Gradiententerm

Gradiententerm, literally "gradient term" in German, refers to a component of an energy functional or Lagrangian that depends on the spatial gradient of a field. It quantifies the energetic cost of spatial inhomogeneity and typically acts as a smoothing or tension term, penalizing rapid variations in the field.

In mathematical terms, for a scalar field φ(x) defined on a domain Ω, a common gradient term appears as E_grad[φ] = ∫Ω (κ/2) |∇φ(x)|^2 dx, with κ > 0 representing a stiffness or gradient coefficient. The term can be extended to multiple fields or to anisotropic media, for example as ∑i (κi/2) |∇φi|^2 or as a quadratic form ∇φ^T A ∇φ, where A may vary with position or direction.

Gradient terms play a central role in variational models. Minimizing an energy that includes a gradient term leads to Euler-Lagrange equations containing a Laplacian term, typically −κ Δφ + ∂F/∂φ = 0, where F(φ) is a local potential. This coupling between local terms and the gradient term governs interface width, smoothness, and pattern formation.

Applications span phase-field models of phase transitions and superconductivity (Ginzburg–Landau theory), gradient elasticity in materials science, and image processing as smoothing priors. Variants include anisotropic gradient terms, spatially varying κ, and higher-order gradient terms such as (Δφ)^2, which impose stronger control on field curvature and microstructure.