The scale-free ideal gas (SFIG) is a physical model assuming a collection of non-interacting elements with an stochastic proportional growth. It is the scale-invariant version of an ideal gas. Some cases of city-population, electoral results and cites to scientific journals can be approximately considered scale-free ideal gases.
In a one-dimensional discrete model with size-parameter k, where k1 and kM are the minimum and maximum allowed sizes respectively, and v = dk/dt is the growth, the bulk probability density function F(k, v) of a scale-free ideal gas follows
where N is the total number of elements, Ω = ln k1/kM is the logaritmic "volume" of the system,
where
Zipf's law may emerge in the external limits of the density since it is a special regime of scale-free ideal gases.