Work Accounting for the Gibbs Mixing Paradox in Isolated Quantum Systems

We calculate the difference in work extracted from an isolated quantum system when observed by two distinct observers, each employing a different type of measurement. We apply this to the situation of the Gibbs mixing paradox, when one observer can distinguish two types of particles (e.g., red and blue), while the other cannot. We find that the difference in extracted work is proportional to the observational entropy difference, and it is similar to Landauer’s bound, N k T ln 2, minus some higher-order corrections that depend on heat capacity. N is the total number of particles, and T is the temperature computed from the observational entropy. We further demonstrate that, although an observer unable to distinguish between particles may associate different entropy values depending on their awareness of the two particle types, they will observe the same entropy increase in both scenarios once the system reaches thermal equilibrium. Consequently, this framework avoids any paradox.