An interactive companion to the lecture notes by Sang Hoon Lee: one Boltzmann factor, \(e^{-E(s)/k_BT}\), and some careful counting of states give you the speeds of molecules, the partition function, and the entropy of a gas.
The probability of a speed is the probability of one velocity vector times the number of vectors with that speed. The first is a Boltzmann factor; the second is the area \(4\pi v^2\) of a sphere in velocity space.
Drag on the plot to move the nearer edge of the shaded band. The shaded area is the probability that a molecule's speed lies between \(v_1\) and \(v_2\).
\(v_\text{max} < \bar v < v_\text{rms}\) always, since \(2 < 8/\pi < 3\). The ratios never change; only the scale √(kBT/m) does.
Start 1,200 argon atoms with an artificial distribution of speeds and let them collide. Every collision conserves energy and momentum, but randomizes direction. Nothing in the rules mentions Boltzmann, yet the histogram settles onto the Maxwell curve for the temperature set by the total energy.
The atoms in their 3D box. Color runs from slow (blue) to fast (red). Drag either view to rotate both.
Velocity space in 3D. Points are densest at the origin, yet the most likely speed is the green shell at \(v_\text{max}\): its area \(4\pi v^2\) holds more velocity vectors.
Histogram of speeds against \(\mathcal{D}(v)\) at the measured temperature.
Collisions are counted per atom per second. Set them to zero with “Motion only along x” and the walls alone never thermalize the gas: reflections flip vx but keep every speed.
For two non-interacting, distinguishable particles every pair \((s_1,s_2)\) is a distinct system state, so \(Z_\text{total}=Z_1Z_2\). For identical particles, \((s_1,s_2)\) and \((s_2,s_1)\) are the same state, so the double sum counts nearly every state twice. Only the diagonal \(s_1=s_2\) is counted once, and it becomes a negligible sliver when there are many more single-particle states than particles.
Each cell is one term of \(\sum_{s_1}\sum_{s_2}\) for two particles. Hover or tap a cell to see its mirror image, the same state for identical particles. Magenta cells are \(s_1=s_2\).
“Not too dense” means M ≫ N. In a real gas M ≈ V/vQ is about 10⁷ per molecule, so Z1N/N! is essentially exact. When it fails, you need the boson and fermion counts of the next chapter.
A molecule in a one-dimensional box of length \(L\) has energies \(E_n = h^2n^2/8mL^2\). Measured in units of the quantum length \(\ell_Q = h/\sqrt{2\pi m k_BT}\), the Boltzmann factors depend on one number only:
Energy levels up to \(5k_BT\), with \(\psi_n=\sin(n\pi x/L)\) drawn on the first few. Fainter levels have smaller Boltzmann factors.
Bars: the terms of the sum. Green area: the integral, exactly \(L/\ell_Q\).
The sum and the integral differ by almost exactly ½, the half-bar the integral counts at \(n=0\). Poisson summation makes this precise: the next correction is \(2(L/\ell_Q)\,e^{-4\pi(L/\ell_Q)^2}\), which is why Pickover's \(\left(10^{-5}\sum e^{-n^2/10^{10}}\right)^2\) matches π to some 42 billion digits.
For \(N\) indistinguishable, non-interacting molecules, \(Z=\frac{1}{N!}\left(VZ_\text{int}/v_Q\right)^N\) with \(v_Q=\ell_Q^3\). Everything follows from \(F=-k_BT\ln Z\). For a monatomic gas (\(Z_\text{int}=1\)):
Sackur–Tetrode entropy per molecule versus temperature at the chosen pressure, for five noble gases. Below the dashed line \(V/N < v_Q\): molecules' wavefunctions overlap and Boltzmann statistics no longer applies.
Why the \(1/N!\) matters. Without it, \(S/Nk_B=\ln(V/v_Q)+\tfrac32\) depends on \(V\) rather than \(V/N\). Pull out a partition between two identical boxes of the same gas and that formula would add \(2Nk_B\ln 2\) of entropy, though nothing has changed. This is the Gibbs paradox; the \(N!\) restores extensivity.
Where \(\tfrac32Nk_BT\) comes from. \(Z_\text{tr}=V/v_Q\propto\beta^{-3/2}\), so \(-\partial\ln Z/\partial\beta\) gives \(\tfrac32 k_BT\) per molecule: the equipartition result, recovered here as the classical limit of the quantum sum.