Polymer localization in random potential

This work re-examines the classical problem of polymer collapse in a random system, using a Gaussian variational formalism to treat the “poor solvent” case. In particular we seek to clarify some of the disputed questions related to symmetry breaking between the replicas used to analyze such quenched systems, and the scaling of the globular collapse with the disorder strength. We map the random system to a chain with attractive interactions in the standard way, and conduct a variational analysis along with a detailed examination of the theory's stability to replica symmetry breaking. The results suggest that replica symmetry is in fact not broken by the collapse of the chain in a random-disordered system, and that inclusion of a positive third, or higher, virial coefficient is crucial to stabilize the theory. For three dimensions, we find the globule square radius falling as (α-2ω)2, where α and ω are proportional to the second and third virial coefficients, respectively. This is in keeping with the earliest results of the replica-symmetric argument advanced by Edwards and Muthukumar [J. Chem. Phys. 89 (1988) 2435], but we anticipate a different scaling with dimensionality in other cases, with the globule square radius scaling as l(α,ω)2/(d-2) for d<4, where l is some linear function. This result agrees a scaling argument regarding the chain in a random potential like that put forth by Cates and Ball [J. Phys. (France) 49 (1988) 2009], but with a repulsive third virial coefficient present.