Inverse temperature β

Definition (and physical meaning)

The inverse temperature $\beta$ is defined by

where $T$ is temperature and $k_B$ is the Boltzmann constant .

Physical interpretation: $\beta$ measures “how costly energy is” in equilibrium. Large $\beta$ corresponds to low temperature, meaning high-energy configurations are strongly suppressed; small $\beta$ corresponds to high temperature, meaning energy differences are comparatively less important.

Thermodynamic meaning

From the thermodynamic definition of temperature in the entropy representation,

it follows immediately that

Thus $\beta$ is the derivative of the dimensionless entropy $S/k_B$ with respect to internal energy at fixed volume and particle number .

Equality of $\beta$ across subsystems is the equilibrium condition for energy exchange, i.e. thermal equilibrium in the zeroth law sense.

Canonical-ensemble role

In the canonical ensemble convention , equilibrium weights take the form

where $E_i$ are microstate energies and $Z(\beta)$ is the partition function.

The Helmholtz free energy can be expressed in terms of $Z$ by

Writing thermodynamic relations in terms of $\beta$ often makes the underlying convex/dual structure explicit; for example, passing between entropy and free energy is a Legendre transform between conjugate variables (energy versus temperature/inverse temperature).

Common convention: natural units

Under the natural units convention with $k_B=1$, the definition reduces to $\beta=1/T$, and entropies are treated as dimensionless.