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Scale invariance usually occurs in extended systems where correlationfunctions decay algebraically in space and/or time. Here we introduce a newtype of scale invariance, occurring in the distribution functions of physicalobservables. At equilibrium these functions decay over a typical scale set bythe temperature, but they can become scale invariant in a sudden quantumquench. We exemplify this effect through the analysis of linear and non-linearquantum oscillators. We find that their distribution functions genericallydiverge logarithmically close to the stable points of the classical dynamics.Our study opens the possibility to address integrability and its breaking indistribution functions, with immediate applications to matter-waveinterferometers.

Scale invariant distribution functions in quantum systems with few degrees of freedom