Diffusive hydrodynamics from long-range correlations

Abstract

In the hydrodynamic theory, the non-equilibrium dynamics of a many-body system is approximated, at large scales of space and time, by irreversible relaxation to local entropy maximisation. This results in a convective equation corrected by viscous or diffusive terms in a gradient expansion, such as the Navier-Stokes equations. Diffusive terms are evaluated using the Kubo formula, and possibly arising from an emergent noise due to discarded microscopic degrees of freedom. In one dimension of space, diffusive scaling is often broken as noise leads to super-diffusion. But in linearly degenerate hydrodynamics, such as that of integrable models, diffusive behaviors are observed, and it has long be thought that the standard diffusive picture remains valid. In this letter, we show that in such systems, the Navier-Stokes equation breaks down beyond linear response. We demonstrate that diffusive-order corrections do not take the form of a gradient expansion. Instead, they are completely determined by ballistic transport of initial-state fluctuations, and obtained from the non-local two-point correlations recently predicted by the ballistic macroscopic fluctuation theory (BMFT); the resulting hydrodynamic equations are reversible. To do so, we establish a regularised fluctuation theory, putting on a firm basis the recent idea that ballistic transport of initial-state fluctuations determines fluctuations and correlations beyond the Euler scale. This extends the idea of ``diffusion from convection'' previously developed to explain the Kubo formula in integrable systems, to generic non-equilibrium settings.

Figures (2)

• Figure 1: Numerical checks for a gas of hard rods with length $a=0.3$: (a) $1/\ell$ correction to $\partial_t\langle q_2(x,t)\rangle$ evaluated at the macroscopic time $t=1$, as a function of the macroscopic position $x$. The initial state is $\rho(x,\theta,t=0)=\exp[-(\theta+\tanh(x))^2/2\sigma]/\sigma\sqrt{2\pi}$. The solid line represents the hard rods data. More precisely, we measure $\partial_t \mathcal{q}_2(x,t;\ell)\simeq(\mathcal{q}_2(x,t+\Delta t;\ell)-\mathcal{q}_2(x,t-\Delta t;\ell))/2\Delta t$ for $\ell\in\{500,600,...,1000\}$, using $\Delta t=0.05$ and averaging over $8\times 10^{10}$ realizations for each value of $\ell$. Subsequently, at each fixed macroscopic point $x$, we perform a fit of $\partial_t\langle\mathcal{q}_2(x,t;\ell)\rangle$ with model $f(\ell)=f_1+f_2/\ell$. The plot shows the function $f_2(x)$, and the associated error is estimated as the standard deviation of the parameter of the fit. The red (green) dots represent the theoretical prediction of Eq. \ref{['eq:mainresult']} (Eq. \ref{['eq:NS']}). The inset on the bottom right shows the distance between the numerics and the theoretical predictions, normalized with the numerical errors. (b) Regularized correlation $E_{0,1}^{\rm sym}$ at macroscopic time $t=1.0$ between points $x$ and $y=0.7$ in the hard-rod simulations (black dots) compared with the analytical prediction (red line). Note in particular the discontinuity at $x=y$. (c) Dynamics of the momentum density $\mathcal{q}_1(x,t)$ (solid line), compared with the Euler scale GHD (dots).
• Figure 2: (a) Plot for the microscopic correlations in a Hard rods gas, with rods length $a=0.3$ and scale $\ell=200$. Respectively, from left to right, we plot $\langle q_0(x,t)q_1(y,t)\rangle^c_{\rm reg}-\delta(x-y)\langle q_1(x,t)\rangle$ at the points $y=\ell/2$, $t=\ell/10$ and $\ell=200$. The Hard rods data (black line) is compared with prediction from Eq. \ref{['eq:paircorrHR1']} (red line). The Hard rod data are averaged over $10^7$ initial states. Eq. \ref{['eq:paircorrHR1']} has been evaluated truncating the summation at $k=50$, such that the truncation error is $<10^{-16}$. The agreement between the prediction and the numerical data is excellent, up to the Hard rods monte carlo noise. We stress that discrepancies induced by long-range correlations are expected to be order $\mathcal{O}(\ell^{-1})$. (b) Plot of $\langle q_1(x,t)\rangle$ at a time $t=\ell/10$ from Hard rods numerics. The figure (c) shows the microscopic correlations $\langle q_0(x,t)q_1(y,t)\rangle^c_{\rm reg}-\delta(x-y)\langle q_1(x,t)\rangle$ at $y=0$ and $t=10/\ell$. At this point, the leading contribution in $\ell$ is expected to be vanishing, having $\langle q_0(x=0,t=\ell/10)\rangle\simeq 0$ (as shown in box (b)), and since $\langle q_0(x,t)q_1(y,t)\rangle^c_{\rm reg}\rangle\propto\langle q_1(x,t)\rangle+\mathcal{O}(\ell^{-1})$. Hence, this plots shows the microscopic structure of the discontinuity in the two-point correlation. We can conclude that the 'jump' is developed in a length scale $\sim a$.