Analytical Charts
Charts appear after a successful calculation. They compare exact and limiting behaviour. Logarithmic views reveal scaling transitions clearly.
Experimental MSD Data Fitting
Paste time and MSD columns below. Optional uncertainty values create weighted fits. Fixed parameters remain unchanged during optimisation.
Monte Carlo Trajectory Simulation
Simulate persistent walkers using correlated directions. Compare sampled MSD with analytical predictions. A seeded generator supports repeatable results.
Formula Used
The continuous model crosses from ballistic to diffusive motion. The discrete model tracks directional memory between steps. Drift, noise, and offsets add separate contributions.
How to Use
- Select a continuous or discrete motion model.
- Choose direct calculation or an unknown variable.
- Enter consistent distance and time units.
- Add drift, noise, or anisotropic axis parameters.
- Submit the form and inspect every reported quantity.
- Use fitting or simulation for experimental comparisons.
- Export the report as CSV or PDF.
Example Data
| Scenario | Time or steps | Persistence or ρ | Speed, D, or step length | Expected behaviour |
|---|---|---|---|---|
| Short continuous run | 0.1 s | 3 s | 1.2 µm/s | Nearly ballistic |
| Crossover run | 3 s | 3 s | 0.8 µm²/s | Mixed scaling |
| Long continuous run | 50 s | 3 s | 0.8 µm²/s | Nearly diffusive |
| Persistent discrete walk | 100 steps | ρ = 0.75 | 0.4 µm | Enhanced displacement |
| Anti-persistent walk | 100 steps | ρ = -0.50 | 0.4 µm | Suppressed displacement |
Frequently Asked Questions
What does MSD measure?
MSD measures average squared displacement from an initial position. It removes directional cancellation found in ordinary averages. Larger values indicate broader particle spreading over measured time.
What is persistence time?
Persistence time describes how long motion retains directional memory. Longer values preserve trajectories before random turning dominates. It controls the transition between ballistic and diffusive regimes.
Why does short-time MSD scale quadratically?
Directions remain strongly correlated during very short intervals. Displacement therefore grows almost linearly with elapsed time. Squaring displacement produces the characteristic quadratic MSD scaling.
Why does long-time MSD become linear?
Directional correlations decay after many persistence intervals. Successive movements then resemble ordinary random diffusion. The resulting MSD grows almost linearly with elapsed time.
What does a negative correlation mean?
Negative correlation favours reversals after each movement step. These reversals suppress net displacement across many steps. The walk becomes anti-persistent rather than directionally persistent.
How is measurement noise included?
Independent position noise adds a constant MSD contribution. The contribution equals twice dimensionality times noise variance. This term matters most at very short lags.
Should drift be removed?
Drift adds a quadratic contribution to raw MSD. Remove it when studying fluctuations around directed movement. Keep it when total observed transport remains important.
What does the scaling exponent show?
The exponent describes local MSD growth on logarithmic axes. Values near two indicate ballistic movement. Values near one indicate ordinary long-time diffusion.
Can fitted parameters be uniquely determined?
Parameter identifiability depends on time range and noise. Short records may confuse persistence, speed, and offset. Wider lag coverage usually improves reliable parameter separation.