Abstract This work investigates how correlated environments influence the relative (T₄₅₅^ (r) T eff (r) ) and center-of-mass (T₄₅₅^ (R) T eff (R) ) effective temperatures using the Langevin description. The procedure is to consider N -identical “independent” particles, each coupled to its own thermal bath at temperature Tᵢ T i (i=1, 2, , N i = 1, 2, …, N), where correlations among the baths mediate indirectly the coupling between particles. Using Jacobi coordinates, expressions for T₄₅₅^ (R) T eff (R) and T₄₅₅^ (rᵢ) T eff (r i) are derived, and it is shown that T₄₅₅^ (R) T eff (R) can always be identified with the equilibrium temperature obtained from the first law of thermodynamics when N baths are put in thermal contact. The analysis demonstrates that cross correlations between the environments lead to fluctuations effects which are visible in the expression for T₄₅₅^ (R) T eff (R), highlighting how microscopic noise correlations reshape macroscopic equilibration. Consequently, the procedure allows one to obtain the kind of bath cross-correlations between Langevin equations which correctly mimics the correlations involved in the first law of thermodynamics.
Wiggers et al. (Tue,) studied this question.