We investigate the Rubinstein-Duke model for polymer reptation by means of density-matrix renormalization group techniques both in absence and presence of a driving field. In the former case the renewal time tau and the diffusion coefficient D are calculated for chains up to N=150 reptons and their scaling behavior in N is analyzed. Both quantities scale as powers of N: $tau sim N^z$ and $D sim 1/N^x$ with the asymptotic exponents z=3 and x=2, in agreement with the reptation theory. For an intermediate range of lengths, however, the data are well-fitted by some effective exponents whose values are quite sensitive to the dynamics of the end reptons. We find 2.7 <z< 3.3 and 1.8 <x< 2.1 for the range of parameters considered and we suggest how to influence the end reptons dynamics in order to bring out such a behavior. At finite and not too small driving field, we observe the onset of the so-called band inversion phenomenon according to which long polymers migrate faster than shorter ones as opposed to the small field dynamics. For chains in the range of 20 reptons we present detailed shapes of the reptating chain as function of the driving field and the end repton dynamics.
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