The Maxwell-Boltzmann distribution is a fundamental probability distribution that characterizes the speeds of particles in a gas.

This distribution is named after the physicists James Clerk Maxwell and Ludwig Boltzmann, who independently formulated it in the 19th century. It describes the likelihood of finding a particle in a gas with a specific speed. The distribution is continuous and is influenced by both the temperature of the gas and the mass of the particles involved.

The mathematical expression for the Maxwell-Boltzmann distribution is given by:

In this equation, $f(v)$ represents the probability density function, $m$ denotes the mass of the particle, $k$ is the Boltzmann constant, $T$ is the temperature of the gas, and $v$ is the speed of the particle.

The distribution features a peak at a specific speed, which varies with the temperature of the gas. As the temperature increases, this peak shifts to higher speeds. Additionally, the distribution exhibits a long tail at higher speeds, indicating a small but non-zero probability of encountering particles with very high velocities.

The Maxwell-Boltzmann distribution plays a crucial role in various fields of physics and chemistry, including the study of gases, plasmas, and liquids. Furthermore, it has practical applications in engineering, particularly in the design of gas turbines and rocket engines.