Full counting statistics after quantum quenches as hydrodynamic fluctuations

The statistics of fluctuations on large regions of space encodes universal properties of many-body systems. At equilibrium, it is described by thermodynamics. However, away from equilibrium such as after quantum quenches, the fundamental principles are more nebulous. In particular, although exact results have been conjectured in integrable models, a correct understanding of the physics is largely missing. In this talk, I will discuss these principles, taking the example of the number of particles within a large interval in one-dimensional interacting systems. These are based on simple hydrodynamic arguments from the theory of ballistically transported fluctuations, and in particular the Euler-scale transport of long-range correlations. This allows to obtain a formula for the full counting statistics (FCS) in terms of thermodynamic and hydrodynamic quantities, whose validity though depends on the structure of hydrodynamic modes. In fermionic-statistics interacting integrable models with a continuum of hydrodynamic modes, such as the Lieb-Liniger model for cold atomic gases, the formula reproduces previous conjectures, but is in fact not exact: more specifically, it gives the correct cumulants up to, including, order 5, while long-range correlations modify higher cumulants. In integrable and non-integrable models with two or less hydrodynamic modes, the formula is expected to give all cumulants.

Ref: Horvath, Doyon, Ruggiero, arXiv:2411.14406 [submitted to PRL]