Mixed conserved and nonconserved order-parameter dynamics

With constant positive kinetic coefficient $\Gamma$ and mobility $M$, two independent relaxation channels giveThe deterministic mobility operator is $\Gamma -M{\nabla }^2$. Each channel separately satisfies microscopic reversibility when its noise obeys the corresponding Model A fluctuation-dissipation relation or fluctuation-dissipation relation for a conserved flux.

Put $u_j={\dot{p}}_j+{\partial }_i{W}_{ij}$. The additive independent noises give the forward actionFor a time-even order parameter and time-odd flux, the backward action replaces $u,W$ by $-u,-W$ while keeping $\mu$ fixed on the corresponding configurations. Common normalization and time-reversal invariance of a path Jacobian then giveAt the two thermal noise strengths, periodic boundary conditions cancel the two spatial terms by integration by parts, leaving $-(F_2-F_1)/(k_{B}T)$. Thus the joint dynamics satisfies microscopic reversibility.