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.^[1]

In a one-dimensional discrete model with size-parameter k, where k₁ and k_M 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

${\displaystyle F(k,v)={\frac {N}{\Omega k^{2}}}{\frac {\exp \left[-(v/k-{\overline {w}})^{2}/2\sigma _{w}^{2}\right]}{{\sqrt {2\pi }}\sigma _{w}}},}$

where N is the total number of elements, Ω = ln k₁/k_M is the logaritmic "volume" of the system, ${\displaystyle {\overline {w}}=\langle v/k\rangle }$ is the mean relative growth and ${\displaystyle \sigma _{w}}$ is the standard deviation of the relative growth. The entropy equation of state is

${\displaystyle S=N\kappa \left\{\ln {\frac {\Omega }{N}}{\frac {{\sqrt {2\pi }}\sigma _{w}}{H'}}+{\frac {3}{2}}\right\},}$

where ${\displaystyle \kappa }$ is a constant that accounts for dimensionality and ${\displaystyle H'=1/M\Delta \tau }$ is the elementary volume in phase space, with ${\displaystyle \Delta \tau }$ 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, Ω,σ_w).

Zipf's law may emerge in the external limits of the density since it is a special regime of scale-free ideal gases.^[2]