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During a spontaneous change, a macroscopic physical system will evolvetowards a macro-state with more realizations. This observation is at the basisof the Statistical Mechanical version of the Second Law of Thermodynamics, andit provides an interpretation of entropy in terms of probabilities. However, wecannot rely on the statistical-mechanical expressions for entropy in systemsthat are far from equilibrium. In this paper, we compare various extensions ofthe definition of entropy, which have been proposed for non-equilibriumsystems. It has recently been proposed that measures of information density mayserve to quantify entropy in both equilibrium and nonequilibrium systems. Wepropose a new "bit-wise" method to measure the information density for offlattice systems. This method does not rely on coarse-graining of the particlecoordinates. We then compare different estimates of the system entropy, basedon information density and on the structural properties of the system, andcheck if the various entropies are mutually consistent and, importantly,whether they can detect non-trivial ordering phenomena. We find that, exceptfor simple (one-dimensional) cases, the different methods yield answers thatare at best qualitatively similar, and often not even that, although in severalcases, different entropy estimates do detect ordering phenomena qualitatively.Our entropy estimates based on bit-wise data compression contain no adjustablescaling factor, and show large quantitative differences with the thermodynamicentropy obtained from equilibrium simulations. Hence, our results suggest that,at present, there is not yet a single, structure-based entropy definition thathas general validity for equilibrium and non equilibrium systems.

Information density, structure and entropy in equilibrium and non-equilibrium systems