Loading MSD Data
Loading msd_data.json
Contains unwrapped coordinates for PS1, PS2, PS3, GRA.
1 Why COM & Unwrapping?
Why Center-of-Mass (COM)? ●
Polymers wiggle internally (Rouse modes). Tracking individual atoms includes this "noise," showing \(t^{0.5}\) behavior at short times. The COM filters this out, revealing only the chain's net translational diffusion.
Why Unwrap Coordinates?
Periodic boundaries cause atoms to "jump" across the box (Pac-Man style). We must unwrap these jumps to get continuous trajectories for true displacement calculations.
2 Concept: The Sliding Window
To calculate MSD from a single trajectory, we re-use the data by shifting the "Time Origin" \((t_0)\).
Trade-off: Larger Lag Times \((\tau)\) mean fewer possible windows \((N)\), leading to poorer statistics at the tail.
Calculated Diffusivity \((D)\)
Data Statistics
━ Grey: Individual Measurement Windows \((|r(t)-r(0)|^2)\)
━ Blue: Mean MSD (Average)
-- Amber: Linear Fit
🎓 Understanding Mean Square Displacement (MSD)
What is MSD? Mean Square Displacement measures how far molecules move from their starting positions over time. For diffusive motion in 3D, the Einstein relation tells us that \(\text{MSD}(\tau) = 6D\tau\), where \(D\) is the diffusion coefficient we want to extract and \(\tau\) is the lag time.
The Challenge: Computing MSD from a single time origin (e.g., measuring displacement only from frame 0) gives extremely noisy results because we get just one measurement per lag time. This is where the sliding window technique becomes crucial.
The Solution: Instead of one origin, we measure displacement starting from every possible frame. For a lag time \(\tau = 100\) frames in a 1000-frame trajectory, we get 900 independent measurements! This dramatically reduces noise and reveals the true linear trend needed to extract accurate diffusivity.
💡 Interactive Learning: Use the controls above to experiment! Try increasing the Max Lag Time slider and watch how the number of measurements (N) decreases, making the curve noisier at the tail. Adjust the Fit Region sliders to see how choosing different portions of the curve affects the calculated diffusivity and \(R^2\) value.
Note on the data: This demo uses pre-calculated center-of-mass (COM) trajectories for polystyrene chains. Using COM instead of individual atoms filters out internal wiggling motions (Rouse modes), revealing only the chain's net translational diffusion. The coordinates are "unwrapped" to handle periodic boundary conditions correctly.