Gibbs Paradox
In statistical mechanics, a semi-classical derivation of the entropy that does not take into account the indistinguishability of particles, yields an expression for the entropy which is not extensive (is not proportional to the amount of substance in question). This leads to a paradox known as the Gibbs paradox, after Josiah Willard Gibbs. The paradox allows for the entropy of closed systems to decrease, violating the second law of thermodynamics. It is possible, however, to take the perspective that it is merely the definition of entropy that is changed to ignore particle permutation (and thereby avert the paradox).
Read more about Gibbs Paradox: Illustration of The Problem, Calculating The Entropy of Ideal Gas, and Making It Extensive, The Mixing Paradox
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