Mathematical definition of the model

Our model consists of two parts. First we have the basic chain, and secondly we have the tube and its motion. So:

• Basic chain
  Rouse chain with parameters N, b and $\gamma$.

• Primitive chain 

  i

  ) The primitive chain has contour length L. The position along the chain is indicated by $s\in [0,L]$. The configurations of the chain are Gaussian; by this we mean that
  

  $$\langle ( \vec{R}(s)-\vec{R}(s^{\prime }))^{2}\rangle =a\vert s-s^{\prime }\vert$$ (8.1)

  
  

  where a is a new parameter having the dimensions of length.

  ii

  ) The primitive chain diffuses along the tube axis with diffusion constant 
  

  $$D_{G}= \frac{kT}{(N+1)\gamma }$$ (8.2)

  
  

  i.e. with the Rouse diffusion constant.

The Gaussian character of the distribution of primitive chain conformations is consistent with the reptation pictured in Fig. (8.2). The chain continuously creates new pieces of tube, which may be chosen in random directions with step length a. Then $\vec{R}(s)-\vec{R}(s^{\prime })$ is a Gaussian vector.

Apparently we have introduced two new parameters, the contour length L and the step length a. Only one of them however is independent, because they are related by $Nb^2=\langle R^2 \rangle = aL$, where the first equality stems from the fact that we are dealing with a Rouse chain, and the second equality follows from an application of Eq. (8.1).