Entropy normalization convention

Definition and what “normalization” fixes

The thermodynamic entropy $S$ is a state function determined (macroscopically) by the second law up to choices that do not affect measurable entropy changes. An “entropy normalization” convention fixes:

1. Units / scale factor (whether $S$ is dimensional or scaled by a constant), and
2. Additive constant (the reference point for “absolute” entropy).

Convention used in this blog

• Logarithm choice: we use the natural logarithm as stated in the logarithm convention . Changing the log base rescales entropy by a constant factor.

• Boltzmann constant explicit by default: entropy is treated as a physical quantity with units of energy per temperature, keeping $k_B$ explicit unless natural units are declared.

• Statistical-mechanical normalization (discrete states): for microstates with probabilities $p_i$, we take the Gibbs/Shannon-form normalization

  $$S = -k_B \sum_i p_i \ln p_i.$$

  The dimensionless quantity $-\sum_i p_i \ln p_i$ is the Shannon entropy (in nats under our log choice), and the thermodynamic entropy is obtained by multiplying by $k_B$.

  In the microcanonical special case ($p_i=1/\Omega$ on $\Omega$ accessible states), this reduces to the Boltzmann form $S=k_B \ln \Omega$.

• Absolute reference (third law): to fix the additive constant, we adopt the third-law convention: for a perfect crystal with a nondegenerate ground state, $S\to 0$ as $T\to 0$. More generally, a residual ground-state degeneracy $g$ corresponds to $S(T\to 0)=k_B \ln g$.

Key implications and useful checks

• Only differences matter in most thermodynamics: statements like the Clausius inequality constrain $\Delta S$ and do not depend on the absolute constant.

• Dimensionless entropy in natural units: if $k_B=1$ units are used, $S$ becomes dimensionless and temperature has energy units; the inverse temperature $\beta$ is then simply $1/T$.

• Connection to relative entropy: with this normalization, entropy differences and free-energy inequalities can often be expressed using Kullback–Leibler divergence (a dimensionless measure), with factors of $k_B$ restoring thermodynamic units.