= Time-reversal invariance of a path Jacobian
{title2=$\mathcal J$}
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, is
$$
\mathcal J_n=\left|\det\left[I+\frac{\Gamma\Delta t}{2}\mathcal H_n\right]\right|,
\qquad \mathcal H_n=\frac{\partial\mu}{\partial p}\bigg|_{(p_n+p_{n+1})/2}.
$$
Here 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 are
$$
f_n=\frac{p_{n+1}-p_n}{\Delta t}+\mathsf D W_n+\Gamma\mu_n,
\qquad N_n=W_n+M\mathsf G\mu_n,
$$
where $\mathsf D,\mathsf G$ represent the <divergence> and <gradient>, and $\Delta=\mathsf D\mathsf G$ represents the <Laplacian>. The <Jacobian matrix> is
$$
\frac{\partial(f_n,N_n)}{\partial(p_{n+1},W_n)}=
\begin{pmatrix}
\frac{I}{\Delta t}+\Gamma\mathcal H_n/2&\mathsf D\\
M\mathsf G\mathcal H_n/2&I
\end{pmatrix}.
$$
Taking its <Schur complement> gives the joint factor
$$
\boxed{\mathcal J_n=\left|\det\left[I+\frac{\Delta t}{2}(\Gamma I-M\Delta)\mathcal H_n\right]\right|.}
$$
For <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.
Back to article page
