The extraordinary importance of self-avoiding behavior in two-dimensional polymers: Insights from large-deviation theory

Some recent work pointed out the usefulness of taking a large-deviation perspective when trying to extract anything resembling a macroscopic order parameter from a computer simulation. In this paper, we note that the end-to-end distance of polymers is such an order parameter. The presence of long-ranged excluded volume interactions leads to significant qualitative differences between the conformations of two- and three-dimensional polymers, some of which are difficult to quantify in computer simulations of realistic (off-lattice) polymer models. However, we show here that phenomena such as the greatly enlarged non-Hooke's-law elasticity present in 2D are straightforward to extract from simulation using a large-deviation framework-even though simulating that nonlinearity is tantamount to simulating a fourth order susceptibility. The large-deviation perspective includes both a set of thermodynamic-like tools suitable for studying finite-size systems and a realization that an accurate description of the system's average behavior needs to be consistent with how improbably large fluctuations would behave in that system. The latter is key because strong correlations are absent in this asymptotic large fluctuation regime, so the regime's far-reaching effects can be analytically incorporated into the analysis of simulation data. That, in turn, allows us to direct the efforts of simulations away from difficult-to-sample rare-event domains. We illustrate this point with two- and three-dimensional Monte Carlo simulations (and exact results) on two models of a single isolated polymer chain: a chain of linked hard spheres, which has long-ranged excluded volume effects, and a discretized worm-like chain, which does not.