I have this probability distribution $\rho$ that's showing up in my Hamiltonian. It's showing up in the form

$\int dx \rho(x) \left(\partial_x log(\rho(x))\right)^2$


Have any of you seen something of this form before? Essentially, this is a term in a hydrodynamic Hamiltonian that I am trying to understand at a deeper level. So any insight into this term is appreciated.