PERM: A Monte Carlo Strategy for Simulating Polymers and Other Things

TL;DR

PERM is a novel Monte Carlo sampling method that efficiently simulates complex polymers and other high-dimensional systems by selectively cloning and killing configurations, avoiding Markov chain limitations.

Contribution

The paper introduces PERM, a new Monte Carlo strategy that builds configurations stepwise with bias correction, enabling large-scale simulations of polymers without relying on Markov processes.

Findings

Confirmed strong logarithmic corrections in theta polymers.

Validated Flory-Huggins mean field theory for unmixing.

Suggested corrections explain experimental deviations from mean field predictions.

Abstract

We describe a general strategy, PERM (Pruned-Enriched Rosenbluth Method), for sampling configurations from a given Gibbs-Boltzmann distribution. The method is not based on the Metropolis concept of establishing a Markov process whose stationary state is the wanted distribution. Instead, it starts off building instances according to a biased distribution, but corrects for this by cloning "good" and killing "bad" configurations. In doing so, it uses the fact that nontrivial problems in statistical physics are high dimensional. Therefore, instances are built step by step, and the final "success" of an instance can be guessed at an early stage. Using weighted samples, this is done so that the final distribution is strictly unbiased. In contrast to evolutionary algorithms, the cloning/killing is done without simultaneously keeping a large population in computer memory. We apply this in large scale simulations of homopolymers near the theta and unmixing critical points. In addition we sketch other applications, notably to polymers in confined geometries and to randomly branched polymers. For theta polymers we confirm the very strong logarithmic corrections found in previous work. For critical unmixing we essentially confirm the Flory-Huggins mean field theory and the logarithmic corrections to it computed by Duplantier. We suggest that the latter are responsible for some apparent violations of mean field behavior. This concerns in particular the exponent for the chain length dependence of the critical density which is 1/2 in Flory-Huggins theory, but is claimed to be $\approx 0.38$ in several experiments.