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The scale-free ideal gas 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 k1 and kM are the minimum and maximum allowed sizes respectively, and v = dk/dt is the growth, the bulk probability density function F 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, w ¯ = ⟨ v / k ⟩ {\displaystyle {\overline {w}}=\langle v/k\rangle } is the mean relative growth and σ w {\displaystyle \sigma _{w}} is the standard deviation of the relative growth. The entropy equation of state is
where κ {\displaystyle \kappa } is a constant that accounts for dimensionality and H ′ = 1 / M Δ τ {\displaystyle H'=1/M\Delta \tau } is the elementary volume in phase space, with Δ τ {\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 by.