Slide 1: Introduction

• Kinetic theory bridges microscopic particle behavior and macroscopic thermodynamics.
• Statistical mechanics governs velocity and energy distributions in ideal gases.

Slide 2: Maxwell-Boltzmann Speed Formula (LaTeX)

The probability density function for molecular speeds \(v\) in thermal equilibrium is given by:

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

• \(m\): molecular mass
• \(k_B\): Boltzmann constant
• \(T\): absolute temperature

Slide 3: Cartesian Component Factorization (LaTeX)

Because velocity components are orthogonal and independent, the joint probability distribution factorizes into three independent Gaussians:

\[f(v_x, v_y, v_z) = f(v_x)f(v_y)f(v_z) = \left(\frac{m}{2\pi k_B T}\right)^{3/2} \exp\left(-\frac{m(v_x^2+v_y^2+v_z^2)}{2k_B T}\right)\]

• Each component \(v_i \sim \mathcal{N}\left(0, \frac{k_B T}{m}\right)\)

Slide 4: Speed Distribution (Ggplot 1)

Slide 5: Temperature Sensitivity (Ggplot 2)

Slide 6: 3D Phase Space Simulation (Plotly)

Slide 7: R Code Implementation

Here is the code used to generate the 3D phase-space simulation above:

set.seed(42)
df3d <- data.frame(
  vx = rnorm(250, 0, 10),
  vy = rnorm(250, 0, 10),
  vz = rnorm(250, 0, 10)
)
plot_ly(df3d, x = ~vx, y = ~vy, z = ~vz, 
        type = 'scatter3d', mode = 'markers',
        marker = list(size = 3, color = ~vz, colorscale = 'Viridis', opacity = 0.7))

Slide 8: Conclusion & Summary

• Micro vs Macro: Statistical mechanics successfully bridges individual molecular kinetic energy to collective thermodynamic behavior.
• Dimensionality: 3D velocity space manifests as an isotropic Gaussian cloud, whereas scalar speed curves skew right due to phase-space volume expansion (\(v^2\)).