Numerical study of continuous and discontinuous dynamical phase transitions for boundary-driven systems

The existence and search for thermodynamic phase transitions is of unfading interest. In this paper, we present numerical evidence of dynamical phase transitions occurring in boundary-driven systems with a constrained integrated current. It is shown that certain models exhibit a discontinuous transition between two different density profiles and a continuous transition between a time-independent and a time-dependent profile. We also verified that the Kipnis-Marchioro-Presutti model exhibits no phase transitions in a range much larger than previously explored.

Figure 1

(a) The diffusion $D$ and (b) the conductivity $\sigma$. The models under inspection: in solid blue, the Mexican flat hat model for $A=1$ and $B=-20$; in dotted red, the KMP model; in dashed yellow, the long-range hopping with exclusion model for $\alpha =1/24$ and $\beta =9$.

Figure 2

Fixing the boundary conditions ${\varrho }_L={\varrho }_R=\widehat{\varrho }$ (green diamond) between a minimal and a maximal point in $\sigma$ (red circles). For $D=1$, we expect the AP density profile to increase as the current is increasing (see Appendix pp1 for some insight). However, at some point the density profile reaches a maximal value of $\sigma$. Further heightening of the density profile is no longer favorable. This calls for a different characteristic solution of the density profile. It could conceivably be a different AP solution or a time-dependent solution.

Figure 3

The sniping method results for the Mexican flat hat and the long-range hopping with exclusion models. We are interested in finding a solution for which $\left|g_{\omega }\left(x=1\right)\right|=0$ for some $\lambda$ and $\omega$ using the sniping method. Identification of numerical zeros is made by going below the numerical error bars implying the existence of a solution to (9) with the boundary conditions (11). Solid lines: the absolute values of $g_{\omega }\left(x=1\right)$ using the sniping method. Dashed lines: the numerically estimated errors on $\left|g_{\omega }\left(x=1\right)\right|$. (a) The Mexican flat hat model for $A=1$ and $B=-20$. (b) The long-range hopping with exclusion model for $\alpha =1/24$ and $\beta =9$. No solutions to (9) were found below $\lambda =5.6$ in (a) and below $\lambda =10.22$ in (b).

Figure 4

The value of $\delta s_{\omega }^2$ for a range of $\omega$ and $\lambda$ above the critical region for the Mexican flat hat and the long-range hopping with exclusion models. Here, red and blue highlight the positive and negative values of $\delta s_{\omega }^2$, respectively. The locations for which $\delta s_{\omega }^2=0$ imply that only the trivial solution $f_{\omega }=g_{\omega }=0$ exists. The negative values of $\delta s_{\omega }^2$ imply a second-order DPT. (a) The Mexican flat hat model for ${\omega }_0=4\pi$. (b) The long-range hopping with exclusion model for ${\omega }_0=6\pi$. The reference points ${\omega }_0$ in (a) and (b) are chosen for aesthetic reasons.

Figure 5

The trajectories of the density profile $\varrho \left(x\right)$ for different values of $J$. Red (light gray) indicates the frowning solutions, and blue (dark gray) indicates the smiling solutions.

Figure 6

The trajectory of the calculation of $J$. The green thin line is the conductivity profile $\sigma$, which plays the part of a potential. The yellow dotted line is the “total energy” $-\frac{1}{2K}$. The purple dot represents the boundary value ${\varrho }_L={\varrho }_R=\widehat{\varrho }$. (a) The trajectory for the smiling solution (in thick blue). (b) The trajectory for the frowning solution (in thick red).

Figure 7

The large deviation function $\mathrm{\Phi }\left(J\right)$ corresponding to the two solutions of $\varrho \left(x\right)$ of Eq. (B1). The solid red line depicts solutions with $\frac{d\varrho }{dx}>0$ at $x=0$ (the frowning solution). The dashed blue line depicts solutions with $\frac{d\varrho }{dx}<0$ at $x=0$ (the smiling solution). Note that the smiling solution is attainable only from $J_{\mathrm{transition}}\simeq 4.906$, where it is the favorable solution. The transition is indicated by the dotted yellow line.