2D Brownian Motion Simulation - Complete Guide

Background

What is Brownian Motion?

Brownian motion is the random movement of microscopic particles suspended in a fluid, caused by constant bombardment from fluid molecules. First observed by botanist Robert Brown in 1827, it provided crucial evidence for the atomic nature of matter.

Modern Applications:

Drug delivery systems
Single-molecule tracking
Financial modeling
Nanoscale transport

Theory

Einstein's Diffusion Equation

For 2D Brownian motion, the Mean Square Displacement (MSD) grows linearly with time:

MSD(t) = ⟨r²(t)⟩ = 4Dt

Where:

D = Diffusion coefficient (m²/s)
t = Time elapsed (s)
r² = x² + y² (squared distance from origin)

Stokes-Einstein Relation

D = kᵦT / (3πηd)

Where:

kᵦ = Boltzmann's constant (1.38 × 10⁻²³ J/K)
T = Temperature (K)
η = Fluid viscosity (Pa·s)
d = Particle diameter (m)

Simulation Method

Algorithm Steps

Calculate diffusion coefficient D from physical parameters
Compute scaling factor: k = √(2Dτ)
Generate random displacements: dx = k × randn(N,1), dy = k × randn(N,1)
Convert to positions: x = cumsum(dx), y = cumsum(dy)
Calculate MSD: r²(t) = x²(t) + y²(t)
Compare with theory: MSD_theory = 4Dt

Why k = √(2Dτ)?

For 2D motion: ⟨(dx)² + (dy)²⟩ = 4Dτ

Since dx and dy are independent with variance σ²: ⟨(dx)² + (dy)²⟩ = 2σ²

Setting 2σ² = 4Dτ gives σ = √(2Dτ) = k

Program Structure

Main Components:
├── Parameter Setup (N, τ, d, η, T)
├── Calculate D and k
├── Generate Random Walk
│   ├── dx, dy (displacements)
│   └── x, y (positions via cumsum)
├── Calculate MSD
├── Statistical Analysis
└── Visualization (6 plots)

Key Variables

Variable	Description	Units
N	Number of time steps	-
tau	Time interval	s
D	Diffusion coefficient	m²/s
k	Scaling factor	m
dx, dy	Step displacements	m
x, y	Positions	m
squaredDisplacement	MSD from origin	m²
msd_theoretical	Theory prediction	m²

How to Run

Save code as brownian_motion_2d.m
Open MATLAB and navigate to file location
Run: brownian_motion_2d or press Run button
View two figure windows with 6 plots total

Modify Parameters

Edit these at the top of the code:

N = 2000;        % Try: 500, 5000, 10000
tau = 0.01;      % Try: 0.001, 0.1
d = 1.0e-6;      % Try: 0.5e-6, 2.0e-6
T = 293;         % Try: 273, 310, 373

Understanding the Results

Figure 1

Plot 1: Particle Trajectory

Shows 2D random walk path
Green circle = start, Red square = end
Should look irregular and "tangled"
No preferred direction

Plot 2: Displacement Distribution

Histogram of dx values
Should be bell-shaped (Gaussian)
Centered at zero
Symmetric

Plot 3: MSD vs Time

Black line = theoretical (smooth, linear)
Blue line = simulated (noisy, fluctuating)
Blue should fluctuate around black
Noise is NORMAL for single particle

Figure 2

Plot 4: Position vs Time

Blue = X position, Red = Y position
Both should wander randomly
No systematic drift
Independent of each other

Plot 5: Step Displacement Distribution

Right-skewed distribution
Chi-squared with 2 degrees of freedom
Peak near zero, long tail

Plot 6: Autocorrelation

Single sharp peak at lag = 0
All other values near zero
Confirms statistical independence
Validates Markov property

Parameters

N (Number of Steps)

Default: 2000
More steps = longer trajectory, better statistics
Recommended range: 500-10000

tau (Time Step)

