Entropy density

The entropy density is the thermodynamic entropy per volume . It is the natural “per-volume” entropy used in continuum thermodynamics and statistical mechanics.

For a homogeneous system with entropy $S$ and volume $V$, the entropy density is

Equivalently, $S = sV$ in a homogeneous equilibrium state.

In local descriptions (e.g., local equilibrium), one may use a field $s(\mathbf{r})$ whose integral over the system volume gives the total entropy.

Physical interpretation

Entropy density tells you how much entropy is carried by each unit volume of matter. Like number density and energy density , it is a specific quantity (per volume) and thus typically intensive for homogeneous systems.

It is also the natural quantity for discussing the second law in spatially extended systems, where entropy can flow and be produced.

Key relations (one-component simple system)

Let $u=U/V$ be the energy density and $n=N/V$ the number density . When $u$ is expressed as a function of $(s,n)$, the differential relation

implies the entropy-representation differential

In particular, the derivative of entropy density with respect to energy density at fixed $n$ is the reciprocal of temperature :

This is the density analogue of the standard thermodynamic identity connecting entropy and energy, and it underlies the use of inverse temperature (see inverse temperature ) once a convention for the Boltzmann constant is chosen.