Equilibrium SSEP quasi-potential on a torus

Let $(u,j)$ be any admissible connection of duration $T$ from $\bar{\rho }$ to $\rho$. By [F4], ${\mathcal{I}}_{[0,T]}(u,j)={\mathcal{H}}_{\bar{\rho }}(\rho )+{\mathcal{I}}_{[0,T]}(u^{\theta },j^{\theta })$, because both endpoints have mean $\bar{\rho }$ and [F3] identifies the endpoint free-energy difference with ${\mathcal{H}}_{\bar{\rho }}(\rho )$. The reversed action is nonnegative by [F1], so every admissible connection has action at least ${\mathcal{H}}_{\bar{\rho }}(\rho )$.

Let ${\rho }_s$ be the Fourier heat profile with datum $\rho$. Under [A1], [F6] applies; because $\rho$ is smooth, its Fourier multiplier profile is smooth up to $s=0$, solves ${\mathrm{\partial }}_s{\rho }_s=\mathrm{\Delta }{\rho }_s$, and preserves mean $\bar{\rho }$. Apply the range-preservation assertion of [F6] to $v=(\rho -\epsilon )\mathrm{/}(1-2\epsilon )$, which lies in $[0,1]$. Linearity of the Fourier multiplier and its fixing constants give ${\rho }_s=\epsilon +(1-2\epsilon )v_s$, hence $\epsilon \le {\rho }_s\le 1-\epsilon$ for every $s\ge 0$.

Taking the infimum over all finite-duration connections in [F2] yields ${\mathcal{V}}_{\bar{\rho }}(\rho )\ge {\mathcal{H}}_{\bar{\rho }}(\rho )$.

On the forward heat path $({\rho }_s,-\mathrm{\nabla }{\rho }_s)$, $0\le s\le S$, the continuity equation holds because ${\mathrm{\partial }}_s{\rho }_s=\mathrm{\Delta }{\rho }_s$ and $\mathrm{\nabla }\cdot (-\mathrm{\nabla }{\rho }_s)=-\mathrm{\Delta }{\rho }_s$, and the action is zero because $-\mathrm{\nabla }{\rho }_s+\mathrm{\nabla }{\rho }_s=0$. Its time reverse on $[0,S]$ has current $+\mathrm{\nabla }{\rho }_{S-t}$, and [F4] gives its action as $\int f(\rho )dx-\int f({\rho }_S)dx$. Shift this reversed path to $[1,S+1]$ by $t\mapsto t-1$; substitution in the action integral shows that its action is unchanged, and the resulting path has density ${\rho }_{S+1-t}$ and current $+\mathrm{\nabla }{\rho }_{S+1-t}$. The heat equation verifies continuity on the shifted interval.

Prepend the connector from [F5] on $[0,1]$. Its endpoint derivatives match the reversed heat segment, so the concatenation is a smooth admissible fixed-mean connection from $\bar{\rho }$ to $\rho$ of duration $S+1$. By [F2], ${\mathcal{V}}_{\bar{\rho }}(\rho )\le {\mathcal{I}}_{[0,1]}({\sigma }^S,{\kappa }^S)+\int f(\rho )dx-\int f({\rho }_S)dx$.

As $S\to \mathrm{\infty }$, [F5] gives ${\mathcal{I}}_{[0,1]}({\sigma }^S,{\kappa }^S)\to 0$ and ${\rho }_S\to \bar{\rho }$ uniformly; continuity of $f$ on $[\epsilon ,1-\epsilon ]$ then gives $\int f({\rho }_S)dx\to f(\bar{\rho })$. By [F7], the fixed-mean identity in [F3] is ${\mathcal{H}}_{\bar{\rho }}(\rho )={\int }_{{\mathbb{T}}^d}[\rho (\log \rho -\log \bar{\rho })+(1-\rho )(\log (1-\rho )-\log (1-\bar{\rho }))]dx={\int }_{{\mathbb{T}}^d}[\rho \log (\rho \mathrm{/}\bar{\rho })+(1-\rho )\log ((1-\rho )\mathrm{/}(1-\bar{\rho }))]dx$. Thus ${\mathcal{V}}_{\bar{\rho }}(\rho )\le {\mathcal{H}}_{\bar{\rho }}(\rho )$; combining with the lower bound in step 2.1 proves the claimed equality. The constructed horizons tend to infinity; no finite-time minimizer is asserted.