Scale-free ideal gas

Short description: Ideal gas with no physical scale

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

[math]\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}, }[/math]

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

[math]\displaystyle{ S=N\kappa\left\\{\ln\frac{\Omega}{N}\frac{\sqrt{2\pi}\sigma_w}{H'}+\frac{3}{2}\right\\}, }[/math]

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

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

## References

1. ↑ Hernando, A.; Vesperinas, C.; Plastino, A. (2010). "Fisher information and the thermodynamics of scale-invariant systems". Physica A: Statistical Mechanics and Its Applications 389 (3): 490–498. doi:10.1016/j.physa.2009.09.054. Bibcode: 2010PhyA..389..490H.
2. ↑ Hernando, A.; Puigdomènech, D.; Villuendas, D.; Vesperinas, C.; Plastino, A. (2009). "Zipf's law from a Fisher variational-principle". Physics Letters A 374 (1): 18–21. doi:10.1016/j.physleta.2009.10.027. Bibcode: 2009PhLA..374...18H.

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