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We investigated the roughening of elastic interfaces under random forces. For d=1, we found anomalous scaling invalidating standard models, linking it to Brownian functionals.

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Area of Science:

  • Condensed Matter Physics
  • Statistical Mechanics
  • Nonlinear Dynamics

Background:

  • Studying the roughening of d-dimensional directed elastic interfaces is crucial for understanding material properties.
  • Quenched random forces and elastic energy density, including harmonic and anharmonic terms, are key factors influencing interface behavior.

Purpose of the Study:

  • To analyze the roughening of elastic interfaces under quenched random forces with both harmonic and anharmonic elastic energy densities.
  • To determine scaling exponents and crossover length scales for arbitrary dimensions and anharmonicity.
  • To investigate anomalous scaling in the d=1 case and its relation to Brownian functionals.

Main Methods:

  • Heuristic scaling arguments were employed to derive global roughness exponent (ζ), dynamic exponent (z), and crossover length scale.
  • Numerical calculations were performed to validate theoretical predictions, particularly in d=1.
  • Connections to Brownian functionals were established to explain the d=1 anomalous scaling.

Main Results:

  • An upper critical dimension d_c(n)=4n was determined for the system.
  • Precise agreement between heuristic arguments and numerical calculations was found in d=1.
  • Anomalous "faceted" scaling was observed in d=1, with spectral roughness exponent ζ_s > ζ > 1, invalidating standard scaling assumptions.

Conclusions:

  • The study provides a comprehensive framework for understanding interface roughening with nonlinear elastic energies.
  • The d=1 case reveals complex scaling behavior beyond standard models, linked to fundamental Brownian functionals.
  • Results offer insights for experimental applications in nonlinear elastic interface dynamics, such as in random flows.