Diffusion in a random catalytic environment, polymers in random media, and stochastically growing interfaces.

Abstract

We study a family of equivalent models which includes the polymer in a random medium, the stochastically growing interface with spatially random deposition of particles, and the diffusion in a random catalytic environment ${\ensuremath{\partial}}_{t}$\ensuremath{\psi}(x,t)=D${\ensuremath{\nabla}}^{2}$\ensuremath{\psi}+U(x)\ensuremath{\psi}. $U( x vec---) denotes a frozen Gaussian random potential of strength ${u}_{0}$ and correlation length a. The intrinsic length scales of the problem, ${l}_{0}$=(D/${u}_{0}$${)}^{2/(4\mathrm{\ensuremath{-}}d)}$ and a, are both assumed to be small in comparison with the diffusion length \ensuremath{\surd}Dt and the system size L (L${<10}^{8}$a). Flory-Imry-Ma--type arguments show that for dimensions d<4 the polymer of contour length t\ensuremath{\rightarrow}\ensuremath{\infty} is both collapsed (\ensuremath{\nu}=0) and localized, in agreement with previous results for a\ensuremath{\ll}${l}_{0}$ [Edwards and Muthukumar, J. Chem. Phys. 89, 2435 (1988); and Cates and Ball, J. Phys. (Paris) 49, 2009 (1988)]. The sample-to-sample variations of the polymer free-energy scale as ${t}^{\ensuremath{\chi}}$ with \ensuremath{\chi}=1. For d>4, a collapsed, localized or a Gaussian, delocalized polymer is found for strong or weak disorder, respectively. The disorder becomes irrelevant for self-avoiding polymers. For growing interfaces, the roughness exponent \ensuremath{\chi}/\ensuremath{\nu} and dynamical exponent 1/\ensuremath{\nu} are both equal to unity, but scaling is modified by logarithmic corrections.The results are supported by a renormalization-group (RG) approach. Vertex corrections destroy the relation \ensuremath{\chi}=2\ensuremath{\nu}-1. For d>4 we find a phase transition as a function of disorder strength. For d<4 there is no stable fixed point. In particular, for d\ensuremath{\le}2 the fixed point found is unstable with respect to infinitely many nonlinear terms generated under the RG. It is argued that this corresponds to the appearance of a nonanalytic elastic term in the polymer Hamiltonian. The results are substantiated by mapping the problem onto the localization of a quantum particle in a random potential. When considering activated dynamics for the polymer we find that it cannot relax to the absolutely optimal environment present in the medium during observational time scales \ensuremath{\tau} but will be localized in a typical well. Due to this effect, the lnL terms occurring in the static theory have to be replaced by ln${L}_{\ensuremath{\tau}}$ with an additional dynamical length scale ${L}_{\ensuremath{\tau}}$\ensuremath{\ll}L.