Irreversible evolution towards equilibrium of coupled quantum harmonic oscillators. A coarse grained approach

The dynamics of an integrable linear set of a very great number N (102 to 105) of quantum harmonic oscillators coupled linearly in the rotating wave approximation and subject to the initial condition that only one oscillator is in an excited coherent state whereas the other ones are in the ground state, is numerically studied in the framework of a coarse grained approach. It appears that the coarse grained statistical entropy of this integrable system increases during a certain time characteristic of the system, and then fluctuates around some constant average value which depends on the energy cell width Δε of the analysis. Then, a study of the energy distribution of the oscillators inside these energy cells shows that this distribution is exponentially decreasing as for ergodic systems. The Boltzmann-like parameter B of the exponential distribution is found to fluctuate around a constant value that is inversely proportional to the initial excitation energy. Again, the relative fluctuations of the B parameter of this microcanonical system, appear to be inversely proportional to the square root of N, as for a canonical ensemble. All properties for this system are those of a statistical equilibrium.