Thermal Velocity

Particles in a plasma follow a Maxwell-Boltzmann distribution of speeds. Several characteristic speeds describe this distribution.

Thermal Velocity

The thermal velocity (root-mean-square of one velocity component):

$v_{th} = \sqrt{\frac{k_B T}{m}}$

Most Probable Speed

The speed at the peak of the Maxwell-Boltzmann distribution:

$v_p = \sqrt{\frac{2 k_B T}{m}}$

Mean Speed

The average speed of particles:

$\bar{v} = \sqrt{\frac{8 k_B T}{\pi m}}$

Where $k_B = 1.381 \times 10^{-23}$ J/K, and the masses are:

• Electron: $m_e = 9.109 \times 10^{-31}$ kg
• Proton: $m_p = 1.673 \times 10^{-27}$ kg

Comparing Electrons and Ions

Since $v_{th} \propto 1/\sqrt{m}$, electrons are much faster than ions at the same temperature:

$\frac{v_{th,e}}{v_{th,i}} = \sqrt{\frac{m_i}{m_e}} \approx 43 \text{ (for hydrogen)}$

Physical Significance

• Thermal velocities determine transport properties (diffusion, conductivity)
• Particles with $v > v_{th}$ can escape confinement (loss cone in mirrors)
• Comparing $v_{th}$ with wave phase velocities determines wave-particle interactions (Landau damping)

Implement the three thermal speed functions — all constants must live inside the function bodies.