# Maxwell-Boltzmann distribution

In thermodynamics, Maxwell-Boltzmann distribution refers to the Gaussian-shaped distribution of particle speeds of an ideal gas system that accords to the Boltzmann chaos assumption, at equilibrium, according to which velocities are uniquely distributed per each degree of temperature. The Maxwell-Boltzmann distribution is the centerpiece of the kinetic theory of gases.

History
The distribution originated in the 1857 paper “On the Nature of the Motion which we Call Heat” by German physicist Rudolf Clausius, in which the speeds of atoms were calculated.

In 1860, Scottish physicist James Maxwell published “On the Dynamical Theory of Gases”, showing how, through kinetic theory, to obtain physical properties of gases from the underlying distribution of velocities, which was thus an elaboration of Clausius’ theory, taking into account not simply the average speed of the atoms, but their distribution of speeds greater or smaller than the average.  Maxwell derived, on somewhat abstract and not entirely persuasive grounds, according to thermodynamics biographer David Lindley, a mathematical form for this distribution, in effect a graph of the typical speeds of atoms in a volume of gas at any given temperature.

In 1868, Austrian physicist Ludwig Boltzmann published a more convincing physical explanation for the formula Maxwell had derived. Boltzmann is said to have did this by analyzing what would happen to a volume of gas rising in the Earth’s gravitational field, meaning that pressure would decrease (or volume would increase) with height, according to which Boltzmann showed that Maxwell’s formula correctly predicted how the number of atoms or molecules with a particular energy would change.  Boltzmann followed this up with his 1872 derivation of his minimum theorem, later to be called an H-theorem, a function that quantified an approached to equilibrium at which point the Maxwell distribution of velocities would exist and that the negative value of this function, i.e. -H, was said to be a representative measure of the entropy of the gas.

Human distributions
In 1971, Australian mechanical engineer Roy Henderson monitored the movements of college students on a campus and children on a playground, finding that in both cases their movements fit the Maxwell-Boltzmann distribution, as reported in his high-cited paper “The Statistics of Crowd Fluids”. The abstract for Henderson’s article reads “the speed (velocity) distribution functions have been measured for three crowd fluids in the gaseous phase. Good agreement is obtained with Maxwell-Boltzmann theory except for a significant deviation near the frequency mode of each distribution. This is attributed to sexual inhomogeneity.” 

Henderson also noted a difference between the children and college students, according to Fisher, in that the children had much more energy, or rather kinetic energy, and consequently moved at much higher average speeds. 

References
1. Lindley, David. (2001). Boltzmann’s Atom: the Great Debate that Launched a Revolution in Physics (pgs. 17-18). The Free Press.
2. Henderson, LeRoy F. (1971). “The Statistics of Crowd Fluids”, Nature, 229: 381-83.
3. Fisher, Len. (2009). The Perfect Swarm: the Science of Complexity in Everyday Life (Maxwell-Boltzmann distribution of students, pg. 24). Basic Books.
4. Maxwell, James. (1859). “On the Dynamical Theory of Gases”, presented to the meeting of the British Association for the Advancement of Science, Sep.; Phil. Mag. 19:434-36, 1860; in: Scientific Letters, Vol. I. (pg. 616).