  Thermodynamics makes definite predictions about the thermal behavior of
macroscopic systems in and out of equilibrium. Statistical mechanics aims to
derive this behavior from the dynamics and statistics of the atoms and
molecules making up these systems. A key element in this derivation is the
large number of microscopic degrees of freedom of macroscopic systems.
Therefore, the extension of thermodynamic concepts, such as entropy, to small
(nano) systems raises many questions. Here we shall reexamine various
definitions of entropy for nonequilibrium systems, large and small. These
include thermodynamic (hydrodynamic), Boltzmann, and Gibbs-Shannon entropies.
We shall argue that, despite its common use, the last is not an appropriate
physical entropy for such systems, either isolated or in contact with thermal
reservoirs: physical entropies should depend on the microstate of the system,
not on a subjective probability distribution. To square this point of view with
experimental results of Bechhoefer we shall argue that the Gibbs-Shannon
entropy of a nano particle in a thermal fluid should be interpreted as the
Boltzmann entropy of a dilute gas of Brownian particles in the fluid.
