Non-statistical thermodynamic optimization

CALPHAD方法通过同时处理相和多相混合物的热力学性质以及相平衡条件的实验数据，建立相的热力学模型。尽管有热力学的味道，该方法实际上是一种非线性最小二乘技术，导致模型参数的统计最优向量以及它们的协方差矩阵。对所获得的解决方案的分析要求输入数据具有一定的统计特征。特别是，有必要将合理的实验误差归因于每一次测量，并确保其内涵与数理统计中采用的一致。在现实中，对实验误差的艰苦分析经常被它们的估计所取代，这些估计反映了提出这些估计的专家的个人意见和经验。与其猜测这种普遍存在的做法是好是坏，不如让我们认识到，它剥夺了“实验误差”的统计学意义。还应该注意的是，在许多当代出版物中，观测的准确性没有得到重视，实验数据表越来越频繁地被将这些数据可视化的数字所取代。扫描和数字化这些插图并不是问题；问题是扫描和数字化导致了在统计上没有意义的数量。它的目的并不是要开始一场绝望的斗争，反对今天传播实验知识的一种值得谴责的方式。基于列昂尼德·V·坎托罗维奇提出的对实验信息进行非统计处理的思想和算法，考虑用一种新的评估技术来增强CALPHAD方法，并成功地用于解决经济问题。新的程序不会产生模型参数的矢量，而是它们的值的一个区域(不一定是凸的)，在这个区域内，人们可以尝试满足额外的标准，例如在相边界上没有倒置的混溶间隙或虚假的拐点。对于这一区域的任何点，它保证了反计算量被限制在与所有实验观测相关的非统计走廊内的“最终误差”内。

CALPHAD method is used for constructing thermodynamic models of phases via a simultaneous treatment of experimental data on the thermodynamic properties of phases and multiphase mixtures, and on the conditions of phase equilibria. Despite a thermodynamic flavour, the method is in fact a non-linear least squares technique resulting in a statistically optimal vector of models' parameters along with their covariance matrix. An analysis of the solution obtained requires that input data possess certain statistical characteristics. In particular, it is necessary to ascribe a reasonable experimental error to each measurement, and to ensure that its connotation conforms to that adopted in mathematical statistics. In reality, a painstaking analysis of experimental errors is often replaced with their estimations reflecting a personal opinion and experience of an expert proposing these estimations. Instead of speculating whether this widespread practice is good or bad, let us realize that it strips the "experimental errors" of their statistical meaning. It should also be noticed that in many contemporary publications, an accuracy of observations is not paid attention, that tables of experimental data are more and more frequently replaced with figures visualizing these data. It is not a problem to scan and digitize these illustrations; the problem is that scanning and digitizing lead to statistically meaningless quantities. It is not intended to commence a hopeless struggle against a reproachable way in which experimental knowledge is disseminated today. It is contemplated to augment the CALPHAD method by a new assessment technique based on the ideas and algorithms of a non-statistical treatment of experimental information formulated by Leonid V. Kantorovich and successfully employed for solving economic problems. The new procedure will yield not the vector of models' parameters, but a region (not necessarily convex) of their values within which one can try to satisfy additional criteria such as an absence of inverted miscibility gaps or false inflection points along phase boundaries. For any point from this region, it guaranteed that back-calculated quantities are confined within a non-statistical corridor of "ultimate errors" associated with all experimental observations.