[Paper Review] The scaling limit of polymer pinning dynamics and a one dimensional Stefan freezing problem

Hubert Lacoin|arXiv (Cornell University)|Apr 5, 2012

Stochastic processes and statistical mechanics12 references4 citations

TL;DR

This paper establishes the diffusive scaling limit of a 1+1-dimensional stochastic polymer interface pinned to a substrate, showing that the limit is governed by the heat equation with Dirichlet boundary conditions in the unpinned phase and a one-dimensional Stefan freezing problem with contracting boundaries in the pinned phase. The key contribution is proving existence and regularity of the solution to this free-boundary problem until the moving boundaries collide.

ABSTRACT

We consider the stochastic evolution of a 1+1-dimensional interface (or polymer) in presence of a substrate. This stochastic process is a dynamical version of the homogeneous pinning model. We start from a configuration far from equilibrium: a polymer with a non-trivial macroscopic height profile, and look at the evolution of a space-time rescaled interface. In two cases, we prove that this rescaled interface has a scaling limit on the diffusive scale (space rescaled by $L$ in both dimensions and time rescaled by $L^2$ where $L$ denotes the length of the interface) which we describe: when the interaction with the substrate is such that the system is unpinned at equilibrium, then the scaling limit of the height profile is given by the solution of the heat equation with Dirichlet boundary condition ; when the attraction to the substrate is infinite, the scaling limit is given a free-boundary problem which belongs to the class of Stefan problems with contracting boundary, also referred to as Stefan freezing problems. In addition, we prove the existence and regularity of the solution to this problem until a maximal time, where the boundaries collide.

Motivation & Objective

• To understand the macroscopic evolution of a 1+1-dimensional polymer interface far from equilibrium under diffusive scaling.
• To determine how the nature of the scaling limit depends on the strength of interaction with the substrate (pinned vs. unpinned phase).
• To rigorously establish the emergence of a free-boundary problem of Stefan type as the scaling limit in the strongly pinned regime.
• To prove existence and regularity of the solution to this Stefan-type problem up to the time when the moving boundaries collide.

Proposed method

• Analyzes the stochastic heat-bath dynamics of a polymer interface interacting with a substrate, with interaction strength parameterized by λ.
• Uses space-time rescaling (space ~ L, time ~ L²) to derive the hydrodynamic limit.
• For the unpinned case (λ < ∞), proves convergence to the heat equation with Dirichlet boundary conditions via comparison and stochastic domination techniques.
• For the infinite-pinning case (λ = ∞), derives a free-boundary problem where the interface evolves according to the heat equation in a moving interval, with boundary motion governed by the normal derivative.
• Employs convexification and concavification techniques to control the initial profile and prove short-time existence of solutions.
• Applies probabilistic coupling and large deviation estimates (e.g., corner-flip dynamics) to control the height profile and ensure stability under rescaling.

Experimental results

Research questions

• RQ1 Does the scaling limit of the polymer pinning dynamics depend on whether the system is pinned or unpinned at equilibrium?
• RQ2 Can the hydrodynamic limit in the strongly pinned regime be described by a free-boundary problem of Stefan type with contracting boundaries?
• RQ3 What is the regularity and existence time of the solution to this Stefan-type problem arising in the scaling limit?
• RQ4 How does the interface evolve in the unpinned phase under diffusive scaling, and does it converge to the solution of the heat equation with Dirichlet conditions?
• RQ5 What techniques are required to control the dynamics and prove convergence when the interface is pinned to the substrate?

Key findings

• In the unpinned phase (λ < ∞), the scaling limit of the rescaled polymer height profile is the solution of the heat equation with Dirichlet boundary conditions on the interval [-1,1].
• In the infinite-pinning case (λ = ∞), the scaling limit is described by a one-dimensional Stefan freezing problem with a moving interval (l(t), r(t)) where the interface is pinned outside and evolves via the heat equation inside.
• The free-boundary problem features boundary motion governed by the second derivative of the profile: l′(t) = -fxx(l(t),t) and r′(t) = fxx(r(t),t), with Neumann-like conditions on the derivative.
• The solution to the Stefan problem exists and is regular until the time when the boundaries l(t) and r(t) collide, which corresponds to full relaxation to equilibrium.
• For initial profiles that are 1-Lipschitz and compactly supported in (-1,1), the solution maintains regularity and the moving boundaries contract monotonically.
• The paper establishes a sharp lower bound on the decay of the area under the profile, which is crucial for proving convergence in the infinite-pinning case.