Ring polymers in melts and solutions: scaling and crossover

We propose a simple mean-field theory for the structure of ring polymer melts. By combining the notion of topological volume fraction and a classical van der Waals theory of fluids, we take into account many body effects of topological origin in dense systems. We predict that although the compact statistics with the Flory exponent $\nu=1/3$ is realized for very long chains, most practical cases fall into the crossover regime with the apparent exponent $\nu = 2/5$ during which the system evolves toward a topological dense-packed limit.