  During a spontaneous change, a macroscopic physical system will evolve
towards a macro-state with more realizations. This observation is at the basis
of the Statistical Mechanical version of the Second Law of Thermodynamics, and
it provides an interpretation of entropy in terms of probabilities. However, we
cannot rely on the statistical-mechanical expressions for entropy in systems
that are far from equilibrium. In this paper, we compare various extensions of
the definition of entropy, which have been proposed for non-equilibrium
systems. It has recently been proposed that measures of information density may
serve to quantify entropy in both equilibrium and nonequilibrium systems. We
propose a new "bit-wise" method to measure the information density for off
lattice systems. This method does not rely on coarse-graining of the particle
coordinates. We then compare different estimates of the system entropy, based
on information density and on the structural properties of the system, and
check if the various entropies are mutually consistent and, importantly,
whether they can detect non-trivial ordering phenomena. We find that, except
for simple (one-dimensional) cases, the different methods yield answers that
are at best qualitatively similar, and often not even that, although in several
cases, different entropy estimates do detect ordering phenomena qualitatively.
Our entropy estimates based on bit-wise data compression contain no adjustable
scaling factor, and show large quantitative differences with the thermodynamic
entropy obtained from equilibrium simulations. Hence, our results suggest that,
at present, there is not yet a single, structure-based entropy definition that
has general validity for equilibrium and non equilibrium systems.
