Energy density

The energy density is the internal energy per volume . It is an intensive “per-volume” form of energy that is especially natural in continuum and local-equilibrium settings.

For a homogeneous system with internal energy $U$ and volume $V$, the energy density is

Equivalently, $U = uV$ in a homogeneous equilibrium state.

When the system is not homogeneous, one often considers a local field $u(\mathbf{r})$ whose volume integral gives the total energy.

Physical interpretation

Energy density measures how much internal energy is stored in a unit volume of material. It is a specific quantity (per volume), and—like other densities—remains unchanged if you scale the system size at fixed bulk state, consistent with the thermodynamic limit .

Key relations (one-component simple system)

Energy density becomes particularly transparent when paired with entropy density and number density . Define $s=S/V$ and $n=N/V$. Then extensivity of $U$ (via the Euler relation ) implies the per-volume Euler form

with $T$, $p$, and $\mu$ the conjugate intensive variables (temperature , pressure , chemical potential ). Rearranging gives a useful identity for pressure in density variables:

Moreover, treating $u$ as a function of $(s,n)$ gives the differential (“Gibbs”) form

which is the density version of the first law for homogeneous matter expressed in natural variables.