Analytical derivation of the dynamic exponent z = 5

Derive analytically the dynamic exponent z=5 governing the ultraslow coarsening produced by charge-selective topological locking in the quadrupolarly coupled two-dimensional magnetoelastic theory.

Background

The paper reports numerically that quadrupolar magnetoelastic coupling produces defect densities decaying as t^{-2/5}, defect velocities decaying as t^{-4/5}, and a corresponding dynamic exponent z approximately equal to 5. The authors associate this anomalously slow coarsening with charge-selective magnetic–elastic defect locking and enhanced, spatially extended elastic friction.

Although the dissipative-friction analysis provides a physical interpretation of the slowdown, the paper does not derive the exponent analytically from the coupled relaxational equations. Establishing such a derivation would explain why z=5 emerges in both same-sign and opposite-sign charge-locking regimes.