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 "k"1 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
where "N" is the total number of elements, Ω = ln "k"1/"k""M" is the logaritmic "volume" of the system, formula_2 is the mean relative growth and formula_3 is the standard deviation of the relative growth. The entropy equation of state is
where formula_5 is a constant that accounts for dimensionality and formula_6 is the elementary volume in phase space, with formula_7 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.