- References
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 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, 
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, Ω,σw).
References
2. ^{{cite journal |first=A. |last=Hernando |first2=D. |last2=Puigdomènech |first3=D. |last3=Villuendas |first4=C. |last4=Vesperinas |first5=A. |last5=Plastino |title=Zipf's law from a Fisher variational-principle |journal=Physics Letters A |volume=374 |issue=1 |year=2009 |pages=18–21 |doi=10.1016/j.physleta.2009.10.027 |arxiv = 0908.0501 |bibcode = 2009PhLA..374...18H }}
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