Classical statistical mechanics

The Maxwell–Boltzmann distribution

For a classical ideal gas at equilibrium, Maxwell–Boltzmann statistics gives molecular velocities; the speed distribution includes the density-of-states factor.

f(v)=4\pi\left(\frac{m}{2\pi k_B T}\right)^{3/2}v^2e^{-mv^2/(2k_B T)}

Definition: Quantities and meaning

The three-dimensional velocity-vector distribution is Gaussian in each component; multiplying by the spherical-shell factor 4πv² gives the speed density f(v). The most probable, mean, and rms speeds are √(2kBT/m), √(8kBT/πm), and √(3kBT/m). The model assumes a dilute gas, thermal equilibrium, and the classical regime.

Phân bố tốc độ và nhiệt độ

Quantitative relation

v_{\rm mp}=\sqrt{2k_BT/m}, \quad v_{\rm rms}=\sqrt{3k_BT/m}

Example: Worked example

At the same temperature, replace molecules of mass m with molecules of mass 4m. How does rms speed change?

Solution

Since v_rms=√(3kBT/m), quadrupling the mass halves the rms speed.

Example: Example: estimating nitrogen molecule speeds

At T=300 K, take the mass of an N₂ molecule as m≈4.65×10⁻²⁶ kg. The most probable speed is v_mp=√(2kBT/m)≈422 m·s⁻¹, while v_rms=√(3kBT/m)≈517 m·s⁻¹. They differ because the speed distribution is skewed: its high-speed tail raises the rms speed above the most probable speed. Ludwig Boltzmann helped develop the statistical view of gases at equilibrium.

The velocity-vector distribution is a probability density in three-dimensional velocity space, whereas f(v)dv is the probability that the speed lies between v and v+dv. The v² factor makes the speed density vanish at v=0 even though each component distribution peaks at zero. Raising temperature broadens the curve and moves its peak to higher speed; increasing molecular mass at fixed T shifts it lower. This is a classical equilibrium law and does not describe quantum degeneracy effects. Because f(v) is normalized, its area from zero to infinity is one, and about Nf(v)dv molecules occupy a narrow speed interval. Mean and most-probable speeds are distinct statistics and should not be interchanged. Measuring many molecules reproduces the distribution even though each molecule’s trajectory changes continually through collisions. Velocity statistics connect molecular motion to measurable pressure and heat transport. This is a foundation of kinetic theory.

Quick check

At fixed T, if molecular mass quadruples, how does rms speed change?

Which factor in the Maxwell speed distribution comes from the 3D velocity-space density of states?

References

Charles Kittel and Herbert Kroemer (1980). Thermal Physics
L. D. Landau and E. M. Lifshitz (1980). Statistical Physics