Abstract
Generalized hydrodynamics (GHD) is a large-scale theory for the dynamics of many-body integrable systems. It consists of an infinite set of conservation laws for quasi-particles traveling with effective (“dressed”) velocities that depend on the local state. We show that these equations can be recast into a geometric dynamical problem. They are conservation equations with state-independent quasi-particle velocities, in a space equipped with a family of metrics, parametrized by the quasi-particles' type and speed, that depend on the local state. In the classical hard rod or soliton gas picture, these metrics measure the free length of space as perceived by quasi-particles; in the quantum picture, they weigh space with the density of states available to them. Using this geometric construction, we find a general solution to the initial value problem of GHD, in terms of a set of integral equations where time appears explicitly. These integral equations are solvable by iteration and provide an extremely efficient solution algorithm for GHD.
Highlights
Generalized hydrodynamics (GHD) is a large-scale theory for the dynamics of many-body integrable systems
It was generalized to include inhomogeneous force fields [7]. It is seen as emerging in integrable classical systems such as the hard rod fluid [8,9] and soliton gases [10,11,12,13,14], and a classical molecular dynamics solver has been developed for the general form of GHD [14]
In order to explicitly check that solving the integral equations by iteration coincides with directly solving the GHD equation (7), we focus on the “bump-release problem” studied for instance in [20]
Summary
In the hard rod fluid, the metric has a clear interpretation: it measures the free space available between the rods, a notion that was used in [18] in a derivation of the exact solution to the domain wall initial problem The observation that this generalizes to soliton gases, still described by GHD, suggests the metric construction proposed here. It is worth noting some similarity, in spirit, to Einstein’s theory of general relativity, where currents are conserved in a metric that is determined by the matter content We use this geometric construction in order to provide a system of integral equations that solve the initial-value problem of GHD in full generality.
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