§ 41. Brownian Motion & Einstein–Smoluchowski Diffusion (Grade 8–10)
Random walk trajectory of a suspended microparticle under molecular impacts and linear growth of mean-square displacement ⟨r²⟩ = 6Dt. The visible trembling of a Brownian grain is direct macroscopic proof of the unceasing thermal motion of invisible molecules.
Theory Summary & Scientific Mechanism
Brownian motion (discovered by Robert Brown in 1827) is the perpetual, erratic random walk executed by microscopic suspended particles (r sim 0.1–1 μm) immersed in a fluid. Because the particle’s surface area is small, the incessant thermal impacts of surrounding fluid molecules do not cancel out instantaneously, subjecting the particle to a fluctuating Langevin force F_fl(t).
By isotropy, the mean displacement vector vanishes (langle r(t) rangle = 0), yet the particle steadily wanders away from its origin! In 1905, Albert Einstein and Marian Smoluchowski proved that the mean-squared displacement (MSD) grows linearly with time t: langle r²(t) rangle = 2 d_dim D t (langle r² rangle = 4Dt in a 2D plane; langle r² rangle = 6Dt in 3D space).
The diffusion coefficient D obeys the Stokes–Einstein relation D = (k_B T)/(6π η r) = (R T)/(6π η r N_A), linking thermal energy k_B T to viscous dissipation 6πη r. Jean Perrin’s 1908 measurements of langle r²(t) rangle yielded a precise value of Avogadro’s number N_A and provided definitive proof of the physical reality of atoms.
Everyday Analogy
Look through a microscope at a tiny pollen grain in water and you’ll see it jittering and dancing endlessly! Invisible water molecules bombard the grain unevenly from all sides, direct visual proof that atoms and molecules are in constant thermal motion!
Interactive Question & Derivation
How does the root-mean-square distance √(langle r² rangle) of a Brownian particle from its origin depend on observation time t and fluid viscosity η?
- A: Proportional to the square root of time √(t) and inversely proportional to √(η), since langle r² rangle = 6Dt = (k_B T)/(π η a)t
- B: Grows linearly with time t, as in uniform rectilinear motion
- C: Remains identically zero at all times
Study Hint: From the Einstein–Smoluchowski relation langle r² rangle = 2dDt with D = (k_B T)/(6πη a): how does thermal jittering change when temperature T increases or particle radius a decreases?
Proof & Explanation: Because molecular impacts from different directions are statistically independent, their variances add linearly: langle r² rangle = 6Dt, where the Einstein diffusion coefficient D = (k_B T)/(6πη a) increases with temperature T and decreases with particle radius a and viscosity η.
Key Equations & 3D Simulation Parameters
langle r² rangle = 6 D t, D = (k_B T)/(6π η a)
- Absolute liquid temperature T: 250 … 450 K (default: 310 K)
- Brownian particle radius a: 0.5 … 2.5 μm (default: 1 μm)
- Dynamic fluid viscosity η: 0.5 … 3 mPa·s (default: 1 mPa·s)