Classical ideal gas and the Poisson distribution

A strange noise, as if a whole regiment sneezed simultaneously. 

Let us try to understand deeper the nature of the order parameter. As usually, we start with a gas as a simplest one-component system. An important role in theoretical physics belongs to a model of classical ideal gas in which molecules (particles) obey the laws of Newtonian mechanics and do not interact with each other. 

Consider physically small volume v. Due to discreteness of the matter distribution in space a number of particles Ny in a given volume is a random variable Ny = 0,1,2 — However, on the average each volume contains Ny) = nv particles. Define now microscopic, local density of the particle 

It is useful to find a quantity that could serve us as a measure of these density fluctuations. Its simplest characteristic is the dispersion of a number of particles N in some volume V i.e., (N ) — N). The distinctive feature of the classical ideal gas is a simple relation between the dispersion and macroscopic density N ) - N) — N) = nV. Moreover, all other fluctuation characteristics of the ideal gas, related to the quantity (iV ), could also be expressed through (N) or density n. Therefore, in the model of ideal gas the density n is the only parameter characterizing the fluctuation spectmm. Such the particle distribution is called the Poisson distribution. It could be easily generalized for the many-component system, e.g., a mixture of two ideal gases. Each component is characterized here by its density, ua and ne density fluctuations of different components are statistically independent, (Nfi,Ns) = Na) Nb). 

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