Table 2 Summary of results.

From: Mixing indistinguishable systems leads to a quantum Gibbs paradox

  Quantum Classical Quantum Quantum Classical
Limit (no limit) (no limit) (dn) (dn 1) (dn 1)
ΔSinfo \(2{\mathrm{ln}}\,\left(\begin{array}{c}n\,+\,d\,-\,1\\ n\end{array}\right)\,\\ -\,2{\mathrm{ln}}\,\left(\begin{array}{c}n\,+\,d/2\,-\,1\\ n\end{array}\right)\) \(2{\mathrm{ln}}\,\left(\begin{array}{c}n\,+\,d\,-\,1\\ n\end{array}\right)\,\\ -\,2{\mathrm{ln}}\,\left(\begin{array}{c}n\,+\,d/2\,-\,1\\ n\end{array}\right)\) \(\approx\!2n\,{\mathrm{ln}}\,2\) \(\approx \!2n\,{\mathrm{ln}}\,2\)
ΔSigno \({\sum }_{J}{p}_{J}{\mathrm{ln}}\,{d}_{J}^{B}\,\\ -\,2{\mathrm{ln}}\,\left(\begin{array}{c}n\,+\,d/2\,-\,1\\ n\end{array}\right)\) \({\mathrm{ln}}\,\left(\begin{array}{c}2n\,+\,d\,-\,1\\ 2n\end{array}\right)\,\\ -\,2{\mathrm{ln}}\,\left(\begin{array}{c}n\,+\,d/2\,-\,1\\ n\end{array}\right)\) \(\approx\!{{\Delta }}{S}_{\text{info}}\,-\,H({\bf{p}})\) \(\approx\!2n\,{\mathrm{ln}}\,2\) ≈ 0
  1. Entropy changes ΔSinfo, ΔSigno for the informed and ignorant observers and their limits are expressed for bosons with n = m. For fermions, replace the dimension of the symmetric subspace \(\left(\begin{array}{c}n\,+\,d\,-\,1\\ n\end{array}\right)\) with that of the antisymmetric one \(\left(\begin{array}{c}d\\ n\end{array}\right)\) and \({d}_{J}^{B}\) by \({d}_{J}^{F}\) (both of which are defined in Eq. (13)).