Eigenfunction parameters

The eigenfunction method computes the distribution using a prescription given by Spakowitz and Wang^{1, 2}. It computes the distribution as a sum over a set of eigenfunctions in the space of orientations, which are ordered by an index 'l' which ranges from zero to infinity. The calculation considers only eigenmodes of a certain l-value or less; the number of eigenmodes that are used grows quickly with this l-value. Higher modes decay faster with increasing contour length, which is why this type of calculation is well suited to long polymers. The calculation performs integrations over the angular eigenfunctions, as well as a transform from Fourier space to real displacement space.

The parameters that determine the order of the computation are:

• l_max: the maximum l-value of the eigenmodes to consider. Generally speaking, shorter polymers require higher l-values. For the most general computation the number of eigenmodes to compute is proportional to the fourth power of l_max. If twists are being summed then the number of eigenmodes increases as l_max³; if both tangents and twists are being summed then the order of the computation is proportional to l_max.
  
  
• K_step: the step size in the inverse Fourier integration.
  
  
• K_max: the maximum K-value used in the inverse Fourier integration.
  
  
• saved/default: if set at 'saved', then a saved initialization for the above three fields will be used. This makes the initialization much quicker than a manual initialization, but the saved parameters may not be stringent enough, or they may be unnecessarily stringent and slow the calculation down unnecessarily. To adjust these parameters manually, set to 'custom'.
  
  
• num theta steps: the number of integration points in the 'theta' coordinate over the eigenfunctions.
  
  
• num phi steps: the number of integration points in the 'phi' coordinate over the eigenfunctions.

The stringency of the parameters should be increased to the point at which the result converges, meaning that making the parameters more stringent will not affect the final result significantly. The more stringent the parameters, the longer runtime will be. Increasing stringency means: increasing l_max, decreasing K_step, and increasing K_max.

Because of the finite order of any calculation using this method, it is possible to get nonsensical negative values for the probability. Also, the distribution will never exactly compute to zero when it definitely should, such as when the end-to-end distance is greater than the contour length.


1. Andrew J. Spakowitz and Zhen-Gang Wang, "End-to-end distance vector distribution with fixed end orientations for the wormlike chain model", Phys. Rev. E 72, 041802 (2005)
2. Andrew J. Spakowitz and Zhen-Gang Wang, "Wormlike chain statistics with twist and fixed ends", Europhys. Lett., 73 (5), p. 684 (2006)