# B3 Hydrodynamic Equations for Integrable Many-Body Systems

**Project Leaders**

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Prof. Dr. Christian B. Mendl, Prof. Dr. Herbert Spohn

Summary

In 1+1 spacetime dimensions, there are integrable field theories, either quantum or classical, in the sense that their number of local conservation laws is proportional to the system size. Generic examples are the Toda lattice and the Korteweg–de Vries equation, as classical models, and the Lieb–Liniger *δ*-Bose gas and the XYZ chain, as quantum models. Since the early 1970s the study of such integrable systems is

a flourishing field connecting diverse mathematical strands. From the side of condensed matter physics, roughly six years ago, a major advance has been the discovery of a novel theory, now called generalized hydrodynamics (GHD). The basic insight leading to this theory is easily described. For a large, but confined, non-integrable system one expects thermalization in the long-time limit. In sharp contrast, the dynamics of an integrable field theory is highly constrained through conservation laws and one expects convergence to a generalized Gibbs ensemble (GGE), which is a family of statistical states depending on infinitely many parameters. To write down the respective hydrodynamic equations, the required main input are GGE-averaged conserved fields *and* associated GGE-averaged currents.

For the Toda chain, a variational formula for the generalized free energy has been proved recently. One part of the proposal is to use this advance in tackling GGEs for other integrable systems, for example the Calogero–Moser model of classical particles interacting through a 1/sinh^{2} potential. A further line of research is to investigate qualitative properties of GGEs.

The second pillar of the proposal aims at a numerical validation of GHD. Here classical models are preferred because large system sizes and long times are well accessible through molecular dynamics simulations. Such numerical studies of the microscopic dynamics have to be compared with the respective GHD solutions. GHD is an infinite system of hyperbolic conservation laws, which has to be solved numerically by an iterative scheme. Only through numerical techniques the domain of validity for GHD can be tested.