Residual entropy and the scope of the third-law postulates

Residual entropy here means a finite entropy limit at low temperature under a model's specified entropy convention. In a finite canonical model, let a finite set of microstates have energies $E_i$ in joules, assign $p_i(T)\propto \exp [-E_i\mathrm{/}(k_{\mathrm{B}}T)]$ for $T>0$, and use the Gibbs entropy $-k_{\mathrm{B}}{\sum }_i{p}_i(T)\log p_i(T)$. If the ground energy has finite degeneracy $g$ and there is a positive gap to all excited states, this model-specific entropy tends to $k_{\mathrm{B}}\log g$; Residual entropy in a finite canonical model ↗ supplies the full finite-state calculation. Here $g$ is the number of ground states and $\log$ is the natural logarithm of The natural logarithm as the inverse of the exponential function.

This formula belongs to that specified microscopic preparation and entropy convention. It does not identify every Gibbs entropy with the macroscopic $S_c$, and it does not establish a universal value $S_c(0)=0$. The uniform Nernst postulate Uniform Nernst heat-theorem postulate on a compact family constrains differences across its declared family, not the common entropy offset: adding one fixed constant to every $S_K(t,a)$ leaves every displayed difference unchanged. Thus that postulate alone neither selects an absolute zero nor forces a unique ground state. The Nernst heat theorem, zero-entropy convention for a unique ground state, and operational unattainability remain distinct model statements.