Maxwell–Boltzmann distribution

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The Maxwell–Boltzmann distribution describes particle speeds in gases, where the particles do not constantly interact with each other but move freely between short collisions. It describes the probability of a particle's speed (the magnitude of its velocity vector) being near a given value as a function of the temperature of the system, the mass of the particle, and that speed value. This probability distribution is named after James Clerk Maxwell and Ludwig Boltzmann.

The Maxwell–Boltzmann distribution is usually thought of as the distribution for molecular speeds, but it can also refer to the distribution for velocities, momenta, and magnitude of the momenta of the molecules, each of which will have a different probability distribution function, all of which are related. Unless otherwise stated, this article will use "Maxwell–Boltzmann distribution" to refer to the distribution of speed. This distribution can be thought of as the magnitude of a 3-dimensional vector whose components are independent and normally distributed with mean 0 and standard deviation a. If Xi are distributed as $X \sim N(0, a^2)$, then

is distributed as a Maxwell–Boltzmann distribution with parameter a. Apart from the scale parameter a, the distribution is identical to the chi distribution with 3 degrees of freedom.

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Physical applications of the Maxwell–Boltzmann distribution

The Maxwell–Boltzmann distribution applies to ideal gases close to thermodynamic equilibrium, negligible quantum effects, and non-relativistic speeds. It forms the basis of the kinetic theory of gases, which explains many fundamental gas properties, including pressure and diffusion.