FIG. 4.

Effect of memory in coarse-grained metastable states. (a) Double-well potential (Eq. (68)) exactly coarse-grained into states A and B with boundary along the green dashed line. We also show example paths (solid blue and red lines) that both exit B (circles) but start from different initial conditions within B (squares). (b) Length distributions ρ(ℓ) of paths (with β = 10) that exit B but start at different initial conditions (at the central barrier, corresponding to the blue square in (a), or in the low-energy basin, corresponding to the red square in (a)), along with the power law ℓ^−3/2 for comparison. For paths that exit the coarse-grained state B, (c) the mean time ${\overline{t}}^{\left(1\right)}$ and CVs ${\overline{\ell }}^{\left(\mathrm{cv}\right)}$, ${\overline{t}}^{\left(\mathrm{cv}\right)}$ of length and time, and (d) skewness ${\overline{\ell }}_{\mathrm{std}}^{\left(3\right)}$, ${\overline{t}}_{\mathrm{std}}^{\left(3\right)}$ and kurtosis ${\overline{\ell }}_{\mathrm{std}}^{\left(4\right)}$, ${\overline{t}}_{\mathrm{std}}^{\left(4\right)}$ of length and time, all as functions of β. All calculations use a discretized square lattice with Δx = 0.05 over the space (x, y) ∈ [ − 2, 2] × [ − 2, 2].