Threshold effects in the transport of energy from a hot to a cold oscillator: Theory with analog and digital simulation

We study a system consisting of two oscillators coupled to each other through a Hamiltonian interaction. The oscillator of interest and the irrelevant oscillator are also coupled, via rigorous Hamiltonian interactions, to two independent thermal baths at the temperatures T1 and T2, respectively. Since T1<T2, these two oscillators are also termed cold and hot oscillators, respectively. In a preceding short communication [L. Fronzoni and P. Grigolini, Phys. Lett. 106A, 289 (1984)] it was noted that when T1=0, the vanishing effective temperature of the cold oscillator does not change upon increase of T2 until a certain critical value Tc is reached. The threshold T2=Tc is characterized by a critical slowing down. This theory is extended so as to take into account the influence of the finite time scale of the hot oscillator and the effect of the higher-order perturbation terms, both neglected in the preceding short report. This allows us to make predictions also on the statistical behavior of the hot oscillator, i.e., on how its dynamics depends on that of the cold oscillator. It is shown that in the white-noise limit our system is indistinguishable from the van der Pol oscillator. The quantitative determination of the threshold now relies on a more reliable criterion, based on the evaluation of the norm of the energy distribution. The freezing state of the cold oscillator corresponds to an infinite value of this norm and the regime where the heating up of the cold oscillator is made possible means a finite value of it. The analytical expression for the threshold transition from the former to the latter regime is accordingly determined. The role of a weak and spurious additive noise causing the breakdown of a neat threshold is discussed. To get rid of these disturbances, unavoidably affecting analog simulation, an experiment of digital simulation is carried out. The agreement with the theoretical predictions is proven to be excellent.