Classical Resolution of the Gibbs Paradox from the Equal Probability Principle: An Informational Perspective

The Gibbs paradox is a conventional paradox in classical statistical mechanics, typically resolved by invoking quantum indistinguishability through the 1/N! correction. In this letter, we present a resolution within classical ensemble theory, which relies solely on the equal probability principle and does not invoke the 1/N! correction. Our resolution can be naturally interpretated from a purely informational perspective, where the Gibbs entropy is explicitly regarded as the Shannon entropy, quantifying ignorance rather than disorder. From this informational perspective, we also clarify the connection between information and extractable work in the gas mixing processes. Our work opens a new avenue to reconsider the role of information in statistical mechanics.

💡 Research Summary

The paper tackles the long‑standing Gibbs paradox without invoking the quantum‑mechanical 1/N! correction. Instead, it stays entirely within classical statistical mechanics, relying only on the equal‑probability principle that underlies both microcanonical and canonical ensembles. The authors first review the paradox: when two identical ideal gases, each containing N particles in volume V, are separated by a partition, the classical entropy formula S_id(N,V,T) (derived from the canonical distribution ρ∝e^{-βH}) is non‑extensive. Removing the partition yields an apparent entropy increase ΔS = 2Nk ln 2, which contradicts the intuition that nothing thermodynamically changes for identical gases. Traditional resolutions introduce a factor 1/N! in the partition function to enforce particle indistinguishability, thereby restoring extensivity.

The novel resolution proceeds by calculating the total partition function directly from the equal‑probability principle, without any ad‑hoc corrections. For two different gases (type A and B) the phase space factorizes, giving a total entropy S(1)_t = 2 S_id(N,V,T) before mixing and S(2)_t = S_id(2N,2V,T) after mixing, reproducing the conventional mixing entropy ΔS = 2Nk ln 2. The key insight appears when the gases are identical. Classical particles are distinguishable, but before the wall is removed the observer does not know which particle resides on which side. Consequently the full 6N‑dimensional phase space splits into (2N)!/(N!)² disconnected regions, each corresponding to a distinct assignment of particles to the left or right half. The equal‑probability principle requires each region to be equally weighted, which introduces an extra combinatorial factor into the partition function. The resulting entropy before mixing is

 S(1)_t = 2 S_id(N,V,T) + 2Nk ln 2.

After the wall is removed the combinatorial restriction disappears, and the entropy becomes S(2)_t = S_id(2N,2V,T). Using Stirling’s approximation one finds S(2)_t – S(1)_t = 0. Thus, the apparent paradox disappears: the extra term 2Nk ln 2 represents the information loss about particle assignments when the wall is removed, and this loss exactly cancels the increase that would otherwise be attributed to mixing.

The authors then adopt an informational perspective, treating the Gibbs entropy as a Shannon entropy. They decompose total ignorance into (1) uncertainty about which particles belong to which subsystem (S_d) and (2) uncertainty about the kinetic states of each subsystem (S_A, S_B). For different gases, S_d = 0, so mixing adds only kinetic‑state uncertainty, yielding the familiar ΔS = 2Nk ln 2. For identical gases, S_d = k ln