In this talk equilibrium and non-equilibrium formalisms to treat anharmonic systems at high temperatures are described.
In equilibrium case, lattice dynamics at high-temperatures is described by an effective harmonic theory obtained from sampling of the phase space in the canonical ensemble. This could be done self-consistently based on a model anharmonic hamiltonian and is known as the self-consistent phonon theory. In the non-equilibrium case, while previous approaches used the Keldysh formalism, we have derived a simpler classical formalism based on the equation of motion method and Langevin thermostats attached to the central anharmonic device. In contrast to previous results, we find that the leading term which needs to be included is the quartic anharmonic term. Cubic terms are then a second-order correction to the latter, which can either be added perturbatively or self-consistently.Finally, our approach leads to a current-conserving approximation, which is not always guaranteed in non-linear models.