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::::::::To find that out just from knowledge of the whole-system instantaneous microstate, we need to follow its trajectory for a good length of time, even for a rather long time. Overwhelmingly, not remotely as long as the Poincaré recurrence time, but still much longer than the time needed to make a measurement of, say, local temperature or wall pressure. To verify thermodynamic equilibrium or non-equilibrium, we need time to make very many measurements well separated in time.
::::::::Equilibrium is characterized by all measurements of every particular supposed state variable hovering around their respective means. The whole-system instantaneous microstate shows no drift over time, however long, practically 'covering', but not necessarily 'filling', the whole of <math>R_0</math> practically uniformly over time. Thermodynamic entropy gives a precise measurement of how the practically uniform 'covering' of <math>R_0</math> actually 'fills' it over infinite time, a sort of time-averaged logarithmic ''density'' × d ''area'' integral. Such an integration is job for mathematicians. They have an arsenal of definitions of various entropies. Our IP mathematician friend is expert in this, and thinks it is the underlying basis of the general concept of 'entropy'; he has a good case.
::::::::Statistical mechanics provides a sort of Monte Carlo procedure to estimate that integral, using ergodic theorems.
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