A method to investigate acoustic Hawking radiation is proposed, where entanglement entropy and mutual information are measured from the fluctuations of the number of particles. The rate of entropy radiated per one-dimensional (1D) channel is given by
$dot{S}=kappa/12$, where $kappa$ is the sound acceleration on the sonic horizon. This entropy production is accompanied by a corresponding formation of mutual information to ensure the overall conservation of information. The predictions are confirmed using an emph{ab initio} analytical approach in transonic flows of 1D degenerate ideal Fermi fluids.
An hydrodynamic description of a one-dimensional flow of an ideal Fermi fluid is constructed from a semiclassical approximation. For an initially fully degenerate fluid, Euler and continuity hydrodynamic equations are dual to two uncoupled inviscid B
urgers equations. Yet the price for the initial simplicity of the description is paid by the complexity of non-linear instabilities towards possible turbulent evolutions. Nevertheless, it is shown that linear long-wavelength density perturbations on a stationary flow are generically stable. Consequently, linear sound obeys a wave equation with analogy to gravity. The results have applications for ultra-cold atomic gases.
