Fig. 3 | Nature Communications

Fig. 3

From: Generalized laws of thermodynamics in the presence of correlations

Fig. 3

Violation of the zeroth law. In the presence of correlations, the notion of equilibrium is not an equivalence relation. Consider 3-party state ρ Bρ AC with all marginals thermal states. The thermal equilibria \({\mathrm{A}} \mathbin{\lower.3ex\hbox{$\buildrel\textstyle\leftharpoonup\over\\ {\smash{\rightharpoondown}}$}} {\mathrm{B}}\) and \({\mathrm{B}} \mathbin{\lower.3ex\hbox{$\buildrel\textstyle\leftharpoonup\over\\ {\smash{\rightharpoondown}}$}} {\mathrm{C}}\) imply that A, B and C share the same temperature. But, in the presence of correlations between A and C, that does not assure the equilibrium \({\mathrm{A}} \mathbin{\lower.3ex\hbox{$\buildrel\textstyle\leftharpoonup\over\\ {\smash{\rightharpoondown}}$}} {\mathrm{C}}\). Therefore, the transitive property of the equivalence relation is violated. This is justified, on the right, as \(F\left( {\rho _{{\mathrm{AC}}}} \right) >F\left( {\rho _{\mathrm{A}} \otimes \rho _C} \right)\). Thus, the generalized zeroth law has to overcome these limitations

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