Dynamical phase transition to localized states in the two-dimensional random walk conditioned on partial currents

The study of dynamical large deviations allows for a characterization of stationary states of lattice gas models out of equilibrium conditioned on averages of dynamical observables. The application of this framework to the two-dimensional random walk conditioned on partial currents reveals the existence of a dynamical phase transition between delocalized band dynamics and localized vortex dynamics. We present a numerical microscopic characterization of the phases involved and provide analytical insight based on the macroscopic fluctuation theory. A spectral analysis of the microscopic generator shows that the continuous phase transition is accompanied by spontaneous ${\mathbb{Z}}_2$-symmetry breaking whereby the stationary solution loses the reflection symmetry of the generator. Dynamical phase transitions similar to this one, which do not rely on exclusion effects or interactions, are likely to be observed in more complex nonequilibrium physics models.

Figure 1

Microscopic and macroscopic large-deviation analysis of the partial current statistics in the two-dimensional random walk. (a) (Main) Rescaled SCGF $L^2\theta \left(s\right)$ (green circles) and average partial current $L^2{\langle J\rangle }_s$ (magenta squares) for linear size $L=32$, relative slit length $h=1/2$ and field strength $E=5$. (Inset) Density field ${\rho }^s\left(n\right)$ (gray color map, darker color indicates a larger number of particles) and current field ${\mathbb{j}}^s\left(n\right)$ (blue arrows) for $s=0$ (natural dynamics). The slit is highlighted in red. (b) Density field ${\rho }^s\left(n\right)$ and current field ${\mathbb{j}}^s\left(n\right)$ for $s=-1$ (corresponding to $L^2{\langle J\rangle }_s=2.42$), $s=-3$ ($L^2{\langle J\rangle }_s=13.09$), $s=1$ ($L^2{\langle J\rangle }_s=0.23$), and $s=3$ ($L^2{\langle J\rangle }_s=-20.33$), with format as in the inset of panel (a). (c) Macroscopic fluctuation theory functional for the flat solution $G_{\text{flat}}\left(J\right)$, the inner-band solution $G_{\text{i.b.}}\left(J\right)$, the outer-band solution $G_{\text{o.b.}}\left(J\right)$, and the vortex solution $G_{\text{vort.}}\left(J\right)$. The minimum value among them corresponding to each $J$, denoted $G_{\text{min.}}\left(J\right)$, is also shown, as well as its convex envelope $G_{\text{Maxw.}}\left(J\right)$. See text for definitions and the Appendices for details.

Figure 2

Characterization of the DPT: SCGF and average current. (a) Density-rescaled SCGF $L^2\theta \left(s\right)$. (b) Same content as in panel (a) but enlarged around small values of $\left|s\right|$ and also including macroscopic SCGF (red discontinuous line). (c) SCGF $\theta \left(s\right)$. (d) Average current ${\langle J\rangle }_s$ multiplied by $L^2$ (main panel), by $L/h$ (top right inset) and without rescaling (lower inset). See text for justifications of rescalings. In all panels, the colors and symbols represent different system sizes $L$ according to the legend in panel (a).

Figure 3

Characterization of the DPT: spectral gaps and inverse participation ratio. (a) Rescaled spectral gaps $L^2{\mathrm{\Delta }}_1\left(s\right)$ (continuous lines, large symbols) and $L^2{\mathrm{\Delta }}_2\left(s\right)$ (dotted lines, small symbols); see text for definitions. (b) Inverse participation ratio, without (main panel) and with rescaling (inset). As $L$ increases, the range of $s$ explored decreases, due to convergence issues of the eigenvalue algorithm. In all panels, the colors and symbols represent different system sizes $L$ according to the legend in Fig. 2.

Figure 4

Dynamical regimes for the two-dimensional random walk conditioned on different values of the partial current $J$ across a slit (vertical strip in the center). (a) Flat profile. (b) Inner band. (c) Outer band. (d) Vortices. These regimes are idealizations of those observed in the microscopic analysis based on the Doob transform.

Figure 5

Macroscopic functionals for the inner band, the outer band, and the vortex solution as functions of the characteristic length $\ell$ and the partial current $J$. (a) Inner-band functional $G_{\text{i.b.}}(J,\ell )$. (b) Outer-band functional $G_{\text{o.b.}}(J,\ell )$. (a) Vortex functional $G_{\text{vort.}}(J,\ell )$. In each case, the location of the minimum value of the functional for each $J$ is shown as a gray continuous line. In panels (a) and (b), the red discontinuous line corresponds to the average partial current $\langle J\rangle$.

Figure 6

Comparison of the MFT functional for the different dynamical regimes under consideration. The MFT functional of the flat solution $G_{\text{flat}}\left(J\right)$, the inner-band solution $G_{\text{i.b.}}(J,{\ell }^{*}\left(J\right))$, the outer-band solution $G_{\text{o.b.}}(J,{\ell }^{*}\left(J\right))$, and the vortex solution $G_{\text{vort.}}(J,{\ell }^{*}\left(J\right))$ are displayed. The minimum value among them corresponding to each $J$, which is denoted $G_{\text{min.}}\left(J\right)$, is also shown, as well as its convex envelope $G_{\text{Maxw.}}\left(J\right)$. Panels (a) and (b) contain the same results with significantly different axis ranges, thus making it possible to inspect different aspects of them (see text for an explanation). Figure 1 displays the same results with yet another choice of axis ranges.