  We consider an isolated system in an arbitrary state and provide a general
formulation using first principles for an additive and non-negative statistical
quantity that is shown to reproduce the equilibrium thermodynamic entropy of
the isolated system. We further show that the statistical quantity represents
the nonequilibrium thermodynamic entropy when the latter is a state function of
nonequilibrium state variables; see text. We consider an isolated 1-d ideal gas
and determine its non-equilibrium statistical entropy as a function of the box
size as the gas expands freely isoenergetically, and compare it with the
equilibrium thermodynamic entropy S_{0eq}. We find that the statistical entropy
is less than S_{0eq} in accordance with the second law, as expected. To
understand how the statistical entropy is different from thermodynamic entropy
of classical continuum models that is known to become negative under certain
conditions, we calculate it for a 1-d lattice model and discover that it can be
related to the thermodynamic entropy of the continuum 1-d Tonks gas by taking
the lattice spacing {\delta} go to zero, but only if the latter is
state-independent. We discuss the semi-classical approximation of our entropy
and show that the standard quantity S_{f}(t) in the Boltzmann's H-theorem does
not directly correspond to the statistical entropy.
