# Theory of Faults (ToF): Numerical Quality Management in Complex Systems
## 3 Development
### Entropy
Internal variables (IV), denominated \(\chi_{\alpha}\), are partly unknown; they are the system's local state variables, where \"\(\alpha\)\" is a chosen set of the known variables that is used in modeling efforts to gain an understanding of entropy. The first two laws of thermodynamics written for that \"\(\alpha\)\" set [52] are as follows:  
1. \(d\mathbb{E}=W+Q\), \(W=\sum_{\alpha}A_{\alpha}d\chi_{\alpha}\), \(Q=\sum_{\alpha}B_{\alpha}d\chi_{\alpha}\)
2. \(dS=\frac{\mathrm{d}Q}{\theta}\geq 0,S=S(\chi_{\alpha}),\theta=\theta(\chi_{ \alpha}),\theta>0\)  
In this presentation, \(\mathbb{E}\), \(W\), and \(Q\) are extensive variables; the assumption of extensivity is not correct for interacting statistical systems [53]. The energy \(d\mathbb{E}\) invested in the system generates some work (W), and the rest is dissipated in the system (\(Q\)), translated to the system's extensive state variables [\(\chi_{\alpha}\)]. \(d\mathbb{E}\) is the \"free\" energy; it is related to open faults, added states, and entropy inflation; faults are change _agents_. \(Q\) is the path-related dissipative effect, directly related to the state variables and entropy. These equations portray a transformation of faults to morphology. The second law states that entropy is not limited; there is a systemic minimum to entropy, but there is no maximum to its function. From the above equations, we gather that \(S=S\) (\(m\), \(t\), \(\chi_{\alpha}\)). Entropy and \(\theta\) (Temperature) are thermodynamically conjugated variables of the storage function; a high \(\theta\) enables a reduced translation of free energy to entropy; in our systems, \(\theta\) is related to variance and quality, and entropy to risk, thus the conjugate may hint at a negative correlation between quality and risk. In dissipative systems, Equation (17) is an inequality; a part of the energy influx is dissipated. The nominal magnitude of entropy is an outcome of decisions about the time scale and the traits chosen (\(\alpha\)). It bears little meaning [39]; its relevance is within a system, and even within a system caution is advised because systems evolve, and the relative value of entropy erodes. Entropy trending and entropy change (\(dS\)) are our parameters of interest. Entropy is a real-time means to monitor trendlines, flag bifurcation points, and predict avalanches. In the system transfer from of Experiment A.4, a reduction in entropy is a sign for small-world collapse. In Case Study 1, entropy was calculated for successive network time snapshots to generate entropy trending. Entropy peak and collapse is traced to the collapse of QC. Entropy combined with descriptive statistics is a potent tool for responsibility allocation, and in real time, it is a tool to direct mitigative efforts. Notes: Shannon entropy was invariably used here, although entropy additivity in scale-free networks is contested [54], MaxEntropy yields exponential magnitude probability distributions rather than power functions, and the applicability of generalized entropies is suggested [53]. For example, total entropy trajectory projections cannot be distributed between site and precast subsystem, for precast entropy causes site entropy and its effects are complex (non-extensivity; in Equation (17), \(\mathbb{E}\) is not proportional to the quantity of \(\chi_{\alpha}\)); on the other hand, there may be an over-notation owing to joint entropy between the systems (the non-additive is manifested by the mutual information I: S(A,B) = S(A) + S(B) - I(A;B)); ergodicity breakdown is manifested in Experiment A.4. An analogy to the additivity of historic [accounting] costs may be misleading; while the causes of cost overflow have complex effects, the accounting is not complex. The relationship between entropy and the phase parameter is left for further study. It is noted, though, that the series representation of \(ln\left\{p=ax^{-b}\middle|x>0\right\}\) in Shannon's entropy has a discontinuity in \(x=1\); at the same time, our studies showed continuity for all \(x>0\); a generalized entropy is called for.