The Maxwell-Boltzmann distribution describes the distribution of speeds of particles in a gas, assuming the gas is in thermal equilibrium and its molecules are moving randomly. In a three-dimensional space, the distribution of molecular speeds depends on the velocity components in the $x$, $y$, and $z$ directions.

To better understand the probability distribution function for the molecular speed $u$, consider the following:

1. Velocity Components in the Maxwell-Boltzmann Distribution:

The total speed $u$ of a molecule is related to its velocity components $u_x$, $u_y$, and $u_z$ by: $u = \sqrt{u_x^2 + u_y^2 + u_z^2}$ where $u_x$, $u_y$, and $u_z$ are the velocity components in the $x$, $y$, and $z$-directions, respectively.

Since the velocity components are independent, the total probability distribution function $F(u) \, du$ can be written as the product of the probability distributions for each velocity component: $F(u) \, du = f(u_x) \, du_x \, f(u_y) \, du_y \, f(u_z) \, du_z$ where $f(u_x)$, $f(u_y)$, and $f(u_z)$ are the probability densities for the velocity components in each direction.

2. Gaussian Nature of the Velocity Components:

For a homogeneous and isotropic gas, the velocity components $u_x$, $u_y$, and $u_z$ follow independent normal (Gaussian) distributions centered around zero. The probability density function for each velocity component can be written as: $f(u_i) = \left( \frac{m}{2\pi k_B T} \right)^{1/2} \exp \left( - \frac{m u_i^2}{2 k_B T} \right)$ where:

• $m$ is the mass of a gas molecule,
• $k_B$ is Boltzmann's constant,
• $T$ is the absolute temperature,
• $u_i$ is the velocity component (i.e., $u_x$, $u_y$, or $u_z$).

3. Maxwell-Boltzmann Speed Distribution:

To find the distribution of speeds (magnitudes of velocity) rather than individual velocity components, we need to integrate over the spherical symmetry of velocity space. The speed distribution is derived from the product of the Gaussian distributions for $u_x$, $u_y$, and $u_z$. The Maxwell-Boltzmann speed distribution is: $F(u) = 4\pi \left( \frac{m}{2\pi k_B T} \right)^{3/2} u^2 \exp \left( - \frac{m u^2}{2 k_B T} \right)$ Here:

• $u$ is the speed of the molecule (the magnitude of the velocity vector),
• The factor $4\pi u^2$ accounts for the spherical shell in velocity space.

Summary:

• The Maxwell-Boltzmann distribution for molecular speed $u$ is the product of the distributions for the velocity components $u_x$, $u_y$, and $u_z$.
• Each velocity component follows a Gaussian distribution, and the combined distribution for the magnitude of speed is the Maxwell-Boltzmann distribution.

This function gives the probability of finding a molecule with speed $u$ at a given temperature.

Would you like further details or examples? Here are some related questions:

1. How do you derive the most probable speed from the Maxwell-Boltzmann distribution?
2. What is the difference between the most probable, average, and root-mean-square speed?
3. How does temperature affect the shape of the Maxwell-Boltzmann distribution?
4. Can we apply the Maxwell-Boltzmann distribution to liquids and solids?
5. How does quantum mechanics influence the classical Maxwell-Boltzmann distribution?

Tip: The area under the entire Maxwell-Boltzmann distribution curve represents the total probability, which must equal 1.