gradientenergy

Gradient energy refers to the portion of a field's energy that depends on spatial variations of the field variable, typically represented by the squared norm of its spatial gradient. In continuum theories, the total free energy F[φ] includes a bulk term plus a gradient term: F[φ] = ∫ [ f_bulk(φ) + (κ/2) |∇φ|^2 ] dV, where φ(r) is a scalar field and κ > 0 is a gradient-energy coefficient. The term (κ/2)|∇φ|^2 penalizes nonuniform configurations and thus favors spatially uniform states on short scales, while f_bulk(φ) can drive phase separation or ordering.

The gradient energy arises from microscopic interactions when a system is coarse-grained to a continuum description. In magnets, it appears as exchange stiffness that resists spatial variation of the magnetization; in fluids and alloys, it is related to capillary or interfacial energy between phases. The gradient term plays a central role in phase-field models of microstructure evolution and in the Ginzburg-Landau theory of superconductivity.

Physically, the gradient term controls interface properties: a larger κ broadens interfaces and increases surface tension, while a smaller κ allows sharper transitions. In practice, one studies gradient-flow or time-dependent equations such as ∂φ/∂t = -M δF/δφ, which lead to dynamics like the Cahn-Hilliard or Allen-Cahn equations.

Gradient energy is used across condensed matter physics, materials science, and cosmology to describe energy costs associated with spatial inhomogeneities in scalar or vector fields.