Brownian Motion Calculator
Compute mean square displacement and RMS displacement of a Brownian particle.
Input Data
Diffusion Coefficient M2Per S Input
m²/s
Time S
s
Results
⟨x²⟩ = 2Dt.
0m²
⟨r²⟩ = 6Dt.
0m²
√(2Dt).
0.66181568μm
√(6Dt).
1.14629839μm
√(2Dt/π).
0.37338952μm
Confirmed D.
0m²/s
At a glance:MSD_1D = 2Dt, MSD_3D = 6Dt; RMS = √(2Dt). Complements stokes-einstein-relation and ficks-first-law.
Formula
MSD = 2Dt (1D).
RMS = √(2Dt).
$$\langle x^2 \rangle = 2Dt, \quad \langle r^2 \rangle = 6Dt, \quad \langle |x| \rangle = \sqrt{\frac{2Dt}{\pi}}$$How to Use
- Enter D and time.
- The tool returns MSD and RMS displacements.
Case Studies
D=1e-9, t=1
RMS_1D ≈ 44.7 µm.
FAQ
Why √t?
Random walk displacement grows as √time, not linearly.
Related Tools
References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.