Reduced density matrices of a system of N coupled oscillators: 4. The eigenstructure of the p-particle matrices for the 2-temperature canonical ensemble

It is proved that the reduced p-particle matrix (p?N) corresponding to a 2-temperature equilibrium state of a system of N coupled oscillators (the two temperatures T_tr and T_re1 relating to the translational and relative motion of the latter) coincides with the density matrix of a two-temperature canonical ensemble of an effective system of p coupled oscillators, the effective temperature T?_re1 being equal to T_re1 while $\text{T}\text{?}{}_{\mathrm{tr}}\text{≠T}{}_{\mathrm{tr}}$, and in particular $\text{T}\text{?}{}_{\mathrm{tr}}\text{>0}$ for T_tr=0.