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

Abstract This work investigates how correlated environments influence the relative ( $$T_{\mathrm{eff}}^{(r)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>eff</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> ) and center-of-mass ( $$T_{\mathrm{eff}}^{(R)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>eff</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> ) effective temperatures using the Langevin description. The procedure is to consider N -identical “independent” particles, each coupled to its own thermal bath at temperature $$T_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> ( $$i=1,2,\ldots ,N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> ), where correlations among the baths mediate indirectly the coupling between particles. Using Jacobi coordinates, expressions for $$T_{\mathrm{eff}}^{(R)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>eff</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> and $$T_{\mathrm{eff}}^{(r_i)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>eff</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> are derived, and it is shown that $$T_{\mathrm{eff}}^{(R)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>eff</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> 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 $$T_{\mathrm{eff}}^{(R)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>eff</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> , 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.