= Fluctuation-dissipation relation for a conserved flux
{title2=$\sigma_N^2=2Mk_BT$}
For the flux $W_{ij}=-M\partial_i\mu_j+N_{ij}$ of a <vector order parameter>, <microscopic reversibility> requires independent <Gaussian white noise> components with covariance
$$
\langle N_{ij}(\mathbf r,t)N_{kl}(\mathbf r',t')\rangle
=2Mk_BT\,\delta_{ik}\delta_{jl}\delta(\mathbf r-\mathbf r')\delta(t-t').
$$
The current changes sign under time reversal. The resulting path-probability ratio contributes $-(k_BT)^{-1}\int W_{ij}\partial_i\mu_j$ to its logarithm; <integration by parts> converts this to the free-energy loss from the conserved dynamics.
Back to article page
