# Scale-free ideal gas

The scale-free ideal gas (SFIG) is a physical model assuming a collection of non-interacting elements with an 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(kv) 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/kM 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 (NVT) 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. Bibcode:2010PhyA..389..490H. arXiv:. doi:10.1016/j.physa.2009.09.054.
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. Bibcode:2009PhLA..374...18H. arXiv:. doi:10.1016/j.physleta.2009.10.027.