Now, by Liouville's theorem, time-reversal of all microscopic processes implies that the amount of information needed to describe the exact microstate of an isolated system (its information-theoretic joint entropy) is constant in time. This joint entropy is equal to the marginal entropy (entropy assuming no correlations) plus the entropy of correlation (mutual entropy, or its negative mutual information). If we assume no correlations between the particles initially, then this joint entropy is just the marginal entropy, which is just the initial thermodynamic entropy of the system, divided by Boltzmann's constant. However, if these are indeed the initial conditions (and this is a crucial assumption), then such correlations form with time. In other words, there is a decreasing mutual entropy (or increasing mutual information), and for a time that is not too long—the correlations (mutual information) between particles only increase with time. Therefore, the thermodynamic entropy, which is proportional to the marginal entropy, must also increase with time (note that "not too long" in this context is relative to the time needed, in a classical version of the system, for it to pass through all its possible microstates—a time that can be roughly estimated as formula_5, where formula_6 is the time between particle collisions and S is the system's entropy. In any practical case this time is huge compared to everything else). Note that the correlation between particles is not a fully objective quantity. One cannot measure the mutual entropy, one can only measure its change, assuming one can measure a microstate. Thermodynamics is restricted to the case where microstates cannot be distinguished, which means that only the marginal entropy, proportional to the thermodynamic entropy, can be measured, and, in a practical sense, always increases.