Monte Carlo Simulations of Polymer Dynamics on a 2D Lattice: Rouse Constraints, Heterogeneous Update Rules and Emergent Global Behavior

In this report, we study polymer dynamics using a discrete, lattice-based analogue of the Rouse model, where monomers move subject to a nearest-neighbor distance constraint (we analyze the polymers both with and without self-avoidance). Through Monte Carlo simulations, we quantify chain evolution via the end-to-end distance, radius of gyration and mean-squared displacement, observing a subdiffusive-to-diffusive crossover consistent with Rouse scaling (when self-avoidance is neglected). Building on this, we construct 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 - we make the polymers 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. We study the block-copolymer model in particular detail. In this model, two halves of the chain evolve under distinct dynamical rules with temporal heterogeneity in addition to geometric heterogeneity. We systematically analyze the diffusion of both halves separately, as well as the diffusion of the center of mass, for different degrees of heterogeneity in the biased update frequency and different chain lengths. In the appendices, we look at a minimal model of polymer collapse where only the end monomers experience an energetic attraction, and finally also reconstruct the mean-field phase diagram of the magnetic polymer model of Garel et al. (1999).