Large fluctuations of a Kardar-Parisi-Zhang interface on a half line: The height statistics at a shifted point

We consider a stochastic interface $h(x,t)$, described by the $1+1$ Kardar-Parisi-Zhang (KPZ) equation on the half line $x\ge 0$ with the reflecting boundary at $x=0$. The interface is initially flat, $h(x,t=0)=0$. We focus on the short-time probability distribution $\mathcal{P}(H,L,t)$ of the height $H$ of the interface at point $x=L$. Using the optimal fluctuation method, we determine the (Gaussian) body of the distribution and the strongly asymmetric non-Gaussian tails. We find that the slower-decaying tail scales as $-\sqrt{t}ln\mathcal{P}\simeq {\left|H\right|}^{3/2}{f}_{-}(L/\sqrt{\left|H\right|t})$ and calculate the function $f_{-}$ analytically. Remarkably, this tail exhibits a first-order dynamical phase transition at a critical value of $L$, $L_c=0.60223\cdots \sqrt{\left|H\right|t}$. The transition results from a competition between two different fluctuation paths of the system. The faster-decaying tail scales as $-\sqrt{t}ln\mathcal{P}\simeq {\left|H\right|}^{5/2}{f}_{+}(L/\sqrt{\left|H\right|t})$. We evaluate the function $f_{+}$ using a specially developed numerical method which involves solving a nonlinear second-order elliptic equation in Lagrangian coordinates. The faster-decaying tail also involves a sharp transition which occurs at a critical value $L_c\simeq 2\sqrt{2\left|H\right|t}/\pi$. This transition is similar to the one recently found for the KPZ equation on a ring, and we believe that it has the same fractional order, $5/2$. It is smoothed, however, by small diffusion effects.

Figure 1

A schematic of the problem.

Figure 2

A phase diagram of the system in the $(L/\sqrt{\nu t},|\lambda |H/\nu )$ plane. The solid lines and dashed lines denote sharp and smooth transitions, respectively. For sufficiently large $L$ the half-line system behaves as the full-line system. Typical (small) height fluctuations are Gaussian, see Sec. 3. For large negative $H$ the solution involves a static or traveling optimal noise soliton and is described in Sec. 4. For large positive $H$ the solution is approximately inviscid and describes a hydrodynamic collapse of an effective gas cloud, see Sec. 5. Both sharp transition lines are given by $L/\sqrt{t}\sim \sqrt{\left|H\right|}$. “Nonperturbative” denotes intermediate regions where there is no analytical theory.

Figure 3

The half-line action (in units of the full-line action) vs $\ell$ for typical fluctuations.

Figure 4

The optimal path $h(x,t)/H$ (a) and $\rho (x,t)/H$ (b) as described by the linear theory [Eqs. (31) and (21), respectively] for $\ell =1$. The $x$ profiles are shown at rescaled times $t=0$, 0.5, 0.75, 0.85, 0.95, and $t=1$ (from bottom to top) for $h$ and at the same times for $\rho$, except that $t=1$ is replaced by $t=0.999$. Notice the corner singularity of $h(x,t=1)$ at $x=\ell$.

Figure 5

The optimal path $h(x,t)$ (a) and $\rho (x,t)$ (b), described by the static solution for the $\lambda H>0$ tail. The parameters are $H=-100$ and $\ell =7$. Shown are numerical results (solid lines) and analytical predictions of Eqs. (32) and (33) (dotted lines) at indicated times. Upper inset of (a): $h(x,t=0.99)$ inside the boundary layer at $x=\ell +\sqrt{c/2}t$. Lower inset of (a): $h(x,t=0.99)$ inside the boundary layer at $x=0$. Inset of (b): $\rho (x=\ell ,t)$. Clearly, $\rho (x,t)$ does not change in time except during narrow transients near $t=0$ and $t=1$. The numerical and analytical curves are only distinguishable in the insets in (a) and during the short transients in (b). The numerical solution captures the boundary layers of the $h$ profile, unaccounted for by Eq. (35).

