Monte Carlo simulations of discrete Rouse dynamics on a 2D lattice: emergence of global behavior in a polymer chain from local constraints

A polymer is a chain of monomers whose connectivity makes its dynamics far richer than those of simple particles. The Rouse model captures this by treating monomers as harmonically coupled Brownian oscillators: each bead feels a restoring force from its neighbors plus thermal noise. In this report, we study a discrete, lattice-based analogue of the Rouse model. On a 2D lattice, we enforce a single microscopic rule - fixed nearest-neighbor distance along the chain - together with self-avoidance, and use Monte Carlo simulations to follow the polymer’s motion. We quantify the dynamics via the squared end-to-end distance, the squared radius of gyration and the monomer mean-squared displacement. Without self-avoidance, the MSD shows the expected Rouse crossover from subdiffusive to diffusive regimes around a timescale that is consistent with Rouse scaling. Even without explicit energies, this minimal distance-preserving rule reproduces essential polymer-dynamical features, highlighting how complex behavior can arise from very simple geometric constraints. To go beyond Rouse dynamics, we then introduce several alternating-rule toy models (alternating copolymers) and a block-copolymer toy model that impose spatially heterogeneous geometric constraints. By changing only which moves different monomers may attempt - without adding forces, potentials or energetic biases - these models break detailed balance and generate a spectrum of nonequilibrium responses. Some remain close to Rouse-like behavior due to geometric suppression of rule heterogeneity, while others exhibit strong nonequilibrium expansion driven by bond-length fluctuations.