Multishocks in driven diffusive processes: Insights from fixed-point analysis of the boundary layers

Boundary-induced phase transitions in a driven diffusive process can be studied through a phase-plane analysis of the boundary-layer equations. In this paper, we generalize this approach further to show how various shapes including multishocks and downward shocks in the bulk particle density profile can be understood by studying the dependence of the fixed points of the boundary-layer equation on an appropriate parameter. This is done for a particular driven interacting particle system as a prototypical example. The present analysis shows the special role of a specific bifurcation of the fixed points in giving rise to different kinds of shocks.

Figure 1

Current $j$ is plotted as a function of $\rho$. Lines a and b represent an upward and a downward shock, respectively.

Figure 2

Qualitative phase diagrams for different values of $r=\epsilon -{\epsilon }_J$. LD represents the low-density phase with the boundary layer at $x=1$. Thick and thin solid lines represent first-order and continuous phase transitions, respectively. Dashed line represents a surface transition line across which the boundary layer near $x=1$ changes from a particle-rich (R) one to a particle-depleted (D) one. $c_1$ and $c_2$ represent critical points. (a) Tricritical point, J. (b) Shaded area represents a region where the density profile has a downward shock. The double-shock phase (2s) appears above the single-shock (1s) phase. The surface transition line is not shown explicitly here. (c) Merging of critical lines (critical points $c_1$ and $c_2$ in $\gamma$-$\alpha$ plane) in the $\gamma$-$r$ plane. The separation between the two critical lines $\Delta \gamma$ vanishes with a specific scaling exponent as $r\to 0$ from below. There is another critical line through J for $r>0$.

Figure 3

$C_0$ plotted as a function of ${\rho }_{1b}$, with $r=-0.2$ and $u=2.2$.

Figure 4

Flow behavior of fixed points.

Figure 5

Fixed points are plotted for different values of $C_0$ with $r=-0.2$ and $u=2.2$. ${\rho }_{1\pm }^{*}$ and ${\rho }_{2\pm }^{*}$ are different fixed points mentioned in the text. Four different parts of the two-lobed curve correspond to four different fixed points ${\rho }_{1\pm }^{*}$ and ${\rho }_{2\pm }^{*}$ as given in Eqs. (18) and (19).

Figure 6

(a) Possible variations of density as one moves along the density profile from the $x=0$ end are shown on the $C_0$-${\rho }_1^{*}$ plane. Dashed lines or curves with open arrows show the variation of the density. Open arrows point in the direction of increasing $x$. Dotted lines mark the boundary conditions ${\alpha }_1$ and ${\gamma }_1$. Solid lines with arrows are the flow trajectories of the fixed points. (b) Numerical solutions for density ${\rho }_1$ for various ${\alpha }_1$ with ${\gamma }_1=0.34$. Inset: A view of the entire density profile. From a zoom-in view of the density profile near $x=1$, it can be seen that the profile has no boundary-layer near $x=1$.

Figure 7

(a) Same as Fig. 6a except that here we explicitly show possible density variations for two different left boundary conditions, specified by ${\alpha }_1$ and ${\alpha }_1^{\prime }$. (b) Numerical solutions for ${\rho }_1(x)$ for various ${\alpha }_1$ with ${\gamma }_1=0.23$. Zoom-in view of boundary layers near $x=0$. Inset: Entire density profile over the entire lattice. (c) Zoom-in view of the same density profiles near $x=1$, showing the particle-depleted boundary layers near $x=1$.

Figure 8

(a) Same as Fig. 6a. (b) Plot of density profiles ${\rho }_1(x)$ for various large negative values of ${\alpha }_1$ with ${\gamma }_1=0.32$. No boundary layer is formed at $x=0$ or $x=1$. A few density profiles toward the right have double shocks.

Figure 9

(a) Trajectory of the density on the $C_0$-${\rho }_1$ plane. Five possible trajectories are shown, distinguished by different numbers. The alphabetical sequence represents the variation of the density along increasing $x$. (b) Plot of density profiles ${\rho }_1(x)$ for various ${\alpha }_1$ values with ${\gamma }_1=-0.28$. (c) Zoom-in views of the boundary layers of the density profiles in (a) near $x=1$.