Equilibrium temperatures of discrete nonlinear systems

During the evolution of discrete nonlinear systems with dynamics dictated by the discrete nonlinear Schr\"odinger equation, two quantities are conserved: system energy (Hamiltonian) and system density (number of particles). It is then possible to analyze system evolution in relation to an energy-density phase diagram. Previous works have identified a ``thermalization zone'' on the phase diagram where regular statistical mechanics methods apply. Based on these statistical mechanics methods we have now assigned a specific equilibrium temperature to every point of the thermalization zone. Temperatures were derived in the grand canonical picture through an entropy-temperature relation, modified to suit the nonlinear lattice systems. Generally, everywhere in the thermalization zone of the phase diagram, temperatures along a fixed system-density line, grow monotonously from zero to infinity. Isotherms on the phase diagram are concave.