Nonconserved order-parameter dynamics (Model A dynamics)

Nonconserved relaxational order-parameter dynamics has ${\partial }_t\psi =-\Gamma \delta F/\delta \psi$ plus thermal noise. Its zero-wavenumber mode may decay.

For quadratic free energy with positive $a,\kappa$ and kinetic coefficient $\Gamma$, a deterministic initial Dirac delta distribution evolves under nonconserved order-parameter dynamics to the meanTaking the expectation removes the centered Gaussian white noise; the resulting diffusion-reaction equation is solved by the heat kernel. The integrated mean is $A{e}^{-\Gamma at}$, so spreading does not conserve the initial excess. The connected Fourier mode covariance grows from zero to $k_{B}T/(a+\kappa q^2)$ as $1-e^{-2\Gamma (a+\kappa q^2)t}$ by the Itô isometry. A realization retains equilibrium fluctuations after its mean has decayed. Pointwise continuum white-noise quantities require a coarse-graining cutoff.

For nonconserved order-parameter dynamics, the positive kinetic coefficient $\Gamma$ converts the functional derivative of the free energy into a relaxation rate through $\dot{p}=-\Gamma \delta F/\delta p$. Microscopic reversibility fixes the accompanying Gaussian white noise variance by the Model A fluctuation-dissipation relation.

For additive Gaussian white noise of variance ${\sigma }^2$, a Model A trajectory has path weight proportional toComparing a path with its time reverse yields the free-energy change and hence the thermal fluctuation-dissipation relation.

For additive Gaussian noise, a stochastic trajectory has probability proportional to the exponential of minus the squared noise history required to generate it. Rewriting that history in terms of the trajectory gives an Onsager--Machlup action; comparing it with the time-reversed action yields a path-probability ratio.

The Onsager–Machlup functional is the action in the exponential of an additive Gaussian stochastic path probability. For $\dot{x}=b(x)+\sigma \Lambda$, its elementary discretization contains $\int |\dot{x}-b(x){|}^2/(2{\sigma }^2)dt$, with convention-dependent Jacobian terms for multiplicative or state-dependent drift.

For additive Gaussian white noise and a time-even order parameter, a midpoint discretization of the Onsager–Machlup functional evaluates drift derivatives at the midpoint configurations. Reversing a trajectory visits the same midpoints in reverse order, so the noise-to-path Jacobian determinant is invariant under reversal. It may depend on the trajectory, and must be distinguished from the constant normalization of the noise measure.

For nonconserved order-parameter dynamics with $\dot{p}=-\Gamma \mu [p]+f$, the factor at one time step, apart from a path-independent power of the step size, isHere the fields have first been restricted to a finite spatial grid, and ${\mathcal{H}}_n$ is the Hessian matrix of the free energy.

For mixed conserved and nonconserved order-parameter dynamics, the required noises for a joint configuration and flux trajectory arewhere $\mathsf{D},\mathsf{G}$ represent the divergence and gradient, and $\Delta =\mathsf{DG}$ represents the Laplacian. The Jacobian matrix isTaking its Schur complement gives the joint factorFor periodic boundary conditions, discretizing the two spatial operators compatibly gives $\mathsf{D}=-{\mathsf{G}}^{\mathsf{T}}$, so $\Gamma I-M\Delta$ is the positive relaxation operator when $\Gamma ,M>0$. Reversal leaves ${\mathcal{H}}_n$ unchanged and reverses the sign of the flux. The product of the joint factors is consequently the same for both histories, justifying its cancellation from the joint path probability of an order parameter and its flux. The functional chain rule used in the action ratio holds in the continuum limit of this midpoint convention.

For nonconserved order-parameter dynamics, microscopic reversibility with equilibrium weight proportional to $e^{-F/(k_{B}T)}$ requiresIndeed the Onsager--Machlup path probability gives $\log (P_F/P_B)=-2\Gamma \int \dot{p}\delta F/\delta pdt/{\sigma }^2=-2\Gamma (F_2-F_1)/{\sigma }^2$. Equating this to the detailed balance value proves the relation. The functional chain rule is understood with the same midpoint convention as the path action.

After a sudden change from quadratic kernel $K_I(q)$ to $K_F(q)$, each Fourier mode is an Ornstein-Uhlenbeck process with decay rate $r(q)=\Gamma K_F(q)$. Its equal-time covariance interpolates exponentially between $k_{B}T/K_I(q)$ and $k_{B}T/K_F(q)$.