Scale-free ideal gas

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 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

: 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 logarithmic "volume" of the system, \overline{w}=\langle v/k \rangle is the mean relative growth and \sigma_w is the standard deviation of the relative growth. The entropy equation of state is

: S=N\kappa\left{\ln\frac{\Omega}{N}\frac{\sqrt{2\pi}\sigma_w}{H'}+\frac{3}{2}\right},

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