The scale-free ideal gas (SFIG) is a physical model assuming a collection of non-interacting elements with a 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 k<sub>1</sub> and k<sub>M</sub> 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, é = ln k<sub>1</sub>/k<sub>M</sub> is the logarithmic "volume" of the system, is the mean relative growth and is the standard deviation of the relative growth. The entropy equation of state is
where is a constant that accounts for dimensionality and is the elementary volume in phase space, with the elementary time and M the total number of allowed discrete sizes. This expression has the same form as the one-dimensional ideal gas, changing the thermodynamical variables (N, V, T) by (N, é,ÃÂ<sub>w</sub>).
Zipf's law may emerge in the external limits of the density since it is a special regime of scale-free ideal gases.