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The study of large, complicated systems employing statistics and probability theory to extract average properties and to provide a connection between mechanics and thermodynamics.

Statistical mechanics (SM) is a branch of physics that aims to predict the properties of large, complicated systems by employing the mathematical framework of statistics and probability theory.

For typical microscopic systems, the SM approach is the only one possible. Indeed, a typical macroscopic body contains several molecules of the order of Avogadro's number, $$N_A \approx 6 \cdot 10^{23}$$. If we wanted to predict the trajectory of each particle of a system of $$N$$ particles exactly, we would need to solve $$3 \cdot N \approx 10^{24}$$ coupled equations, which is infeasible even for modern computers. Moreover, even if a computer was able to solve such a large number of equations in a short time, we would need to write the $$6 \cdot N \approx 10^{24}$$ initial conditions, which is definitely infeasible. Therefore, we renounce a complete knowledge of the system and try to get an average knowledge by applying the tools of statistics and probability theory.

SM was pioneered in the late 1800s / early 1900s by the works of Maxwell, Boltzmann, and Gibbs.

Maxwell formulated the first-ever statistical law in physics, the Maxwell distribution of molecular velocities. Boltzmann developed Maxwell's ideas further, investigating the kinetics of gases, the link between thermodynamics and mechanics, and the origins of macroscopic irreversibility. Gibbs, who coined the term "statistical mechanics", gave a rigorous and coherent formulation of SM, introducing concepts such as the phase space and the statistical ensemble.