Out-of-equilibrium dynamics of repulsive ranked diffusions: The expanding crystal

We study the nonequilibrium Langevin dynamics of $N$ particles in one dimension with Coulomb repulsive linear interactions. This is a dynamical version of the so-called jellium model (without confinement) also known as ranked diffusion. Using a mapping to the Lieb-Liniger model of quantum bosons, we obtain an exact formula for the joint distribution of the positions of the $N$ particles at time $t$, all starting from the origin. A saddle-point analysis shows that the system converges at long time to a linearly expanding crystal. Properly rescaled, this dynamical state resembles the equilibrium crystal in a time-dependent effective quadratic potential. This analogy allows us to study the fluctuations around the perfect crystal, which, to leading order, are Gaussian. There are however deviations from this Gaussian behavior, which embody long-range correlations of purely dynamical origin, characterized by the higher-order cumulants of, e.g., the gaps between the particles, which we calculate exactly. We complement these results using a recent approach by one of us in terms of a noisy Burgers equation. In the large-$N$ limit, the mean density of the gas can be obtained at any time from the solution of a deterministic viscous Burgers equation. This approach provides a quantitative description of the dense regime at shorter times. Our predictions are in good agreement with numerical simulations for finite and large $N$.

Figure 1

(a)–(c) Summary of the values of the dimensionless final time $c^{2}t$ used in (${\mathrm{a}}^{\prime }$)–(${\mathrm{c}}^{\prime }$) and (${\mathrm{a}}^{″}$)–(${\mathrm{c}}^{″}$). (d) Size of the gas as a function of time (in ln-ln scale), which shows a crossover between regimes I and II and time $t_1^{*}=T/{\left(Nc\right)}^2$, which is indicated by a vertical dashed line. Symbols are the results of numerical simulations. The pink solid line at short time represents the prediction in regime I, ${\ell }_T=\sqrt{2Tt}$ (here $T=1$), and the green one at longer time corresponds to the prediction in regime II, $\ell =2Nct$ [see Eq. (1)]. (${\mathrm{a}}^{\prime }$)–(${\mathrm{c}}^{\prime }$) Examples of trajectories $x_i\left(t\right)$ vs $t$ of $N=500$ particles evolving via the Langevin equation (4). One can identify three different regimes determined by the value of the dimensionless time $c^{2}t$. (${\mathrm{a}}^{″}$)–(${\mathrm{c}}^{″}$) Corresponding densities of particles $\rho (x,t_f)$ at the final time $t=t_f$ for each of (a)–(c). They are obtained by averaging over ${10}^4$ realizations of the noise and using 100 bins to construct the histograms. Here, for convenience, we chose $t_f=50$ and varied $c$. These three values fall in each of the three regimes I–III discussed in the text.

Figure 2

Probability distribution $P(y,t)$ of the relative coordinate $y\left(t\right)=x_2\left(t\right)-x_1\left(t\right)$ for two particles for different times (a) $t=5000$, (b) $t=15000$, and (c) $t=50000$. The blue dots are obtained by the simulation with $c=0.01$, $T=1$, and averaging over ${10}^6$ realizations. The orange solid lines represent the analytical expression for the distribution derived in Eq. (14). One can see that the distribution becomes bimodal at long time.

Figure 3

Plot of first four cumulants ${\langle \left(\right|y|-2ct)}^k{\rangle }_c$, with (a) $k=1$, (b) $k=2$, (c) $k=3$, and (d) $k=4$, as a function of dimensionless time $c^{2}t$ (in the scale of the natural logarithm) from the numerical solution of the Langevin equation, compared to the analytical prediction at long time from Eqs. (17, 18, 19, 20) (black dashed line). We chose $c=0.1$ and averaged over $2\times {10}^8$ realizations of the noise.

Figure 4

Plot of the distribution of the centered and scaled gap $\tilde{G}=\frac{G^{\text{RD}}-2ct}{2\sqrt{t}}$ for times $t=500$ and 3000, from the numerical simulation of the Langevin equation (4) for both the midgap $G^{\text{RD}}\equiv G_{\mathrm{mid}\mathrm{gap}}^{\text{RD}}$ and the edge gap $G^{\text{RD}}\equiv G_{\mathrm{edge}\mathrm{gap}}^{\text{RD}}$. Here $N=500$ and $c=0.1$ (note that $c^{2}t$ is the dimensionless time). Shown for comparison is the normal Gaussian distribution $e^{-x^2/2}/\sqrt{2\pi }$ as predicted in (48) and (54) (black solid line). The inset shows $ln\mathcal{P}$ as a function of ${\tilde{G}}^2$ using the same numerical data. The black solid line is the normal Gaussian distribution.

Figure 5

Plot of the density $\rho (x,t)$ evaluated numerically from the Langevin equation, at three different times (a) $t=10$, (b) $t=100$, and (c) $t=200$ for $\gamma =1$ and $T=1$. The initial condition is that all particles are at the origin at time zero. For each time we simulated the density profile for three different combinations of $N$ and $c$ with fixed $\gamma =Nc=1$. The averaging is done over ${10}^4$ realizations of the noise. The numerical data are compared with the analytical prediction with a $\delta$-function initial condition from Eq. (87) (black solid lines).

Figure 6

Comparison of numerical simulation of the boundary layer at the right edge with the result from Eq. (90). In the simulation $\gamma =cN=7$ with $N=1000$, $c=0.007$, and $T=1$. We chose four different times $t\in \{20,50,100,500\}$, which are inside regime II. The averaging is done over $10000$ realizations of the noise and at $t=0$ all particles are at $x=0$.

Figure 7

Rank field $r(x,t)$ computed numerically (dots and crosses) and from the analytical expression (solid lines) for two different initial conditions. Green crosses represent the square initial condition $\rho (x,0)=\frac{1}{2l}\theta (l-|x\left|\right)$ for $l=10$. The numerical data are compared with the analytical expression (88) (green solid line connecting the crosses). The purple dots show the numerical data for the initial condition where all the particles start from the origin. The analytical expression (85) is represented by the purple solid line connecting the circles. The inset shows the pointwise difference between numerical data $r_{\mathrm{num}}(x,t)$ and the analytical expression $r(x,t)$ for both $\delta$ and square initial conditions. The deviations are of order ${10}^{-3}$. For both plots the rank field is measured at $t=10$ with $\gamma =2$, $T=1$, and $N=1000$. Here the data are averaged over ${10}^3$ different realizations, which shows a small systematic deviation, due to the finite-$N$ effects (see Fig. 8).

Figure 8

Fluctuations of the rank field as measured by $\mathrm{\Sigma }(N,t)$ defined in (91). We plotted the numerical data for $\mathrm{\Sigma }$ as a function of $N$ in ln-ln scale for $n=500$ and two different times $t=100$ and 300. We have checked that the number of points $n$ on the grid [see Eq. (91)] does not affect significantly the data. Here we fixed $\gamma =1$ and $T=1$ and we averaged over ${10}^4$ realizations of the noise. At time zero we start from the square initial condition with $l=10$. The dashed lines are guides to the eyes and have slopes of $-\frac{1}{2}$ and $-1$, respectively.