[Paper Review] Pinning of interfaces in random media

Nicolas Dirr, Patrick Dondl+1 more|arXiv (Cornell University)|Nov 22, 2009

Theoretical and Computational Physics5 references8 citations

TL;DR

This paper proves the existence of a stationary positive supersolution for curvature-driven interface motion in random media, demonstrating that pinning and rate-independent hysteresis emerge even under viscous dynamics. By constructing a supersolution using Delaunay surfaces and gluing techniques, it establishes a finite critical force above which the interface remains pinned, explaining the macroscopic emergence of hysteresis from microscopic viscous laws.

ABSTRACT

For a model for the propagation of a curvature sensitive interface in a time independent random medium, as well as for a linearized version which is commonly referred to as Quenched Edwards-Wilkinson equation, we prove existence of a stationary positive supersolution at non-vanishing applied load. This leads to the emergence of a hysteresis that does not vanish for slow loading, even though the local evolution law is viscous (in particular, the velocity of the interface in the model is linear in the driving force).

Motivation & Objective

• To establish the existence of a stationary positive supersolution for interface evolution in random media under viscous dynamics.
• To demonstrate that rate-independent hysteresis can emerge from a viscous, curvature-sensitive interface model with random obstacles.
• To resolve the challenge of analytical pinning in sparse random environments where traditional methods fail.
• To bridge the gap between microscopic viscous kinetics and macroscopic rate-independent behavior in phase transitions or plasticity.

Proposed method

• Constructs a supersolution using rotationally symmetric Delaunay surfaces with constant mean curvature for local interface profiles.
• Employs a gluing function to patch local supersolutions with flat regions, ensuring global supersolution properties.
• Uses viscosity solution theory to verify the supersolution condition for the nonlinear mean curvature operator.
• Applies scaling arguments to match the radial derivative of the Delaunay surface with the gluing function's derivative at the interface.
• Imposes conditions on obstacle strength, distribution, and curvature to ensure the supersolution remains below the interface evolution.
• Leverages comparison principles to prove that the interface remains pinned below the critical force.

Experimental results

Research questions

• RQ1 Can a stationary supersolution exist for curvature-driven interface motion in a random medium with viscous dynamics?
• RQ2 Does pinning and hysteresis persist in the presence of random, sparse obstacles even when the local evolution law is viscous?
• RQ3 What conditions on obstacle distribution and strength are necessary to ensure the existence of a supersolution?
• RQ4 How does the supersolution construction handle the nonlinearity of the mean curvature operator in random environments?
• RQ5 Can the macroscopic emergence of rate-independent behavior be rigorously derived from microscopic viscous kinetics?

Key findings

• A stationary positive supersolution exists for the forced mean curvature flow in random media under suitable conditions on obstacle distribution and strength.
• The supersolution is constructed using Delaunay surfaces and a gluing function, ensuring global validity and viscosity solution properties.
• The critical force required for depinning is finite and strictly positive, even though the local dynamics are viscous.
• Hysteresis persists for arbitrarily slow loading due to the pinning mechanism, confirming rate-independent behavior at the macroscopic level.
• The construction remains valid in the lattice obstacle case for n=1, extending the result to discrete random configurations.
• The analysis confirms that macroscopic rate-independent kinetics can emerge from microscopic viscous laws through the pinning mechanism.