Hydrodynamic Projections and the Emergence of Linearised Euler Equations in One-Dimensional Isolated Systems

Benjamin Doyon

doi:10.1007/s00220-022-04310-3

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^* 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.

Highlights

The passage from short-scale, microscopic motion to large-scale, emergent collective behaviours is at the heart of some of the deepest questions in modern theoretical physics
We consider correlation functions in an appropriate limit of large wavelengths and long times, and we show that hydrodynamic projections occur (Theorem 3.2)
We describe the main results of this work, as specialised to the context of quantum spin chains

Summary

Introduction

The passage from short-scale, microscopic motion to large-scale, emergent collective behaviours is at the heart of some of the deepest questions in modern theoretical physics. The strongest effect on any local probe that is far enough away – say a nearby floating leaf – occurs when the surface wave, propagating out of the local disturbance, hits it. The surface wave is an emergent behaviour, with its own, new dynamics. In this case, it is obtained by linear response from the Euler equations with boundary conditions at the surface. In a large class of manybody systems, strong correlations are expected to occur along trajectories associated with the propagation of ballistic, or slowly decaying modes, such as surface water waves or sound waves, and hydrodynamics is their emergent theory [1]

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