FIG. 4.

(Color online) Chain motion on the lattice. In the panels, different colors indicate the different planes to which the sites belong: Blue for the central plane, red for the plane above, and green for the plane below. We label the central site “0” and we number the twelve neighboring sites from 1 to 12. Site “13” is an example of a next nearest neighbor of the central site “0” that is a nearest neighbor of sites “5” and “6.” (a) Initial configuration of a polymer chain comprising three monomers (m₁,m₂,m₃) sitting on the nodes of a three-dimensional lattice. To illustrate our algorithm for the motion of the chain, we consider the motion of the monomer m₂. (b) Projection onto the central plane of the polymer configuration and the neighboring lattice sites. The color code is the same as in (a). Monomer m₂ can move with equal probability to any of the ten empty neighboring sites. (c) m₂ moves to site “7” (indicated by the black arrow). This site is a nearest neighbor of site “1” in which m₃ sits, but it is not a nearest neighbor of site “5” which is occupied by m₁. Since consecutive monomers in the polymer chain must occupy neighboring sites in the lattice, m₁ must “reptate,” i.e., it must move to a neighboring site which is also a neighbor of site “7.” These are sites {“0,” “6,” “11,” “13”} indicated by boxes. Purple boxes show sites which cannot be occupied {“0,” “6”}, because the final configuration would violate stiffness constraints. Black boxes indicate acceptable sites {“11,” “13”}. With equal probability, monomer m₁ can move to either of the two acceptable sites. The two possible final configurations are shown in panels (e) and (d).