## 3 Prehistorical Description of the Optimal Fluctuations
### Interpretation of the PPD
#### 3.4.4 At Equilibrium  
We say the system in Equation (1) is at equilibrium if it is stationary and its law is time-symmetrical, i.e., \(\mathcal{L}_{\mathbf{x}}=\overline{\mathcal{L}}_{\mathbf{x}}\). Thus, we can obtain the following condition of detailed balance:  
\[2\mathbf{A}^{-1}(\mathbf{x})\Big{(}\mathbf{b}(\mathbf{x})-\frac{\varepsilon}{ 2}\
abla\cdot\mathbf{A}(\mathbf{x})\Big{)}=\varepsilon\
abla\ln(\rho_{s}( \mathbf{x})) \tag{86}\]  
If expressing the stationary density as \(\rho_{s}(\mathbf{x})=\exp(-U(\mathbf{x},\varepsilon)/\varepsilon)\), we can obtain  
\[2\mathbf{A}^{-1}(\mathbf{x})\Big{(}\mathbf{b}(\mathbf{x})-\frac{\varepsilon}{ 2}\
abla\cdot\mathbf{A}(\mathbf{x})\Big{)}=-\
abla U(\mathbf{x},\varepsilon) \tag{87}\]  
That is, it requires \(\mathbf{A}^{-1}(\mathbf{x})\big{(}\mathbf{b}(\mathbf{x})-\frac{\varepsilon}{ 2}\
abla\cdot\mathbf{A}(\mathbf{x})\big{)}\) to be a gradient field. We can thus divide it into the following three categories:\(\mathbf{A}(\mathbf{x})=\mathbf{I}\) and \(\mathbf{b}(\mathbf{x})=-\
abla u_{0}(\mathbf{x})\), such that \(u_{0}(\mathbf{x}_{q})=0\). In this case, the quasi-potential \(V(\mathbf{x}_{eq},\mathbf{x})=2u_{0}(\mathbf{x})\), as it satisfies the Hamilton-Jacobi equation (Equation (17)). The potential at equilibrium is \(U(\mathbf{x},\varepsilon)=2u_{0}(\mathbf{x})+const\), as the normalization condition \(\int\rho_{s}(\mathbf{x})d\mathbf{x}=1\) should be satisfied. Thus, the quasi-potential coincides with the potential up to a constant. The optimal fluctuation field will be as follows: \[\mathbf{b}(\mathbf{x})+\mathbf{A}(\mathbf{x})\
abla_{\mathbf{x}}V(\mathbf{x} _{eq};\mathbf{x})=-\mathbf{b}(\mathbf{x})\] (88) That is, the OP will follow the time reversal of the deterministic average dynamics. As a result, time symmetry for the optimal fluctuations and relaxation motions (in the absence of noise) under the time-reversed transformation holds.
2. \(\mathbf{A}^{-1}(\mathbf{x})\
abla\cdot\mathbf{A}(\mathbf{x})=\
abla u_{1}( \mathbf{x})\) and \(\mathbf{A}^{-1}(\mathbf{x})\mathbf{b}(\mathbf{x})=-\
abla u_{0}(\mathbf{x})\), such that \(u_{0}(\mathbf{x}_{eq})=0\). In this case, Equation (87) will hold for any choice of \(\varepsilon\). Similarly, the quasi-potential satisfies \(V(\mathbf{x}_{eq},\mathbf{x})=2u_{0}(\mathbf{x})\), and Equation (88) is still valid, but \(U(\mathbf{x},\varepsilon)=2u_{0}(\mathbf{x})+\varepsilon u_{1}(\mathbf{x})+const\). The deviation of the metastable states (i.e., minimum points of \(U\)) from the fixed points of the average system will emerge due to the non-homogeneity of the diffusion coefficient.
3. \(\mathbf{A}^{-1}(\mathbf{x})\mathbf{b}(\mathbf{x})\) is not a gradient field. In this case, if Equation (87) is valid, it holds only for an isolated point \(\varepsilon=\varepsilon_{0}\). This is the case where neither \(\mathbf{A}^{-1}(\mathbf{x})\mathbf{b}(\mathbf{x})\) nor \(\mathbf{A}^{-1}(\mathbf{x})\
abla\cdot\mathbf{A}(\mathbf{x})\) is a gradient field, while their linear combination is. If \(\varepsilon_{0}\) is small enough, we can also approximate the potential by breaking it down into two parts \(U(\mathbf{x},\varepsilon_{0})=U_{0}(\mathbf{x})+\varepsilon_{0}U_{1}(\mathbf{ x},\varepsilon_{0})+const\) such that \(U_{0}(\mathbf{x}_{eq})=0\), where the \(\varepsilon_{0}\)-independent part \(U_{0}(\mathbf{x})\) is nothing but the quasi-potential, with \(\exp(-U_{1})\) being the prefactor of the WKB approximation (Equation (31)). The deviation of the metastable states from the fixed points of the average system still exists, but the optimal fluctuation field will not coincide with \(-\mathbf{b}(\mathbf{x})\). This is very different from the case of additive noise.