f_{F} = \dot{p} + Γ \frac{δ F}{δ p} .

P_{F} [p] = N_{F} exp [- \frac{1}{2 σ ^{2}} \int_{t_{1}}^{t_{2}} d t \int d r \dot{p} + Γ \frac{δ F}{δ p}^{2}] .

Assume the order-parameter field is even under time-reversal symmetry and its free-energy functional is time-reversal invariant. The reversed path is

p^{R} (t) = p (t_{1} + t_{2} - t) .

P_{B} [p^{R}] = N_{B} exp [- \frac{1}{2 σ ^{2}} \int_{t_{1}}^{t_{2}} d t \int d r - \dot{p} + Γ \frac{δ F}{δ p}^{2}] .

For additive Gaussian white noise, the trajectory-to-noise Jacobian is the same in the two directions. We assume it and all path-independent normalization factors are absorbed into equal constants

N_{F} = N_{B}

. A time-reversal-odd order parameter would require the corresponding parity transformation as well.

Solution

After the sudden parameter change, the functional derivative is

\frac{δ F}{δ p} = a_{F} p - κ \nabla^{2} p .

Each Cartesian component of each Fourier mode consequently obeys

\overset{p}{˙}_{q, i} = - r (q) p_{q, i} + f_{q, i}, r (q) = Γ (a_{F} + κ q^{2}) .

p_{q, i} (t) = p_{q, i} (t_{1}) e^{- r (q) (t - t_{1})} + \int_{t_{1}}^{t} d t^{'} f_{q, i} (t^{'}) e^{- r (q) (t - t^{'})} .

Because both coefficients are positive, every mode has positive decay rate.

Evolution from the earlier time to the later time multiplies the earlier field by the deterministic decay factor, plus fresh noise independent of that field. The unequal-time correlation function is therefore