Diffusion & Brownian Motion

Brownian Motion

Brownian motion describes the random thermal motion of particles suspended in a fluid. In 1905, Einstein derived the relationship between this random motion and the diffusion coefficient.

Stokes-Einstein Relation

For a spherical particle of radius r in a fluid of viscosity η at temperature T:

$D = \frac{k_B T}{6\pi\eta r}$

• k_B = 1.381 × 10⁻²³ J/K (Boltzmann constant)
• T = temperature in Kelvin
• η = dynamic viscosity in Pa·s (water ≈ 1 × 10⁻³ Pa·s at 20°C)
• r = hydrodynamic radius in meters

Mean Squared Displacement (MSD)

The MSD grows linearly with time, a hallmark of diffusion:

$\text{MSD} = 2d \cdot D \cdot t$

where d is the number of spatial dimensions (d=1 for 1D, d=3 for 3D). The root mean square displacement is:

$r_{\text{rms}} = \sqrt{\text{MSD}}$

Time to Diffuse a Distance

Rearranging MSD, the characteristic time to diffuse a distance L is:

$t = \frac{L^2}{2d \cdot D}$

This scales as L², meaning diffusion becomes extremely slow over long distances — a 10× larger cell takes 100× longer to transport molecules by diffusion alone.

Biological Examples

Your Task

Implement three functions:

1. diffusion_coefficient_m2_s(T_K, eta_Pa_s, r_m) — Stokes-Einstein diffusion coefficient in m²/s
2. msd_m2(D_m2_s, t_s, d=3) — Mean squared displacement in m² (default 3D)
3. diffusion_time_s(L_m, D_m2_s, d=3) — Time to diffuse distance L in seconds (default 3D)