  We demonstrate that the Gibbs-Shannon entropy is applicable to
non-equilibrium systems of any size and boundary conditions. The change in
microscopic entropy can be attributed to the stochastic nature of dynamic
processes and to the inherent uncertainties of thermodynamic systems. The
latter predicts that entropy production is nonnegative on average and varies
with different trajectories according to the fluctuation theorem. By contrast,
heat is affiliated with stochastic processes underlying particle motions and
the ensemble average over all possible trajectories leads to the Clausius
inequality. The Jarzynski/Crooks equations can be readily derived by applying
the fluctuation theorem to heat variation over different trajectories linking
equilibrium states.
