Crossover from the macroscopic fluctuation theory to the Kardar-Parisi-Zhang equation controls the large deviations beyond Einstein's diffusion

We study the crossover from the macroscopic fluctuation theory (MFT), which describes one-dimensional stochastic diffusive systems at late times, to the weak noise theory (WNT), which describes the Kardar-Parisi-Zhang (KPZ) equation at early times. We focus on the example of the diffusion in a time-dependent random field, observed in an atypical direction which induces an asymmetry. The crossover is described by a nonlinear system which interpolates between the derivative and the standard nonlinear Schrodinger equations in imaginary time. We solve this system using the inverse scattering method for mixed-time boundary conditions introduced by us to solve the WNT. We obtain the rate function which describes the large deviations of the sample-to-sample fluctuations of the cumulative distribution of the tracer position. It exhibits a crossover as the asymmetry is varied, recovering both MFT and KPZ limits. We sketch how it is consistent with extracting the asymptotics of a Fredholm determinant formula, recently derived for sticky Brownian motions. The crossover mechanism studied here should generalize to a larger class of models described by the MFT. Our results apply to study extremal diffusion beyond Einstein's theory.

Figure 1

For $\xi =(0,1,2,3,4,5)$ we plot the following. (Top left (a)) The derivative rate function ${\mathrm{\Psi }}^{\prime }\left(z\right)$ from Table 1 as a function of $z$, with ${\mathrm{\Psi }}^{\prime }(+\infty )=0$ and ${\mathrm{\Psi }}^{\prime }(-\infty )=1$ (all the branches are shown). For $\xi >{\xi }_1$ and $z\in [z_{c1},z_{c2}]$ the function is multivalued (see text). (Inset) First-order transition: at $z=z^{*}$ such that the areas of the two shaded regions become equal the value of (the optimal) ${\mathrm{\Psi }}^{\prime }\left(z\right)$ jumps from one branch to the other, shown for $\xi =4$. (Top right (b)) The function $z=z\left(H\right)$ from the Legendre transform (35). The reciprocal function $H\left(z\right)$ is multivalued for $\xi >{\xi }_1$ and $z\in [z_{c1},z_{c2}]$. (Bottom left (c)) The large deviation rate function $\mathrm{\Phi }\left(H\right)$ versus $H$, obtained using the parametric representation (35) and Table 1. As $\xi$ increases, the location $H_{\mathrm{typ}}$ of the minimum at $\mathrm{\Phi }\left(H_{\mathrm{typ}}\right)=0$ is shifted toward negative values. (Bottom right (d)) The rate function $\widehat{\mathrm{\Phi }}\left(Z\right)$ versus $Z$. For $\xi =0$, it is symmetric around $Z=0.5$ and one recovers the result of Ref. [40] (in general, the symmetry is $Z\left(\xi \right)\leftrightarrow 1-Z(-\xi )$). For large values $\xi >{\xi }_1$, $\widehat{\mathrm{\Phi }}\left(Z\right)$ develops a concave part which is responsible for the first-order phase transition.

Figure 2

Plot of the Riemann surface of the logarithm $z\mapsto lnz$. This surface is composed of different sheets continuously connected is a staircase manner.

Figure 3

Schematic plot of the argument of the logarithm in Eq. (F5) in the complex $q$ plane for various values of $\tilde{z}$. (Top left) $\tilde{z}⩾-1$, $\tilde{z}\in \mathbb{R}$. (Top right) $\tilde{z}⩽-1$, $\tilde{z}\in \mathbb{R}$. (Bottom) $\Re \left(\tilde{z}\right)<-1$, $\Im \left(\tilde{z}\right)=0^{+}$. The black crosses correspond to the locations where the argument $A_{\mathrm{KPZ}}\left(q\right)$ is zero and the red curves correspond to the branch cuts of $ln{A}_{\mathrm{KPZ}}\left(q\right)$.

Figure 4

Schematic behavior of the zeros $p_i$ of the equation $f_{z,\xi }\left(p_i\right)=0$ as a function of $z<0$ for $\xi >0$. (Top left) The case $\xi <{\xi }_1=\sqrt{8}$ in which case there is a single zero, which changes sign at $z=z_c$. (Top right) The case ${\xi }_1<\xi <{\xi }_2$ in which case there are three zeros $p_1>p_2>p_3$ in the interval $z\in (z_{c1},z_{c2})$ and $z_{c2}<z_c$. (Bottom) The case $\xi >{\xi }_2$, same but now $z_{c2}>z_c$. The zeros $p_2,p_3$ coalesce and annihilate at $z=z_{c2}$ and $p_2,p_3$ at $z=z_{c1}$. The points $z_{c1}$ and $z_{c2}$ also serve as turning points in the definition of the function $\mathrm{\Psi }\left(z\right)$.

Figure 5

Schematic plots of the argument of the logarithm $A\left(q\right)$ given in Eq. (G1) as a function of $q$ in the complex plane for $\xi >0$ and $z<0$. The crosses indicate the positions of the zeros of $A\left(q\right)$ and the red lines are the branch cuts. (Top left) $z_c<z<0$ for all $\xi >0$. No branch cut crosses the real axis. (Top right) $z<z_c$ for $0<\xi <{\xi }_1=\sqrt{8}$, and $z_{c2}<z<z_c$ for ${\xi }_1<\xi <{\xi }_2$. In that case one branch cut crosses the real axis. The integration contour in $q$ in Eq. (G9) can be deformed (dotted lines) to avoid the branch cut. (Bottom left) $z_{c1}<z<z_{c2}$ for ${\xi }_1<\xi$. It is still possible to avoid the branch cut. (Bottom right) $\xi >{\xi }_1$ and $z<z_{c1}$. In that case the branch cuts have met and form a cross, and there is no way to deform the integration contour to avoid them.

Figure 6

Plots for various values of $\xi =0,1,2,3,4,5$. (Left) Large deviation rate function $\mathrm{\Phi }\left(H\right(z\left)\right)$ as a function of $z$ using the definition (35). The function is symmetric for $\xi =0$ and becomes asymmetric for nonzero values of $\xi$. It satisfies the symmetry ${\mathrm{\Phi }\left(H\left(z\right)\right)|}_{-\xi }{=\mathrm{\Phi }\left(H(-z)\right)|}_{\xi }$. (Right) Large deviation rate function $\mathrm{\Psi }\left(z\right)$. The minimum branch of $\mathrm{\Psi }\left(z\right)$ defines the “optimal” solution to the Legendre inversion of Eq. (8). For large negative values of $z$, the function becomes almost linear, i.e., $\mathrm{\Psi }\left(z\right)\simeq z$.

Figure 7

Plot of $\tilde{z}=-z/z_c$ as a function of $H_{\mathrm{KPZ}}=H+\frac{{\xi }^2}{4}+ln\left(\frac{\xi }{2}\right)$ for several values of $\xi =4,5,6$ as compared with the asymptotic $\xi =\infty$ KPZ expression. All branches are represented. The convergence to KPZ is excellent. The last two branches in Table 1 correspond to the sharp decrease to $\tilde{z}\to -\infty$ and is pushed to $H_{\mathrm{KPZ}}=+\infty$ as $\xi =+\infty$. It corresponds to events where $Z\approx 1$ which become irrelevant in that limit.