Inside the box#

A litre of air holds something like 2.5 × 10²² molecules. No single one of them is hot or has a pressure: those belong to the crowd.

The simulation assumes no gas law. It moves a few hundred hard discs in a box, adds up the impulse they deliver to the walls, and averages the squares of their speeds. The dashed curve over the histogram is the Maxwell–Boltzmann distribution, and each rose flash marks a real momentum transfer to a wall.

Pressure is a count of impacts#

A wall in this box feels nothing continuous. It gets hit, then nothing, then hit again. Each hit is a molecule arriving with velocity component $v_\perp$, leaving with $-v_\perp$, and handing the wall

$$\Delta p = 2 m |v_\perp|$$

Pressure is that impulse per unit time, per unit length of wall. The simulation simply counts it: every $2m|v_\perp|$, divided by the elapsed time and the box's perimeter. That is the measured bar. The dashed line, $N k_B T/A$, is the prediction, and nothing in the model consults it.

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With pair collisions off, this gas sits at a well-defined 300 K, delivers essentially the right pressure, and every molecule in it moves at 422 m/s forever. So the Maxwell–Boltzmann distribution is something other than the definition of temperature: it is what molecular collisions do to a set of speeds, given a little time.

Temperature is a mean kinetic energy#

The temperature readout has one input, $\langle v^2 \rangle$, computed fresh every frame. In two dimensions each molecule has two independent directions of motion, and equipartition gives each $\tfrac12 k_B T$:

$$\tfrac12 m \langle v_x^2 \rangle = \tfrac12 m \langle v_y^2 \rangle = \tfrac12 k_B T, \qquad \kin{\tfrac12 m \langle v^2 \rangle} = k_B T$$

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What one temperature looks like#

The dashed curve on the histogram is the two-dimensional Maxwell–Boltzmann distribution, a prediction rather than a fit:

$$f(v) = \frac{v}{v_p^2}\, e^{-v^2 / 2v_p^2}, \qquad v_p = \sqrt{k_B T/m}$$

Its long tail pulls the three "typical" speeds apart. For nitrogen at 300 K: $v_p = 298$, $\langle v \rangle = 374$ and $v_{\text{rms}} = 422$ m/s. Pressure comes from $v^2$, so the gas law is built from the largest.

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What this leaves open#

You watched a temperature hold while the speeds spread, doubled a pressure three ways, and crowded a gas until it stopped being ideal. Every collision here conserves kinetic energy, which made temperature easy to define. Real nitrogen is two atoms on a bond that can rotate and stretch, with places to store energy these discs lack, which is why a real gas's heat capacity differs from this box's and changes with temperature. And the piston has been a fixed wall. Let it move during the run and energy crosses the boundary as work: that is where thermodynamic cycles begin.