Default: 0.01 s
Smaller = finer resolution
Larger = coarser resolution
Typical range: 0.001-0.1 s

d (Particle Diameter)

Default: 1.0 × 10⁻⁶ m (1 micron)
Larger particles diffuse slower (D ∝ 1/d)
Typical range: 0.5-10 microns

T (Temperature)

Default: 293 K (20°C)
Higher T = faster diffusion (D ∝ T)
Typical values: 273K (ice), 310K (body), 373K (boiling)

eta (Viscosity)

Default: 1.0 × 10⁻³ Pa·s (water at 20°C)
Higher η = slower diffusion (D ∝ 1/η)
Water at 37°C: 0.7 × 10⁻³ Pa·s

Troubleshooting

MSD doesn't match theory closely

Normal for single particle! Fluctuations are expected
Try N = 5000 for smoother curves
Run multiple times to see variability

Estimated D differs from theoretical D

Check units (d in meters, not microns)
Verify k = sqrt(2Dtau), not sqrt(4Dtau)
Increase N for better statistics

Trajectory looks unrealistic

Ensure using new random numbers each step
Check that D is not zero
Verify k is reasonable (~10⁻⁸ m)

Autocorrelation shows broad peak

If using unmodified code: shouldn't happen
If modified: may have introduced correlations
Each step must use NEW randn() call

Lab Questions

Q1: Does theoretical line match simulated displacement?

Answer: Yes, approximately. The simulated MSD (blue curve) fluctuates around the theoretical line (black). Significant deviations (±50%) are normal for a single particle. The general trend and average slope should match. This is expected statistical behavior, not an error.

Q2: Is the estimated D reliable?

Answer: Moderately reliable for single particle. Typical relative error: 2-10% for N=2000. Standard error is usually 1-3% of D. Reliability increases with:

More samples (larger N)
Ensemble averaging (multiple particles)

For single particle, this level of agreement is acceptable and demonstrates correct simulation.

Q3: Effect of bulk flow on trajectory and D estimation?

Answer: With bulk flow added (constant drift):

Trajectory shows systematic displacement in drift direction
Still shows random fluctuations superimposed
Estimated D is overestimated (flow adds to MSD)
MSD curve shows quadratic component (v²t²)
With large flow: no longer looks like pure Brownian motion

Q4: Ensemble averaging vs single particle?

Answer: YES, ensemble averaging gives better D estimates:

Single particle (N=2000): ~3% standard error, 10-20% deviation
10 particles: ~1% standard error, ~5% deviation
100 particles: ~0.3% standard error, ~2% deviation
Improvement factor: 1/√(number of particles)

Q5: Judging Brownian motion from autocorrelation?

True Brownian Motion:

Single sharp peak at lag = 0
All other values near zero (|r| < 0.2)
Symmetric about zero
No systematic pattern

Non-Brownian Motion:

Broad peak → correlated motion (inertia)
Multiple peaks → periodic motion (trap)
Asymmetric → time-dependent drift
Gradual decay → memory effects (viscoelastic)

Criterion: If |correlation| < 0.2 for all lags except 0, motion is consistent with Brownian motion.

Calculating MSD for Your Report

Your code already calculates MSD automatically in the variable squaredDisplacement. For your report, you should:

1. Present the Plot

Show Plot 3 (MSD vs Time) with both simulated and theoretical curves.

