Analytical measure of temperature for nonlinear dynamical systems

We present an analytical approach for measuring the temperature of nonlinear dynamical systems in the microcanonical ensemble. Via the self-consistent phonon theory, one can analytically obtain the temperature with respect to the internal energy density in a canonical way. We show how that provides a measure of temperature in the microcanonical ensemble, under the hypothesis of ensemble equivalence. Two models, the FPU-β and ϕ^{4} lattices, are studied obtaining results consistent with those derived from time averages along trajectories in the phase space. Furthermore, our approach is corroborated by the fact that temperature obtained in terms of the average energy density after thermalization agrees with the thermostat temperature. The hypothesis is validated via examining the energy distribution for different numbers of particles in the canonical ensemble. Further, we have quantified the corresponding finite size effects. Unlike other existing methods, which require time-consuming computations, our analytical approach performance improves with the number of particles.