🔬 Brownian Motion

🔬 Robert Brown's Discovery (1827)

Robert Brown, a Scottish botanist, noticed that pollen grains in water jittered erratically and continuously under a microscope. He found the same behaviour in dead plant material and finely ground rock dust — ruling out any biological explanation. For nearly 80 years it remained one of physics' most puzzling mysteries.

The phenomenon bore his name, but its cause — molecular bombardment — was only explained in 1905, after the atomic theory of matter had gained enough traction to allow a quantitative treatment.

💢 Einstein's 1905 Theory

In his annus mirabilis, Albert Einstein published a theory showing Brownian motion results from the cumulative effect of molecular collisions. His key prediction: the mean-square displacement (MSD) grows linearly with time:

⟨x²⟩ = 2Dt  (1D)    ⟨r²⟩ = 4Dt  (2D)

where D is the diffusion coefficient. Einstein also derived D from first principles: D = kₛT / 6πηr (Stokes-Einstein equation). Jean Perrin (Nobel 1926) confirmed this in 1908, measuring Avogadro's number from diffusion data — definitive proof that atoms exist.

⚙️ The Langevin Equation

In 1908, Paul Langevin reformulated the theory with an explicit dynamical equation for the particle's velocity:

m dv/dt = −γv + F(t)

The first term −γv is viscous drag (Stokes drag: γ = 6πηr). The second term F(t) is the stochastic random force with ⟨F(t)F(t')⟩ = 2γkₛTδ(t−t') � the fluctuation-dissipation theorem: the same γ that causes drag also sets the noise amplitude.

🌎 Real-World Applications

• Finance (Black-Scholes): Stock prices modelled as geometric Brownian motion — basis of options pricing
• Cell biology: Diffusion of ions, neurotransmitters, and proteins across membranes
• Nanotechnology: Nanoparticle motion in colloidal suspensions; drug delivery design
• Atmospheric science: Aerosol particle dispersion and settling in clouds
• MEMS/NEMS: Thermal noise in micro-mechanical sensors and resonators
• Polymer physics: Rouse model of polymer chain dynamics in solution

Step 1: Measure MSD

Perrin tracked gamboge resin spheres (r = 0.212 μm) in water at T = 293 K. After t = 30 s the MSD was ⟨x²⟩ = 8.0×10²¹ m².

D = ⟨x²⟩/2t = 8.0×10²¹ / 60 ≈ 1.33×10¹³ m²/s

Step 2: Apply Stokes-Einstein

D = kₛT / 6πηr = RT / (Nₐ × 6πηr)

Rearrange: Nₐ = RT / (6πηrD)

With R = 8.314, η(water) = 1.0×10−³ Pa·s, r = 2.12×10−&sup7; m, T = 293 K.

Step 3: Result

Nₐ ≈ 6.0×10²³ mol−¹

Perrin's original value was 6.4×10²³ — within 7% of the modern value of 6.022×10²³. This was the first precision measurement of Avogadro's number, providing definitive proof of atomic theory and winning Perrin the Nobel Prize in 1926.

Try it in the simulator

Set the preset to Water 20°C, enable trails, and watch the MSD grow linearly. Use D = MSD/(2t) and the Stokes-Einstein equation with η = 0.001 Pa·s to estimate kₛ. Then divide R by kₛ to get your own estimate of Nₐ.