Abstract In many-body systems, the dynamics is governed, at large scales of space and time, by the hydrodynamic principle of projection onto the conserved densities admitted by the model. This is formalised as local relaxation of fluctuations in the Ballistic Macroscopic Fluctuation Theory, and is a nonlinear version of the Boltzmann-Gibbs principle. We use it to derive a projection formula, expressing n -point connected correlation functions (cumulants) of local observables at different space-time points, in terms of those of conserved densities. This applies in every $$d\ge 1$$ d ≥ 1 spatial dimensions and under the ballistic scaling of space and time, both in and out of equilibrium. It generalises the well-known linear-response principle for 2-point functions. For higher-point functions, one needs to account for nonlinear fluctuations of conserved densities and, correspondingly, higher derivatives of local averages. Using Malyshev’s formula for the cumulant expansion, and keeping the leading order, the result is a nonlinear projection, expressed as a sum of products of correlation functions of conserved densities with equilibrium multivariances as coefficients. The sum is combinatorially organised via certain covers of the set of space-time points, which we call “minimal connected covers”. We use this in order to get general, explicit formulas for two- and three-point functions in stationary states, expressed in terms of thermodynamic and Euler-scale data.
Abstract In many-body systems, the dynamics is governed, at large scales of space and time, by the hydrodynamic principle of projection onto the conserved densities admitted by the model. This is formalised as local relaxation of fluctuations in the Ballistic Macroscopic Fluctuation Theory, and is a nonlinear version of the Boltzmann-Gibbs principle. We use it to derive a projection formula, expressing n -point connected correlation functions (cumulants) of local observables at different space-time points, in terms of those of conserved densities. This applies in every d≥1 d ≥ 1 spatial dimensions and under the ballistic scaling of space and time, both in and out of equilibrium. It generalises the well-known linear-response principle for 2-point functions. For higher-point functions, one needs to account for nonlinear fluctuations of conserved densities and, correspondingly, higher derivatives of local averages. Using Malyshev’s formula for the cumulant expansion, and keeping the leading order, the result is a nonlinear projection, expressed as a sum of products of correlation functions of conserved densities with equilibrium multivariances as coefficients. The sum is combinatorially organised via certain covers of the set of space-time points, which we call “minimal connected covers”. We use this in order to get general, explicit formulas for two- and three-point functions in stationary states, expressed in terms of thermodynamic and Euler-scale data.
Abstract One of the most profound questions of mathematical physics is that of establishing from first principles the hydrodynamic equations in large, isolated, strongly interacting many-body systems. This involves understanding relaxation at long times under reversible dynamics, determining the space of emergent collective degrees of freedom (the ballistic waves), showing that projection occurs onto them, and establishing their dynamics (the hydrodynamic equations). We make progress in these directions, focussing for simplicity on one-dimensional systems. Under a model-independent definition of the complete space of extensive conserved charges, we show that hydrodynamic projection occurs in Euler-scale two-point correlation functions. A fundamental ingredient is a property of relaxation: we establish ergodicity of correlation functions along almost every direction in space and time. We further show that to every extensive conserved charge with a local density is associated a local current and a continuity equation; and that Euler-scale two-point correlation functions of local conserved densities satisfy a hydrodynamic equation. The results are established rigorously within a general framework based on Hilbert spaces of observables. These spaces occur naturally in the C∗ C ∗ algebra description of statistical mechanics by the Gelfand–Naimark–Segal construction. Using Araki’s exponential clustering and the Lieb–Robinson bound, we show that the results hold, for instance, in every nonzero-temperature Gibbs state of short-range quantum spin chains. Many techniques we introduce are generalisable to higher dimensions. This provides a precise and universal theory for the emergence of ballistic waves at the Euler scale and how they propagate within homogeneous, stationary states.
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.
Abstract Recently, a theoretical framework known as ballistic macroscopic fluctuation theory has been developed to study large-scale fluctuations and correlations in many-body systems exhibiting ballistic transport. In this paper, we review this theory in the context of a one-dimensional gas of hard rods. The initial configurations of the rods are sampled from a probability distribution characterized by slowly varying conserved density profiles across space. Beginning from a microscopic description, we first formulate the macroscopic fluctuation theory in terms of the phase-space density of quasiparticles. In the second part, we apply this framework to compute the two-point, two-time correlation functions of the conserved densities in the Euler scaling limit. We derive an explicit expression for the correlation function which not only reveals its inherent symmetries, but is also straightforward to evaluate numerically for a given initial state. Our results also recover known expressions for space-time correlations in equilibrium for the hard-rod gas.
Abstract We propose a general formalism, within large-deviation theory, giving access to the exact statistics of fluctuations of ballistically transported conserved quantities in homogeneous, stationary states. The formalism is expected to apply to any system with an Euler hydrodynamic description, classical or quantum, integrable or not, in or out of equilibrium. We express the exact scaled cumulant generating function (or full counting statistics) for any (quasi-)local conserved quantity in terms of the flux Jacobian. We show that the “extended fluctuation relations” of Bernard and Doyon follow from the linearity of the hydrodynamic equations, forming a marker of “freeness” much like the absence of hydrodynamic diffusion does. We show how an extension of the formalism gives exact exponential behaviours of spatio-temporal two-point functions of twist fields, with applications to order-parameter dynamical correlations in arbitrary homogeneous, stationary state. We explain in what situations the large-deviation principle at the basis of the results fail, and discuss how this connects with nonlinear fluctuating hydrodynamics. Applying the formalism to conformal hydrodynamics, we evaluate the exact cumulants of energy transport in quantum critical systems of arbitrary dimension at low but nonzero temperatures, observing a phase transition for Lorentz boosts at the sound velocity.
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.