Particle Trajectories
MSD vs Time
Brownian motion describes the ceaseless, random jiggling of small particles suspended in a fluid, caused by countless unseen molecular collisions. Robert Brown first observed it in pollen grains in 1827, but it was Albert Einstein who, in 1905, derived the quantitative relationship now known as the Einstein-Smoluchowski equation: D = kBT / (6πηr), linking the diffusion coefficient D to temperature T, fluid viscosity η, particle radius r, and Boltzmann's constant kB. This single equation underpins everything from drug nanoparticle delivery to estimating Avogadro's number.
This simulator lets you adjust temperature (200–400 K), particle radius (1–100 nm), and fluid type (water, glycerol, or oil) to watch how the mean squared displacement (MSD) evolves over time. The live MSD vs time plot compares your measured trajectory against the theoretical prediction MSD = 4Dt, letting you verify Einstein's formula in real time.
What is the Einstein diffusion coefficient formula?
The diffusion coefficient is D = kBT / (6πηr), where kB = 1.38 × 10−23 J/K is Boltzmann's constant, T is absolute temperature in kelvin, η is the dynamic viscosity of the fluid, and r is the particle radius. For a 10 nm particle in water at 300 K, D ≈ 21 nm²/μs. Doubling the radius halves D; doubling the temperature doubles D.
What is mean squared displacement (MSD) and why does it matter?
MSD is the average of the squared distance a particle has moved from its starting position, calculated over many particles or time windows. In 2D Brownian motion MSD = 4Dt, and in 3D it is MSD = 6Dt, where t is elapsed time. A linear MSD vs time graph is the definitive signature of pure diffusion; deviations indicate confinement (sub-diffusion) or active transport (super-diffusion).
How does viscosity affect diffusion?
Viscosity η appears in the denominator of Einstein's formula, so higher viscosity slows diffusion directly. Glycerol (η ≈ 0.141 Pa·s) is about 141 times more viscous than water (η ≈ 0.001 Pa·s) at room temperature, reducing D by the same factor. This is why nanoparticles diffuse much more slowly in biological gels than in saline solution.
Between 1908 and 1913, Perrin tracked the vertical distribution of resin beads in water using a microscope, applying the barometric formula to deduce kB. Dividing the gas constant R by kB gave NA ≈ 6.4 × 1023. This was the first direct, model-independent determination of Avogadro's number, earning Perrin the 1926 Nobel Prize in Physics.
According to the Stokes-Einstein equation, the drag force on a sphere in a viscous fluid is F = 6πηrv. Smaller radius means less drag, so thermal energy kicks the particle further per unit time. A 1 nm particle diffuses 10 times faster than a 10 nm particle in the same fluid at the same temperature, making nanoscale drug carriers especially mobile inside living cells.
Brownian motion is the continuous-time limit of a discrete random walk. After N steps of length λ, the root-mean-squared displacement is λ√N. In the continuous case λ shrinks and N grows such that λ²N/2 → Dt, recovering the Gaussian displacement distribution with standard deviation σ = √(2Dt). This link is fundamental in financial modelling, where stock prices are often modelled as geometric Brownian motion.
Yes, smoke particles in air exhibit Brownian motion, famously demonstrated with a smoke cell under a microscope. Gas molecules collide far more frequently but with much less momentum than liquid molecules, so the effect is smaller but still measurable. The same Einstein formula applies, using the dynamic viscosity of air (η ≈ 1.8 × 10−5 Pa·s) rather than water.
When MSD scales as tα with α ≠ 1, diffusion is called anomalous. Sub-diffusion (α < 1) occurs in crowded cytoplasm where particles frequently become temporarily trapped. Super-diffusion (α > 1) occurs when particles receive persistent kicks, such as from molecular motors in cells. Brownian motion is the special case α = 1, called normal or Fickian diffusion.
Temperature enters through the thermal energy kBT: at 400 K (hot water), a 10 nm particle in water has D ≈ 28 nm²/μs versus 21 nm²/μs at 300 K. However, water viscosity also decreases at higher temperatures (roughly halving between 20 °C and 70 °C), so the real-world speed increase is even larger than the T factor alone suggests.
Drug delivery: nanoparticles of 10–100 nm diffuse rapidly through tumour interstitium to reach cancer cells. Colloidal stability: charged nanoparticles use electrostatic repulsion to prevent aggregation driven by Brownian collisions. Single-molecule microscopy: optical tweezers trap particles against Brownian fluctuations to measure piconewton forces from molecular motors like kinesin.
With a finite number of particles and a short simulation time, statistical noise causes the measured MSD to fluctuate around the theoretical 4Dt line. The relative uncertainty in MSD scales as 1/√N, so increasing the particle count from 10 to 80 reduces scatter by roughly 2.8 times. Wall reflections in the simulator also introduce slight deviations near the canvas boundary.
The Peclet number Pe = UL/D compares convective transport (velocity U over length L) to diffusive transport. When Pe ≫ 1, flow dominates and Brownian motion is negligible — as in blood flow carrying red blood cells. When Pe ≪ 1, diffusion governs — as in oxygen molecules crossing a cell membrane. Understanding Pe is essential for designing microfluidic lab-on-a-chip devices.