Mesoscopic density grains in the 1d interacting Bose gas from the exact Yang-Yang solution

Number fluctuations in a one-dimensional Bose gas consist of contributions from many smaller independent localized fluctuations, the density grains. We have derived a set of extended integral equations from the Yang-Yang solution for finite temperature that exactly determine all higher order moments of number fluctuations. These moments are closely related to the statistics of the localized (but not zero-range) density grains. We directly calculate the mean occupation of these fluctuations, and the variance, skewness, and kurtosis of their distribution across the whole parameter space of the gas. Findings include: Large mesoscopic density grains with a fat-tailed distribution in the thermal quasicondensate of the dilute gas and in the nonperturbative quantum turbulent regime; Regions of negative skewness and below-Gaussian kurtosis in a part of the fermionized gas, and an unexplained crossover region along $T\sim T_d/\gamma$; The existence of a peak in the density-density correlation function at finite interparticle spacing. We relate these density grain statistics to measurable behavior such as the statistics of coarse imaging bins, and finite-size scaling of number fluctuations. We propose how to experimentally test the relationship between thermodynamically independent density grains and density concentrations visible in single shot images.