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English: This version of Frenkel & Warren's Figure 3 isotherms^[1] is now exact (and discretely-valued) for a 64 binary-spin case, whereas Frenkel & Warren used the weak Stirling-approximation for an infinite (N → +∞) system instead.

From top-left to bottom-left the solid-line isotherms correspond to coldness β ≡ 1/T values above zero in natural units^[2] of {1, 1/2, 1/4, 1/8}[nat/E-unit], while the dotted non-continuous isotherms correspond to coldness values below zero of {-1/8,-1/4,-1/2,-1}[nat/E-unit]. We've not included coldness-values more than 1[nat/E-unit], or less than -1[nat/E-unit], as they result in all spins up or down (i.e. full alignment) for spin-up energy values within the left and right margins of this plot.

In particular there is no vertical infinite-coldness (zero-T) isocontour, because for a finite system 1/T = ΔS/ΔE is always finite. This leads to a simple and robust justification for the 3^rd Law of Thermodynamics that T ≠ 0[K]. Also, the half-up half-down horizontal line (black dashed) at 32 spins-up is the zero-coldness (infinite-T) isocontour where randomness dominates ordered-energy considerations and each state has equal chance of being occupied. This can be approached thermally from the positive T side by contact with a high temperature reservoir, but it is (we think) not so useful for Carnot-cycle construction.

The black arrows correspond to the horizontal adiabatic/isentropic, and curved isothermal, parts of a Carnot-cycle like that discussed in Frenkel & Warren, with isothermal-leg heat flowing in from both negative-temperature (left) and positive-temperature (right) heat-baths, as the magnitude of the external-field (defining the up-spin energy ε) is increased (left) or decreased (right).