# Ideal Gas Kinetics

> Temperature is average kinetic energy and pressure is momentum delivered to the walls. Bounce point particles in a box and PV = NkT falls out.

- Category: Physics › Thermodynamics
- Level: core
- Tags: kinetic-theory, pressure, temperature, particles, maxwell-boltzmann
- Updated: 2026-11-01
- URL: https://gdcs.coody.app/physics/thermodynamics/ideal-gas

## What it is

An ideal gas is a crowd of tiny particles that never interact except by bouncing elastically off the container. Every bounce reverses one velocity component and hands the wall a momentum of `2m|v⊥|`. Add up those impulses per second per square metre and you get **pressure**.

**Temperature** is the average kinetic energy per particle: `½m⟨v²⟩ = (3/2)kT`. Double T and the root-mean-square speed grows by √2, particles hit walls more often and harder, and pressure doubles. Halve the volume and the same particles hit each wall twice as often. That is the whole of `PV = NkT`, recovered from mechanics.

Velocities in equilibrium follow the Maxwell–Boltzmann distribution: each component is a Gaussian with variance `kT/m`. Seeding a sim with Gaussian velocities from a proper PRNG gives the right spread of speeds from the first frame.

For games this is the template for every cheap [particle system](https://gdcs.coody.app/systems/patterns/particle-system) with box collisions: integrate positions at a [fixed timestep](https://gdcs.coody.app/systems/simulation/fixed-timestep), reflect on walls, and measure aggregate quantities instead of tracking every interaction. Real gases add collisions between particles, which needs [spatial partitioning](https://gdcs.coody.app/performance/world/spatial-partitioning).

## Key formulas

- **Ideal gas law:** $PV = N k_B T$
- **Temperature from kinetic energy:** $\tfrac{1}{2} m \langle v^2 \rangle = \tfrac{3}{2} k_B T$
- **RMS speed:** $v_{\text{rms}} = \sqrt{\frac{3 k_B T}{m}}$
- **Pressure from wall impulses:** $P = \frac{1}{A\,\Delta t}\sum_{\text{hits}} 2 m |v_\perp|$
- **Maxwell–Boltzmann velocity component:** $v_x \sim \mathcal{N}\!\left(0, \frac{k_B T}{m}\right)$

## Rules of thumb

- Room-temperature nitrogen: v_rms ≈ 517 m/s. In sim units set m = k_B = 1 so v_rms = √(3T).
- Store unit-temperature velocities and scale by √T when stepping; changing the temperature slider then rescales speeds instantly and correctly.
- Reflect position about the wall (`p = 2b − p`) and flip the velocity; just clamping loses distance and biases pressure.
- Average pressure over ~1–2 s of hits; with 100–300 particles single-frame readings are noisy.
- 300 instanced spheres at 120 Hz is trivial; per-particle collisions are where costs explode (O(N²) without a grid).

## Implementation

Box step that also measures pressure (m = k = 1):

```ts
export function stepGas(pos: Float32Array, vel: Float32Array, n: number,
                        dt: number, T: number, half: [number, number, number]) {
  const s = Math.sqrt(T) // vel holds unit-temperature velocities
  let impulse = 0
  for (let i = 0; i < n * 3; i++) {
    const b = half[i % 3]
    const v = vel[i] * s
    let p = pos[i] + v * dt
    if (p > b || p < -b) {
      p = 2 * (p > 0 ? b : -b) - p // specular reflection
      vel[i] = -vel[i]
      impulse += 2 * Math.abs(v)
    }
    pos[i] = p
  }
  const area = 8 * (half[0] * half[1] + half[1] * half[2] + half[0] * half[2])
  return impulse / (area * dt) // instantaneous pressure; average it
}
```

## Common mistakes

- Seeding random velocities with a weak or correlated generator (e.g. an LCG seeded with the particle index), which gives nearly identical speeds instead of a Maxwellian spread.
- Using uniform random velocities: the mean energy can be right but the speed distribution is wrong.
- Clamping particles to the wall without reflecting, which slowly drains kinetic energy.
- Comparing measured pressure with NkT/V using the slider temperature instead of the gas's actual kinetic temperature for a small N.

## Interactive visual

Particles are coloured by speed from cyan (slow) to orange (fast) inside a translucent box; the right wall is a piston that glows yellow with the measured pressure. Raise temperature or particle count, or narrow the box, and watch the glow and the P = NkT/V readout rise together.

## Prompt

```text
Build an ideal-gas box in TypeScript with Three.js. Create N ≤ 300 particles with uniform positions and Gaussian unit-temperature velocities from a seeded mulberry32 + Box–Muller generator. Each 1/120 s fixed step, advance positions by v·√T·dt, reflect specularly off the walls of a box whose width is adjustable, and return the summed wall impulse 2|v⊥|. Render with one InstancedMesh, colouring each instance from slow (cyan) to fast (orange), with no per-frame allocation. Tests: over 40,000 steps the measured pressure impulse/(area·time) matches N·T_kin/V within 3%, and 2,000 seeded particles have kinetic temperature ≈ 1.
```

## Related

- [Boltzmann Energy Distribution](https://gdcs.coody.app/physics/thermodynamics/boltzmann-distribution.md) — At temperature T a state of energy E is weighted by e^(−E/kT). Metropolis sampling reproduces the distribution with a simple accept/reject rule.
- [Heat Diffusion](https://gdcs.coody.app/physics/thermodynamics/heat-diffusion.md) — Heat flows down temperature gradients at a rate set by the Laplacian. The explicit grid update is four lines of code, and is stable only if αΔt/Δx² ≤ ¼.
- [Particle System](https://gdcs.coody.app/systems/patterns/particle-system.md) — An emitter spawns particles at a rate, each lives for a set time under simple forces, and all of them render in a single instanced draw call.

## Further reading

- [OpenStax University Physics Vol. 2 — Pressure, Temperature, and RMS Speed](https://openstax.org/books/university-physics-volume-2/pages/2-2-pressure-temperature-and-rms-speed)

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Source: Game Development Cheat Sheet (https://gdcs.coody.app)