In the hydrodynamic theory, the nonequilibrium dynamics of a many-body system is approximated, at large scales of space and time, by irreversible relaxation to local entropy maximization. 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 superdiffusion. But in linearly degenerate hydrodynamics, such as that of integrable models, diffusive behaviors are observed, and it has long been 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 nonlocal two-point correlations recently predicted by the ballistic macroscopic fluctuation theory; the resulting hydrodynamic equations are reversible. To do so, we establish a regularized 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 nonequilibrium settings.
In the hydrodynamic theory, the nonequilibrium dynamics of a many-body system is approximated, at large scales of space and time, by irreversible relaxation to local entropy maximization. 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 superdiffusion. But in linearly degenerate hydrodynamics, such as that of integrable models, diffusive behaviors are observed, and it has long been 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 nonlocal two-point correlations recently predicted by the ballistic macroscopic fluctuation theory; the resulting hydrodynamic equations are reversible. To do so, we establish a regularized 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 nonequilibrium settings.
Hydrodynamic noise is the Gaussian process that emerges at larges scales of space and time in many-body systems. It is justified by the central limit theorem, and represents degrees of freedom forgotten when projecting coarse-grained observables onto conserved quantities. It is the basis for fluctuating hydrodynamics, where it appears along with ‘’bare’’ diffusion terms via the Einstein relation. In one dimension of space, nonlinearities may modify the corrections to ballistic behaviours by superdiffusive effects. But in systems where no shocks appear, such as linearly degenerate and integrable systems, it turns out that the diffusive scaling stays intact. Nevertheless, anomalies remain. We show that in such systems, the noise covariance is given by a modification of the Kubo formula, where effects of ballistic long-range correlations – quadratic charges – have been projected out. We further show that nonlinearities are tamed by a point-splitting regularisation. We then obtain a well-defined hydrodynamic fluctuation theory in the ballistic scaling of space-time, as a stochastic PDE. It describes the asymptotic expansion in the inverse variation scale of connected correlation functions, self-consistently organised via a cumulant expansion. The resulting anomalous hydrodynamic equation takes into account both long-range correlations and bare diffusion, generalising recent works. Despite these anomalies, two-point functions satisfy an ordinary diffusion equation, with a normal Kubo formula. In integrable systems, we show that hydrodynamic noise, hence bare diffusion, must vanish, as was conjectured recently, and argue that under an appropriate gauge, this is true at all orders. Thus initial-state fluctuations do not affect coarse-grained currents, and the Ballistic Macroscopic Fluctuation Theory give the all-order hydrodynamic theory for integrable models.
At large scales of space and time, the nonequilibrium dynamics of local observables in extensive many-body systems is well described by hydrodynamics. At the Euler scale, one assumes that each mesoscopic region independently reaches a state of maximal entropy under the constraints given by the available conservation laws. Away from phase transitions, maximal entropy states show exponential correlation decay, and independence of fluid cells might be assumed to subsist during the course of time evolution. We show that this picture is incorrect: under ballistic scaling, regions separated by macroscopic distances "develop long-range correlations as time passes." These correlations take a universal form that only depends on the Euler hydrodynamics of the model. They are rooted in the large-scale motion of interacting fluid modes and are the dominant long-range correlations developing in time from long-wavelength, entropy-maximized states. They require "the presence of interaction" and "at least two different fluid modes" and are of a fundamentally different nature from well-known long-range correlations occurring from diffusive spreading or from quasiparticle excitations produced in far-from-equilibrium quenches. We provide a universal theoretical framework to exactly evaluate them, an adaptation of the macroscopic fluctuation theory to the Euler scale. We verify our exact predictions in the hard-rod gas, by comparing with numerical simulations and finding excellent agreement.
