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Computation
This calculator computes the distribution in three different ways. Each method has complementary strengths and drawbacks.
- Eigenfunction: a method that computes the distribution by summing the eigenstates of the angular-distribution propagator1, 2. This method works well if the polymer's length is much longer than the bending persistence length; it is hard to work for segments shorter than about 2 persistence lengths. The parameters that affect the order of the calculation can be set in fields in the Eigenfunction params fields. Generally speaking, shorter polymers require more stringent values of these parameters that increase runtime. This method runs much faster under three circumstances: 1) the displacements are being summed over; 2) tangents and/or twists are being summed; 3) the initial and final tangents are exactly aligned and the displacement is zero (cyclization).
- Monte Carlo: computes the distribution by randomly generating a host of polymer chains of the specified properties, and sampling the fraction that obey the constraints on the distribution. Monte Carlo works best for short chains, since longer chains require more runtime to propagate. Monte Carlo imposes a significant overhead in generating the chains, but once that is done the distribution can be sampled at many points very quickly, so this method works well when tables are being generated. On the other hand, it is often difficult to sample many regions of a distribution with Monte Carlo with a reasonable number of chains. The parameters that affect chain generation and sampling are set in the Monte Carlo params fields.
- Gaussian: the simplest and crudest way of computing a distribution. If the polymer chain is very long, in the sense that its contour length is much greater than its bending persistence length, the distribution approaches a Gaussian in space and uniform in the angular coordinates (final tangent and twist). The disadvantage is that this method is very crude; the advantage is that it is practically instantaneous to compute, so it can give a quick estimate of the distribution for polymers that are many persistence lengths long.
The spatial units of the result can be set independently of all other units that were chosen, from the following menu:
- Lp: units of the bending persistence length
- nm: nanometers
- A: Angstroms
The angular units are always radians and steradians.
To run the calculator, press the "Calculate" button.
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)