 Conserved order-parameter dynamics

A conserved scalar order parameter evolves by transporting it rather than creating or destroying its total amount. A simple relaxational model has chemical potential $\mu =\delta F/\delta \varphi$, current $\mathbf{J}=-M\nabla \mu$ andwith compatible boundary conditions and optional conserved thermal noise. Integrating gives constant total $\int \varphi$ when boundary flux vanishes.

Linearizing a quadratic Landau-Ginzburg theory gives a decay rate $\Gamma (q)=M{q}^2(a+\kappa q^2)$. Away from criticality, positive $a$ yields ordinary diffusive $q^2$ scaling at small wavevector. At the Gaussian critical point $a=0$, the leading rate is instead proportional to $q^4$. Conservation forces the zero-wavevector mode to remain unchanged but does not by itself fix every critical decay exponent.