Third law of thermodynamics

At zero temperature the system can only be in its ground state. If the ground state is not degenerate, then $p_0 = 1$ and we have $S = 0$. If instead the ground state has degeneracy $g_0$, then we have $g_0$ terms in the sum , all identical to $1/g_0 \ln g_0$, and so the entropy is

$$S(0) = k_{\rm B}\ln g_0.$$ (3.61)

The condition $S(0) = 0$, or in its weaker form , is normally referred to as the third law of thermodynamics. Note that since the entropy is proportional to the number of particles $N$, for a macroscopic system its zero temperature value is always negligible compared to any finite temperature value (unless the degeneracy of the ground state $g_0$ is of the type $n^N$, with $n$ a positive integer).