# QM3 Quantum Matter meets Maths

### Generalized hydrodynamics with dephasing noise

I review recent advances in the development of generalized hydrodynamics, a flexible approach to the out-of-equilibrium dynamics of integrable quantum systems. I explain how this methodology has allowed exact calculations of transport in $1D$ system. Then, I consider the out-of-equilibrium dynamics of an interacting integrable system in the presence of an external dephasing noise. In the limit of large spatial correlation of the noise, we developed an exact description of the dynamics of the system based on a hydrodynamic formulation. This results in an additional term to the standard generalized hydrodynamics theory describing diffusive dynamics in the momentum space of the quasiparticles of the system, with a time- and momentum-dependent diffusion constant. Our analytical predictions are then benchmarked in the classical limit by comparison with a microscopic simulation of the non-linear Schrodinger equation, showing perfect agreement. In the quantum case, our predictions agree with state-of-the-art numerical simulations of the anisotropic Heisenberg spin in the accessible regime of times and with bosonization predictions in the limit of small dephasing times and temperatures.