Geometrical interpretation of dynamical phase transitions in boundary-driven systems

Dynamical phase transitions are defined as nonanalytic points of the large deviation function of current fluctuations. We show that for boundary-driven systems, many dynamical phase transitions can be identified using the geometrical structure of an effective potential of a Hamiltonian, recovered from the macroscopic fluctuation theory description. Using this method we identify new dynamical phase transitions that could not be recovered using existing perturbative methods. Moreover, using the Hamiltonian picture, an experimental scheme is suggested to demonstrate an analog of dynamical phase transitions in linear, rather than exponential, time.

Figure 1

The potential for AMFH model with $B=-20,\epsilon =0.02$ (solid blue line). The extremal points are marked in (red) circles, where ${\rho }_A\left({\rho }_C\right)$ is the lower (higher) peak and ${\rho }_B$ is the local minimum.

Figure 2

Typical density profiles in the AMFH model. (a) Case 1, $\widehat{\rho }={\rho }_A$, the (red) dashed line corresponds to the staying put trajectory and the (blue) solid line corresponds to the second trajectory at large currents (large $\tau$). It begins to saturate ${\rho }_C$ represented by the (green) dashed-dotted line as expected. (b) Case 2, $\widehat{\rho }<{\rho }_A$, the (blue) dotted line corresponds to the short trajectory for low values of the current($\tau <{\tau }_0$) and the (red) solid line corresponds to the long trajectory at large currents ($\tau >{\tau }_0$). It begins to saturate ${\rho }_C$ represented by the (green) dashed-dotted as expected.

Figure 3

The corresponding potentials for various values of $\tilde{E}$ of the WASEP. It can be seen that the basic structure of the potential is completely changed for different values of $\tilde{E}$. For $|\tilde{E}|>2$, there are two maxima at ${\rho }_{\pm }=\frac{1\pm \sqrt{1-2/|\tilde{E}|}}{2}$ and one minimum at $\rho =1/2$, whereas for $|\tilde{E}|<2$ there is a single maximum at $\rho =1/2$. This behavior give rise to the DPTs discussed in the main text.

Figure 4

The corresponding potentials for the Long-range hopping model with $\alpha =\frac{1}{24},\beta =9$ (solid red line) and the KLS model with $\delta =0.45,\epsilon =0.99$ (dashed blue line). The inset shows a zoom-in on the extremal points structure of the Long-range hopping model.

Figure 5

The numerical values for the action $W\left(\tau \right)$ for two possible solutions are considered. (a) corresponds to Case 1 and (b) to Case 3 in Sec. 3. The (red) curve represents the action $W\left(\tau \right)$ in the “staying put” solution and the (blue) circles correspond to the action $W\left(\tau \right)$ of the positive initial momentum ${\mathrm{\Pi }}_0$ solution. In both cases, the positive initial momentum solution becomes dominant as soon as it is feasible [i.e., ${\tau }_0=6.7\left(5.6\right)$ for case 1 (3), correspondingly].