Why does a concentrated cloud spread?
Why does a concentrated cloud spread?
- Start with a narrow cloud and pause after five seconds.
- Double D and compare at equal elapsed time.
- Increase sample count to reduce visual fluctuations.
What to measure
Units and parameter definitions are shown beside every laboratory control and live reading.
The idea underneath
Independent random walks turn a narrow initial concentration into a broader distribution. There is no outward force pushing particles away from the centre: more particles simply leave dense regions than enter them. For an initially Gaussian cloud, variance grows as σ²=σ₀²+2Dt on each axis. The cloud centre fluctuates for a finite sample, so increasing sample count improves statistical stability without changing the diffusion coefficient. The scene has open boundaries; the container grid is a scale reference, not a confining wall.
∂c/∂t=D∇²c · σ²(t)=σ₀²+2DtUnits and parameter definitions are shown beside every laboratory control and live reading.
Build the argument
- Specify the ensemble, particle mass, temperature and boundary conditions.
- Average over many particles before comparing microscopic and macroscopic quantities.
- For independent isotropic diffusion, each coordinate has variance 2Dt, giving mean squared displacement 6Dt in three dimensions.
Worked example 1
σ₀=0.25 m, D=0.08 m²/s, t=5 s gives σ=0.929 m per axis.
Worked example 2
The three-dimensional RMS radius is √(3σ₀²+6Dt).
Applications
Microscopic statistics connect to pressure, diffusion and transport. The visible particles represent an ensemble; their rendered size and playback speed are illustrative.
Common misconception
Temperature is a distribution of molecular kinetic energies, not the speed of every molecule. A stochastic trajectory cannot predict a particular future collision.
A further investigation
Increasing particle count changes the diffusion coefficient?
Check your understanding
Increasing particle count changes the diffusion coefficient?
- No
- It doubles it
- It makes it zero
Explanation
D is a model parameter. More samples improve the estimate, not the underlying diffusion.
Model limits
Independent Gaussian walkers in an unbounded medium; no reactions, advection or concentration-dependent D.