Einstein–Smoluchowski Diffusion Relation formula
| Symbol | Quantity | SI unit | Meaning |
|---|---|---|---|
| L | Diffusion length (RMS distance) | m | Root-mean-square displacement after time t |
| n | Dimensions | — | Number of spatial dimensions in which diffusion occurs |
| D | Diffusion coefficient | m²/s | Diffusion coefficient |
| t | Time | s | Elapsed time |
What is Einstein–Smoluchowski Diffusion Relation?
Einstein–Smoluchowski Diffusion Relation: For a random walk, the mean squared displacement grows linearly with time: ⟨r²⟩ = 2nDt in n dimensions. — Einstein 1905, Smoluchowski 1906
Diffusion is a random walk, so a particle does not travel a distance proportional to time. Its root-mean-square displacement grows with the square root of time. This is the most useful rule of thumb in diffusion: to go 10 times farther takes 100 times longer.
The factor 2n depends on how many directions the particle can move in. Use n = 1 for diffusion along a line or into a slab (the usual "diffusion length" in semiconductors and materials). Use n = 2 for motion on a membrane or surface, and n = 3 for free diffusion in a liquid or gas.
This scaling explains why cells are small, why oxygen needs a circulatory system to travel more than about a millimetre in tissue, and why heat treatment times grow so quickly with case depth.
How to use this calculator
- Choose the quantity you want to find in the Calculate menu. It is highlighted as the result field.
- Enter the known values. Scientific notation such as
1.6e-11is accepted. - Select a unit for each value. Units are converted to SI automatically.
- The answer updates as you type. Open Show calculation steps to see the working, or use Copy result to share a link with your inputs.
Worked example
How far does a small molecule (D = 1 × 10⁻⁹ m²/s) wander in water in one second, in 3D?
- ⟨r²⟩ = 2 × 3 × 1 × 10⁻⁹ × 1 = 6 × 10⁻⁹ m².
- L = √(6 × 10⁻⁹) = 7.7 × 10⁻⁵ m ≈ 77 µm.
To travel 1 cm the same molecule needs t = (0.01)² / (6 × 10⁻⁹) ≈ 16 700 s — about 4.6 hours.
Frequently asked questions
Why do some textbooks use L = √(Dt) or L = 2√(Dt)?
They are all "diffusion lengths" defined slightly differently. √(Dt) is the characteristic length in Fick's second law, 2√(Dt) appears in the erf argument, and √(2Dt) is the 1D RMS displacement. They differ only by constant factors, so state which one you use.
Is this the average distance travelled?
It is the root-mean-square displacement from the starting point, not the path length. The actual path is far longer and very jagged.
What is the diffusion length in semiconductors?
For minority carriers, L = √(Dτ), where τ is the carrier lifetime. Set n = 1 and enter τ as the time, then divide the result by √2 to match that convention.