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Structural modes of a polymer in the repton model

Using extensive computer simulations, the behavior of the structural modes --- more precisely, the eigenmodes of a phantom Rouse polymer --- are characterized for a polymer in the three-dimensional repton model, and are used to study the polymer's dynamics at time scales well before the tube renewal. Although these modes are not the eigenmodes for a polymer in the repton model, we show that numerically the modes maintain a high degree of statistical independence. The correlations in the mode amplitudes decay exponentially with $(p/N)^2A(t)$, in which $p$ is the mode number, $N$ is the polymer length and $A(t)$ is a single function shared by all modes. In time, the quantity $A(t)$ causes an exponential decay for the mode amplitude correlation functions for times $<1$; a stretched exponential with an exponent 1/2 between times 1 and $τ_R\sim N^2$, the time-scale for diffusion of tagged reptons along the contour of the polymer; and again an exponential decay for times $t>τ_R$. Having assumed statistical independence and the validity of a single function $A(t)$ for all modes, we compute the temporal behavior of three structural quantities: the vectorial distance between the positions of the middle monomer and the center-of-mass, the end-to-end vector, and the vector connecting two nearby reptons around the middle of the polymer. Furthermore, we study the mean-squared displacement of the center-of-mass and the middle repton, and their relation with the temporal behavior of the modes.