The scale-free ideal gas (SFIG) is a physical model assuming a collection of non-interacting elements with an stochastic

Stochastic

Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...

 proportional growth. It is the scale-invariant

Scale invariance

In physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...

 version of an ideal gas

Ideal gas

An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

. Some cases of city-population, electoral results and cites to scientific journals can be approximately considered scale-free ideal gases.

If k is the size of the elements, being k₁ and k_M the minimum and maximum allowing sizes respectively, and v = dk/dt is the growth, the bulk probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

 F(k, v) of a scale-free ideal gas follows

where N is the total number of elements, Ω = ln k₁/k_M is the logaritmic "volume" of the system, is the mean relative growth and is the standard deviation of the relative growth. The entropy

Entropy

Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

 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,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.

The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.