Center of mass and equilibrium temperatures of many-identical-particles coupled via bath cross-correlations

This work investigates how correlated environments influence the relative ( Teff⁽ʳ⁾ T eff ( r ) ) and center-of-mass ( Teff(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 Teff(R) T eff ( R ) and Teff(rᵢ) T eff ( r i ) are derived, and it is shown that Teff(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 Teff(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.