We introduce a new universal framework describing fluctuations and correlations in quantum and classical many-body systems, at the Euler hydrodynamic scale of space and time. The framework adapts the ideas of the conventional macroscopic fluctuation theory (MFT) to systems that support ballistic transport. The resulting “ballistic MFT” (BMFT) is solely based on the Euler hydrodynamics data of the many-body system. Within this framework, mesoscopic observables are classical random variables depending only on the fluctuating conserved densities, and Euler-scale fluctuations are obtained by deterministically transporting thermodynamic fluctuations via the Euler hydrodynamics. Using the BMFT, we show that long-range correlations in space generically develop over time from long-wavelength inhomogeneous initial states in interacting models. This result, which we verify by numerical calculations, challenges the long-held paradigm that at the Euler scale, fluid cells may be considered uncorrelated. We also show that the Gallavotti-Cohen fluctuation theorem for non-equilibrium ballistic transport follows purely from time-reversal invariance of the Euler hydrodynamics. We check the validity of the BMFT by applying it to integrable systems, and in particular the hard-rod gas, with extensive simulations that confirm our analytical results.
Hydrodynamic projections, the projection onto conserved charges representing ballistic propagation of fluid waves, give exact transport results in many-body systems, such as the exact Drude weights. Focussing one one-dimensional systems, I show that this principle can be extended beyond the Euler scale, in particular to the diffusive and superdiffusive scales. By hydrodynamic reduction, Hilbert spaces of observables are constructed that generalise the standard space of conserved densities and describe the finer scales of hydrodynamics. The Green-Kubo formula for the Onsager matrix has a natural expression within the diffusive space. This space is associated with quadratically extensive charges, and projections onto any such charge give generic lower bounds for diffusion. In particular, bilinear expressions in linearly extensive charges lead to explicit diffusion lower bounds calculable from the thermodynamics, and applicable for instance to generic momentum-conserving one-dimensional systems. Bilinear charges are interpreted as covariant derivatives on the manifold of maximal entropy states, and represent the contribution to diffusion from scattering of ballistic waves. An analysis of fractionally extensive charges, combined with clustering properties from the superdiffusion phenomenology, gives lower bounds for superdiffusion exponents. These bounds reproduce the predictions of nonlinear fluctuating hydrodynamics, including the Kardar-Parisi-Zhang exponent 2/3 for sound-like modes, the Levy-distribution exponent 3/5 for heat-like modes, and the full Fibonacci sequence.
Abstract Hydrodynamic projections, the projection onto conserved charges representing ballistic propagation of fluid waves, give exact transport results in many-body systems, such as the exact Drude weights. Focussing one one-dimensional systems, I show that this principle can be extended beyond the Euler scale, in particular to the diffusive and superdiffusive scales. By hydrodynamic reduction, Hilbert spaces of observables are constructed that generalise the standard space of conserved densities and describe the finer scales of hydrodynamics. The Green–Kubo formula for the Onsager matrix has a natural expression within the diffusive space. This space is associated with quadratically extensive charges, and projections onto any such charge give generic lower bounds for diffusion. In particular, bilinear expressions in linearly extensive charges lead to explicit diffusion lower bounds calculable from the thermodynamics, and applicable for instance to generic momentum-conserving one-dimensional systems. Bilinear charges are interpreted as covariant derivatives on the manifold of maximal entropy states, and represent the contribution to diffusion from scattering of ballistic waves. An analysis of fractionally extensive charges, combined with clustering properties from the superdiffusion phenomenology, gives lower bounds for superdiffusion exponents. These bounds reproduce the predictions of nonlinear fluctuating hydrodynamics, including the Kardar–Parisi–Zhang exponent 2/3 for sound-like modes, the Levy-distribution exponent 3/5 for heat-like modes, and the full Fibonacci sequence.
We show that hydrodynamic diffusion is generically present in many-body, one-dimensional interacting quantum and classical integrable models. We extend the recently developed generalized hydrodynamic (GHD) to include terms of Navier-Stokes type, which leads to positive entropy production and diffusive relaxation mechanisms. These terms provide the subleading diffusive corrections to Euler-scale GHD for the large-scale nonequilibrium dynamics of integrable systems, and arise due to two-body scatterings among quasiparticles. We give exact expressions for the diffusion coefficients. Our results apply to a large class of integrable models, including quantum and classical, Galilean and relativistic field theories, chains, and gases in one dimension, such as the Lieb-Liniger model describing cold atom gases and the Heisenberg quantum spin chain. We provide numerical evaluations in the Heisenberg XXZ spin chain, both for the spin diffusion constant, and for the diffusive effects during the melting of a small domain wall of spins, finding excellent agreement with time-dependent density matrix renormalization group numerical simulations.