The positivity of the heat capacity is the hallmark of thermal stability of systems in thermodynamic equilibrium. We show that this property remains valid for systems with negative derivative of energy with respect to temperature, as happens to some system described by the microcanonical ensemble. The demonstration rests on considering a trajectory on the Gibbs equilibrium surface, and its projection on the entropy-energy plane. The Gibbs equilibrium surface has the convexity property, but the projection might lack this property, leading to a negative derivative of energy with respect to temperature.

中文翻译：

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微规范集合中的正热容

热容量的正性是系统在热力学平衡状态下热稳定性的标志。我们证明了这种性质对于能量相对于温度具有负导数的系统仍然有效，就像微规范集合描述的某些系统一样。该演示基于考虑吉布斯平衡面上的轨迹及其在熵能平面上的投影。吉布斯平衡表面具有凸性，但投影可能缺少该性质，从而导致能量相对于温度呈负导数。