Nos\'e-Hoover and Langevin thermostats do not reproduce the nonequilibrium behavior of long-range Hamiltonians

We compare simulations performed using the Nos\'e-Hoover and the Langevin thermostats with the Hamiltonian dynamics of a long-range interacting system in contact with a reservoir. We find that while the statistical mechanics equilibrium properties of the system are recovered by all the different methods, the Nos\'e-Hoover and the Langevin thermostats fail in reproducing the nonequilibrium behavior of such Hamiltonian.