  Levy distribution, previously used to describe complex behavior of classical
systems, is shown to characterize that of quantum many-body systems. Using two
complimentary approaches, the canonical and grand-canonical formalisms, we
discovered that the momentum profile of a Tonks-Girardeau gas, -- a
one-dimensional gas of $N$ impenetrable (hard-core) bosons, harmonically
confined on a lattice at finite temperatures, obeys Levy distribution. Finally,
we extend our analysis to different confinement setups and demonstrate that the
tunable Levy distribution properly reproduces momentum profiles in
experimentally accessible regions. Our finding allows for calibration of
complex many-body quantum states by using a unique scaling exponent.
