Boltzmann constant

Definition (and physical meaning)

The Boltzmann constant $k_B$ is the proportionality constant that links thermal energy scales to temperature and converts between a dimensionless entropy (a logarithm of a count or probability) and the thermodynamic entropy measured in J/K.

In SI units,

with this numerical value fixed by the SI definition of the kelvin.

Physical interpretation: $k_B T$ sets the characteristic energy scale of thermal fluctuations at temperature $T$. Whenever you see $k_B T$ in a formula, it is marking the “typical” thermal energy available per microscopic degree of freedom.

Core appearances in thermodynamics and statistical mechanics

• Temperature as an energy scale. In equilibrium statistical mechanics, the inverse-temperature parameter is defined by

  $$\beta=\frac{1}{k_B T},$$

  where $\beta$ is the inverse temperature .

• Entropy as a logarithm times $k_B$. If $\Omega$ counts compatible microstates (in an appropriate coarse-grained sense), the thermodynamic entropy is often written as

  $$S = k_B \ln \Omega.$$

  More generally, $S/k_B$ is a natural dimensionless entropy, comparable in form to Shannon entropy when probabilities are used.

• Thermodynamic derivatives. Since $\partial S/\partial E = 1/T$, one can equivalently write

  $$\frac{\partial}{\partial E}\left(\frac{S}{k_B}\right)=\beta,$$

  connecting $k_B$ directly to the thermodynamic definition of temperature.

Conventions and unit choices

Many formulas simplify under the natural units convention where $k_B=1$. In that convention, temperature has units of energy, $\beta=1/T$, and entropy is treated as dimensionless. If you also adopt a fixed logarithm convention (often natural logarithms ), then restoring SI units amounts to re-inserting the appropriate factors of $k_B$.