The maximum entropy principle consists of two steps: The first step is to find the distribution which maximizes entropy under given constraints. The second step is to calculate the corresponding thermodynamic quantities. The second part is determined by Lagrange multipliers' relation to the measurable physical quantities as temperature or Helmholtz free energy/free entropy. We show that for a given MaxEnt distribution, the whole class of entropies and constraints leads to the same distribution but generally different thermodynamics. Two simple classes of transformations that preserve the MaxEnt distributions are studied: The first case is a transform of the entropy to an arbitrary increasing function of that entropy. The second case is the transform of the energetic constraint to a combination of the normalization and energetic constraints. We derive group transformations of the Lagrange multipliers corresponding to these transformations and determine their connections to thermodynamic quantities. For each case, we provide a simple example of this transformation.

中文翻译：

---

最大熵原理中MaxEnt分布的标定不变性

最大熵原理包括两个步骤：第一步是找到在给定约束下最大化熵的分布。第二步是计算相应的热力学量。第二部分由拉格朗日乘数与温度或亥姆霍兹自由能/自由熵等可测量物理量的关系确定。我们表明，对于给定的 MaxEnt 分布，熵和约束的整个类别导致相同的分布但通常不同的热力学。研究了保留 MaxEnt 分布的两类简单变换：第一种情况是将熵变换为该熵的任意递增函数。第二种情况是将能量约束转换为归一化和能量约束的组合。我们推导出与这些变换相对应的拉格朗日乘数的群变换，并确定它们与热力学量的联系。对于每种情况，我们都提供了这种转换的简单示例。