Maxwell-Boltzmann Distribution

The simulation shows a hard disk gas: particles with finite radius undergoing elastic collisions. We can see how an initial velocity distribution converges toward the Maxwell-Boltzmann equilibrium. The simulation uses dimensionless units with k_B = 1 and m = 1.

The Maxwell-Boltzmann distribution: In 2D, the equilibrium distribution of particle speeds is described by: f(v) = (m/k_BT) · v · exp(−mv²/2k_BT), where v = |v| denotes the speed (magnitude of velocity). This two-dimensional version of the Maxwell-Boltzmann distribution is also called Rayleigh distribution. In our dimensionless units, it simplifies to f(v) = (v/T) · exp(−v²/2T). The distribution peaks at v_peak = √T and has mean speed ⟨v⟩ = √(πT/2).

Entropy and H-functional: The entropy graph displays S(t) = −H(t) for Boltzmann's H-functional: H = Σ_i p_i ln(p_i), where p_i is the fraction of particles with speed in bin i. The famous H-theorem states that H(t) decreases monotonically toward its minimum value (maximum entropy), which is attained for a Maxwell-Boltzmann distribution. Small fluctuations around S_max are expected for finite particle numbers.

Boltzmann's Explanation: While chaotic collisions "drive" the system toward equilibrium, the explanation is ultimately statistical. For the vast majority of possible microscopic configurations (at constant total energy), the distribution of particle speeds is close to the Maxwell-Boltzmann form.

"The ensuing, most likely state, which we call that of the Maxwellian velocity distribution, since it was Maxwell who first found the mathematical expression in a special case, is not an outstanding singular state, opposite to which there are infinitely many more non-Maxwellian velocity distributions, but it is, on the contrary, distinguished by the fact that by far the largest number of possible states have the characteristic properties of the Maxwellian distribution, and that compared to this number, the amount of possible velocity distributions that deviate significantly from Maxwell's is vanishingly small."

— Ludwig Boltzmann, Lectures on Gas Theory (1896)