Quantifying Entanglement for Collections of Chains in Models with Periodic Boundary Conditions

Using the Gauss linking integral we define a new measure of entanglement for a collection of closed or open chains, the linking matrix. For a system employing periodic boundary conditions (PBC) we use the periodic linking number and the periodic self- linking number to define the periodic linking matrix. We discuss its properties with respect to the cell size used for the simulation of a periodic system and we propose a method to extract from it information concerning the homogeneity of the entanglement. Our numerical results on systems of equilateral random walks in PBC indicate that there is a cell size beyond which the dependence of some properties of the periodic linking matrix on cell size vanishes and that the eigenvalues of the linking matrix can measure the homogeneity of the entanglement of the constituent chains.