluated with respect to the overall (system+bath)
equilibrium distribution. Within the second approach, the system partition
function $Z_{S}$ is considered as the fundamental quantity, which is postulated
to be the ratio of the overall (system+bath) and the bath partition functions,
and the standard thermodynamic relation $E_{S}=-d(\ln Z_{S})/d\beta$ is used to
obtain the mean system energy. % ($\beta\equiv 1/(k_{B}T)$, $k_{B}$ is the
Boltzmann constant, %and $T$ is the temperature). Employing both classical and
quantum mechanical treatments, the advantages and shortcomings of the two
approaches are analyzed in detail for various different systems. It is shown
that already within classical mechanics both approaches predict significantly
different results for thermodynamic quantities provided the system-bath
interaction is not bilinear or the system of interest consists of more than a
single particle. Based on the results, it is concluded that the first approach
is superior.
