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

: $F(k,v)=\frac\frac,$

where ''N'' is the total number of elements, Ω = ln ''k''₁/''k''_''M'' is the logaritmic "volume" of the system, $\overline=\langle v/k \rangle$ is the mean relative growth and $\sigma_w$ is the standard deviation of the relative growth. The entropy equation of state is

: $S=N\kappa\left\,$

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.

References

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Information theory
Thermodynamics
Scale-invariant systems