Dilute polymers also alter significantly the distribution of finite-time Lyapunov exponents . In Fig. 4.9 the Cramér rate function is shown for the Newtonian and for the viscoelastic case.
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Since in the former situation the Lyapunov exponent is greater than , were the polymers passive all moments of elongation would grow exponentially fast. However, the feedback can damp stretching so effectively that after polymer addition lies below . This implies a strong reduction of Lagrangian chaos and a decreased mixing efficiency. Moreover, we find that is smaller than for all , a result which guarantees the stationarity of the statistics of in presence of feedback, while imposing a less restrictive condition on than the one proposed in Ref. [54]. The lowering of below unity would seem to contradict the statement that strong feedback takes place only at . Actually there is no inconsistency, since the critical value holds for passive polymers only. For active polymers, the presence of correlations between the conformation tensor and the stretching exponents can indeed lower significantly the threshold. For a discussion of the differences between active and passive transport, see Ref. [80]