  The local quench of a Fermi gas, giving rise to the Fermi edge singularity
and the Anderson orthogonality catastrophe, is a rare example of an
analytically tractable out of equilibrium problem in condensed matter. It
describes the universal physics which occurs when a localized scattering
potential is suddenly introduced in a Fermi sea leading to a brutal disturbance
of the quantum state. It has recently been proposed that the effect could be
efficiently simulated in a controlled manner using the tunability of ultra-cold
atoms. In this work, we analyze the quench problem in a gas of trapped
ultra-cold fermions from a thermodynamic perspective using the full statistics
of the so called work distribution. The statistics of work are shown to provide
an accurate insight into the fundamental physics of the process.