2. Report Key Values

Add this code to extract values at specific times:

fprintf('\n========================================\n');
fprintf('  MSD AT KEY TIME POINTS\n');
fprintf('========================================\n');
fprintf('Time(s)  MSD_sim(m²)      MSD_theory(m²)   Ratio\n');
fprintf('------------------------------------------------------\n');
time_points = [1, 5, 10, 15, 20];
for t = time_points
    idx = round(t/tau);
    if idx <= N
        ratio = squaredDisplacement(idx) / msd_theoretical(idx);
        fprintf('%6.1f   %.4e    %.4e    %.2f\n', ...
                t, squaredDisplacement(idx), msd_theoretical(idx), ratio);
    end
end

3. Include in Report

The calculation formula: MSD(t) = x²(t) + y²(t)
A table of values at key time points
Final MSD value and comparison with theory
Brief discussion of agreement/disagreement

Tips for Lab Report

Structure

Introduction: Background, objectives, theory
Methods: Simulation parameters and algorithm
Results: Plots, tables, numerical values
Analysis: Distributions, autocorrelation, error analysis
Discussion: Agreement with theory, physical interpretation
Conclusions: Summary and validation

Key Statements

"The trajectory exhibits random walk behavior consistent with Brownian motion"
"Displacement distribution is Gaussian with mean zero"
"MSD increases linearly with time, confirming Einstein's equation"
"Autocorrelation shows no correlation between successive steps"
"Estimated D = [value] agrees with theory within [X]% error"

Common Mistakes to Avoid

❌ Expecting exact match between simulated and theoretical MSD
❌ Thinking smooth trajectories are better
❌ Believing large fluctuations indicate errors
✅ Understanding that single-particle noise is physically correct

Further Experiments

Try these variations:

% Temperature effect
T = 273; T = 310; T = 373;

% Particle size effect  
d = 0.5e-6; d = 2.0e-6; d = 10e-6;

% Viscosity effect
eta = 0.7e-3; eta = 1.8e-3; eta = 10e-3;

% Longer simulation
N = 10000;

% Different time resolution
tau = 0.001; tau = 0.1;

Mathematical Details

Step Size Derivation

Starting from MSD = 4Dt for 2D:

⟨Δr²⟩ = 4Dτ (for time step τ)
⟨(Δx)² + (Δy)²⟩ = 4Dτ
Since ⟨(Δx)²⟩ = ⟨(Δy)²⟩ = σ²
2σ² = 4Dτ
σ = √(2Dτ) = k

Chi-Squared Distribution

When X ~ N(0,σ²) and Y ~ N(0,σ²) are independent:

Z = X² + Y² ~ χ²(2)
E[Z] = 2σ² = 4Dτ

This validates that mean(dSquaredDisplacement) = 4Dτ.

References

Essential Papers

Einstein, A. (1905). "Investigations on the Theory of Brownian Movement." Annalen der Physik, 17, 549-560.
Perrin, J. (1909). "Brownian Movement and Molecular Reality." Nobel Prize 1926.

Textbooks

Berg, H.C. (1993). Random Walks in Biology. Princeton University Press.
Reif, F. (2009). Fundamentals of Statistical and Thermal Physics. Waveland Press.
Van Kampen, N.G. (2007). Stochastic Processes in Physics and Chemistry. Elsevier.

Online Resources

MATLAB Documentation: mathworks.com/help/matlab
MIT OpenCourseWare: Statistical Mechanics
Wolfram MathWorld: Random Walk articles

Glossary

Brownian Motion: Random motion of particles in fluid from molecular collisions

Diffusion Coefficient (D): Rate of particle spreading, units m²/s

MSD: Mean Square Displacement, ⟨r²(t)⟩

Autocorrelation: Measure of correlation between variable and its past values

Markov Process: Future depends only on present, not past

Chi-squared Distribution: Distribution of sum of squared normal variables

Isotropic: Same properties in all directions

Overdamped: Regime where friction dominates over inertia

Final Notes

Remember:

Brownian motion is fundamentally random
Single particle trajectories are highly variable
Fluctuations around theory are expected and correct
The beauty is that chaos follows predictable statistical laws

Your simulation successfully demonstrates that:

Random molecular collisions create predictable diffusion
Statistical mechanics works at the microscopic scale
Einstein's 1905 theory accurately describes reality

Good luck with your assignment!

Complete README - Version 1.0

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