Interactive simulations & visualizations
Visualizing the beauty in physics and mathematics
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Two-dimensional Boltzmann gas
🔧 boltzmann_gas_2d.js is based on Three.js
Three-dimensional Boltzmann gas
🔧 boltzmann_gas_3d.js is based on Three.js
⭐ Also available as boltzmann_gas.py
Maxwell-Boltzmann distribution
Maxwell-Boltzmann distribution: 2D vis à vis 3D
Maxwell-Boltzmann velocity distributions for $\vec{v}$
The velocity distribution for $f(\vec{v}) , d^n\vec{v}$ is given by
3D case
\[f(\vec{v}) d^3\vec{v} = \left(\frac{m}{2 \pi k_B T}\right)^{3/2} \exp\Big(-\frac{m v^2}{2 k_B T}\Big) d^3\vec{v}\]2D case
\[f(\vec{v}) d^2\vec{v} = \left(\frac{m}{2 \pi k_B T}\right) \exp\Big(-\frac{m v^2}{2 k_B T}\Big) d^2\vec{v}\]- We have $v^2 = v_x^2 + v_y^2 + v_z^2$ in 3 dimensions and $v^2 = v_x^2 + v_y^2$ in 2 dimensions.
- Normalisation because the integral volume $d^n\vec{v}$ depends on the dimension.
Maxwell-Boltzmann velocity distributions for $|\vec{v}|$
In the graphs we reduce to a radial distribution, so we take the “surface of the cirkel/sphere”:
\[d^n\vec{v} = v^{n-1} dv , d\Omega_n\]where $d\Omega_n$ denotes the angular part. This amounts to an additional factor $v$ for 2D, $v^2$ for 3D.
3D case
\[f(v) dv = 4\pi \left(\frac{m}{2 \pi k_B T}\right)^{3/2} v^2 \exp\Big(-\frac{m v^2}{2 k_B T}\Big) dv\]2D case
\[f(v) dv = \frac{m}{k_B T} , v , \exp\Big(-\frac{m v^2}{2 k_B T}\Big) dv\]- Factor $v^{n-1}$ originates from the velocities with the same speeds (surface of a cirkel/sphere).
- This is what is used in the code, e.g.
value = (v / T) * exp(-v*v / (2 * T))in 2D.
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