Figure 6

The optimal path $h(x,t)$ (a) and $\rho (x,t)$ (b), described by the dynamic solution for the $\lambda H>0$ tail. The parameters are $H=-100$ and $\ell =5$, for which $c_2\simeq 10.68$ and $\tau \simeq 0.44$, see the main text. Shown are numerical results (solid lines) and analytical predictions from Eqs. (45) and (44) (dotted lines) at indicated times. Inset of (a) shows the boundary layer of $h$ at $x=c_{2}t$, captured by the numerical solution at $t=0.6$. Inset of (b) shows the short transients at $t=0$ and $t=\tau$, captured by the numerical solution for $\rho (x=0,t)$. The $\rho$ soliton changes its amplitude in accordance with Eq. (42) as it changes from the static soliton to the traveling one.

Figure 7

The action $s(H\to -\infty ,\ell )$ in units of the full-line action for $H\to -\infty$, $s_{\text{full}}=8\sqrt{2}{\left|H\right|}^{3/2}/3$. The solid line indicates the least action for any $\ell /\sqrt{\left|H\right|}$. The dashed lines are the static and dynamic actions, respectively (see the main text), in the regions where they are not minimal. The static action for very small $\ell /\sqrt{\left|H\right|}$ is not displayed, since the static solution is invalid for $\ell \sim 1/\sqrt{\left|H\right|}$. Evident is a first-order dynamical phase transition at $\ell /\sqrt{\left|H\right|}={\xi }_c=0.602239\cdots$.

Figure 8

The final-time condition (80) for the pressure flow in the Lagrangian coordinate.

Figure 9

The geometry and boundary conditions for the pressure-driven flow in the Lagrangian mass coordinate.

Figure 10

The flow regions of the effective hydrodynamic problem in the Eulerian coordinate for $\ell \simeq 0.876$ (or $t_{★}=0.4$). The solid lines to the left and right of $x=\ell$ are the edges of the compact support of the pressure flow region, $x_{\text{l}}\left(t\right)$ and $x_{\text{r}}\left(t\right)$, respectively. $x_{\text{r}}\left(t\right)$ decreases as a function of time for any $t>0$, while $x_{\text{l}}\left(t\right)$ increases only for $t>t_{★}$, after the gas cloud detaches from $x=0$. The pressure flow region shrinks to zero at $t=1$ as the gas collapses to $x=\ell$. The dashed lines are characteristics of the Hopf equation (38), emanating from the edges of the pressure flow region and carrying with them constant values of the velocity $V(x,t)$ into the Hopf regions. Some of these constant values are indicated.

Figure 11

The numerically found optimal path of the system, corresponding to the $H\gg 1$ tail for $t_{★}=0.4$ (or $\ell =0.876$). Shown are the spatial profiles of $\rho$ (a), $V$ (b), and $h$ (c) at times $t=0,0.25,0.5,0.65$, and 0.8. At $t=t_{★}$ the gas detaches from $x=0$. The regions of pressure flow and Hopf flow are clearly seen in panel (b). As $t$ approaches $\tilde{t}$, the pressure flow solution converges to the (self-similar asymptotic of) full-line solution of Ref. [14]. To remind the reader, $x,t,\rho ,V$, and $h$ scale with $\mathrm{\Lambda }$, as stated in Eq. (59).

Figure 12

Numerical results for $s/s_{\text{full}}$ in the $H\gg 1$ tail as a function of $\ell /\sqrt{H}$. Up to the factor $8\sqrt{2}/15\pi$, $s/s_{\text{full}}$ is the function $f_{+}(\ell /\sqrt{H})$, see Eqs. (64) and (83). Evident is a gradual crossover from $s=s_{\text{full}}/2$ at $\ell =0$ to $s=s_{\text{full}}$ at $\ell ={\ell }_{\text{cr}}\left(H\right)$, and a sharp transition at $\ell ={\ell }_{\text{cr}}\left(H\right)$. For $\ell >{\ell }_{\text{cr}}\left(H\right)$ the action is independent of $\ell$. The numerical value of ${\ell }_{\text{cr}}\left(H\right)/\sqrt{H}$ agrees with $2\sqrt{2}/\pi$ up to less than $0.1\%$.

Figure 13

An example of the two-soliton/two-ramp solution (B2) (a) and (B1) (b) with $N=3$, $c_1=X_1=0$, $c_3=-c_2$, and $X_2=-X_3=c_2\tau$, where $0<\tau <1$. The dashed lines in (a) indicate the nonphysical parts of the solution that are replaced by the trivial solution $h=0$. The boundary layers, where the two solutions match, do not contribute, at leading order, to the action at large $\left|H\right|$.