Computational Walkthrough

Calculation, Step by Step

This page teaches the full computational workflow used in vesicle fluctuation spectroscopy: generate a random quasi-spherical vesicle, compute angular autocorrelation, project the Legendre spectrum, and fit for \(\kappa\) and \(\Sigma\) using \(\chi^2\) minimisation.

Model
Model Description

We restrict the contour to the equator \(\theta=\pi/2\), map fluctuations through spherical harmonics, and project the correlation into Legendre modes for statistically stable \(\kappa\)-\(\Sigma\) fitting.

Equatorial Quasi-Spherical Contour

\[ r(\phi,t)=R\left[1+u\left(\theta=\frac{\pi}{2},\phi,t\right)\right] \]

Spherical Harmonic Expansion

\[ u(\theta,\phi,t)=\sum_{l,m}u_{lm}(t)Y_l^m(\theta,\phi) \]

Equipartition Amplitude

\[ \left\langle |u_{lm}|^2 \right\rangle= \frac{k_B T}{\kappa(l+2)(l-1)\left[l(l+1)+\Sigma\right]} \]

Autocorrelation and Projection

\[ \xi(\gamma)=\left\langle u(\phi)u(\phi+\gamma)\right\rangle, \qquad \xi(\gamma)=\sum_l b_l P_l(\cos\gamma) \]
Model Parameters
Sampling Setup
Projection and Fit Window
Use a fit window where your B_l values are reliable. Very high l modes are often noise dominated in experiments.
The full pipeline is executed in browser JavaScript, mirroring the same sequence as your Python quasi_spherical workflow.
Step 1
Contour Visualization
Generate Quasi-Spherical Contours
Thermal modes are sampled with variance inversely proportional to mode stiffness, then combined into an equatorial contour time-series.
\[\mathrm{var}_l=\frac{1}{\kappa(l-1)(l+2)\left[l(l+1)+\Sigma\right]}\]
Awaiting simulation.
Calculation Log - Thermal Mode Sampling
Run simulation to display numeric substitution.
mode l D_l var_l sqrt(var_l)
Step 2
Autocorrelation Function
Compute Angular Auto-Correlation
For each frame, compute ξ(γ, t) from circular shifts of the contour and average over time.
\[\xi(\gamma)=\frac{1}{2\pi r_{\mathrm{avg}}^2}\int_0^{2\pi}\left[r(\phi+\gamma)-r_{\mathrm{avg}}\right]\left[r(\phi)-r_{\mathrm{avg}}\right]d\phi\]
Calculation Log - Correlation at One γ
Run simulation to display numeric substitution.
i w_i r_i-r_avg r_i+g-r_avg term
Step 3
Legendre Spectrum
Project Legendre Coefficients
The averaged correlation is projected onto P_l(cos(gamma)) to obtain B_l in the selected mode window.
\[b_l=\frac{2l+1}{2}\int_0^{\pi}\xi(\gamma)P_l(\cos\gamma)\sin\gamma\,d\gamma\]
Calculation Log - Legendre Projection
Run simulation to display numeric substitution.
γ (deg) ξ P_l(cos γ) sin(γ) integrand
Step 4
Parameter Estimation
Fit \(\kappa\) and \(\Sigma\) via χ²
Compute χ² over Σ, find the minimum, recover j = kBT/κ, and therefore the fitted κ.
\[\bar{B}_l(j,s)=\frac{j}{p_l+s q_l},\qquad \chi^2=\sum_l\left(\frac{B_l-\bar{B}_l}{\sigma_l}\right)^2\]
κ true
-
κ fitted
-
Σ true
-
Σ fitted
-
Calculation Log - χ² Contributions
Run simulation to display numeric substitution.
l B_l(exp) σ_l B_l(th) residual residual²
  • Run the workflow to populate fit diagnostics.