Natural units convention

Definition (and physical meaning)

The natural units convention in thermodynamics/statistical mechanics is the choice of units in which the Boltzmann constant is set to unity:

With this convention, temperature has the same units as energy, and the thermodynamic entropy is naturally treated as a dimensionless quantity (because it is really $S/k_B$ that appears as a logarithm).

Physical interpretation: the constant $k_B$ is not “removed” from physics; it is absorbed into the unit choice so that thermal energy scales are measured directly in temperature units (or vice versa).

Practical consequences

• Inverse temperature simplifies. The inverse temperature becomes

  $$\beta=\frac{1}{T}.$$

• Entropy is a pure number. Statements like “entropy is the logarithm of the number of accessible microstates” become literally true in formulas: $S=\ln \Omega$ (rather than $S=k_B\ln\Omega$). Restoring SI units is done by multiplying by $k_B$ at the end.

• Free energies and partition functions look cleaner. For example, the canonical relation

  $$F=-k_B T\ln Z$$

  becomes simply $F=-T\ln Z$.

Conventions you must keep consistent

Natural units are usually paired with a specific logarithm convention, often natural logarithms so that entropies are in “nats.” If you instead use $\log_2$, then restoring thermodynamic units requires an extra factor of $\ln 2$, consistent with the chosen entropy normalization convention .

Restoring dimensions (rule of thumb)

If a formula in natural units contains $T$ multiplying an entropy-like quantity, reinsert $k_B$ so that $k_B T$ carries units of energy and $S$ carries units of J/K in conventional thermodynamics.