New definition of the tube diameter

Suppose you know the trajectory of your chains (say from molecular dynamics or single chain stochastic models such as slip-links or slip-springs). How can you calculate the tube diameter?

Here is one possibility: I created a new correlator which calculates the following correlation function:
where r_{ik}(t) is the position of the monomer i of chain k at time t, Nc is number of chains, N is number of monomers in each chain, and T is the total time of your trajectory. In other words, I calculate the average distance from one monomer at particular time to the monomer which was closest to this position some time ago. To start with, I will consider this quantity for middle monomer only, i.e. i=N/2.

If your chain only reptates (for example slithering snake algorithm), this quantity will be zero until time of order reptation time. In reality it will of course be finite, and if it is constant for some time, I suggest this constant will be a good measure of perpendicular thickness of the tube, or tube diameter. This should not be confused with the tube persistence length, which is what Doi and Edwards call "tube diameter" (in their theory these two quantities are assumed to be equal, whereas they can be very different, for example in semiflexible chains).

Here are few results:

This figure shows results from MD simulations of the most entangled chains I can simulate (kb=3, N=200, Nc=400, see our paper for details). The line shows a(t) for the middle monomer, whereas points show (from bottom to top): center of mass dissplacement g3(t), middle monomer displacement g1(N/2,t) and end monomer displacement g1(0,t). One can see that a(t) is almost constant (about 4-5), but not quite - of course I blame constraint release for that. To illustrate this, I calculated the same thing from the slip-springs model (N=48, no CR, Ns=0.5, Ne=4)
The plateau is now very clear at about a^2=0.72.

More investigation is needed, including MD with fixed ends. I would also like to create analogous correlator in parallel to the tube direction - hopefully in the next posts.