Pruned-enriched Rosenbluth method: Simulations ofθpolymers of chain length up to 1 000 000

We present an algorithm for simulating flexible chain polymers. It combines the Rosenbluth-Rosenbluth method with recursive enrichment. Although it can be applied also in more general situations, it is most efficient for three-dimensional $\theta$ polymers on the simple-cubic lattice. There it allows high statistics simulations of chains of length up to ${N=10\mathrm{}}^6$. For storage reasons, this is feasable only for polymers in a finite volume. For free $\theta$ polymers in infinite volume, we present very high statistics runs with $N=10\mathrm{}000$. These simulations fully agree with previous simulations made by Hegger and Grassberger [J. Chem. Phys. 102, 6681 (1995)] with a similar but less efficient algorithm, showing that logarithmic corrections to mean field behavior are much stronger than predicted by field theory. But the finite volume simulations show that the density inside a collapsed globule scales with the distance from the $\theta$ point as predicted by mean field theory, in contrast to claims in the work mentioned above. In addition to the simple-cubic lattice, we also studied two versions of the bond fluctuation model, but with much shorter chains. Finally, we show that our method can be applied also to off-lattice models, and illustrate this with simulations of a model studied in detail by Freire et al. [Macromolecules 19, 452 (1986) and later work].