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Reshaping the Science of Reliability with the Entropy Function
* 1 , 2
1  LUISS University
2  University of Perugia

Abstract: The reliability of machineries and the mortality of individuals are topics of great interest for scientists and common people as well. The reliability theory of aging and longevity is a scientific approach aimed to gain theoretical insights into engineering and biology. However the vast majority of researchers make conclusions about population based on information extracted from random samples; in short theorists follow inductive logic. A mature discipline instead complies with the deductive logic, that is to say theorists derive the results from principles and axioms using theorems. After decades of enquiries, it would be desirable that the reliability theory becomes a mature scientific sector in accordance to the style inaugurated by Gnedenko’s seminal book.   The second law of thermodynamics claims that the entropy of an isolated system will increase as the system goes forward in time. This entails – in a way – that physical objects have an inherent tendency towards disorder, and a general predisposition towards decay. Such a wide-spreading process of annihilation hints an intriguing parallel with the decadence of biological and artificial systems to us. The issues in reliability theory are not so far from some issues inquired by thermodynamics and this closeness suggested us to introduce the entropy function for the study of reliable/reparable systems. We consider that the states of the stochastic system S can be more or less reversible and mean to calculate the state Ai of the system S using the Boltzmann-like entropy Hi where Pi is the probability of Ai. Hi = ln (Pi).   We confine our attention to the reliability entropy Hf of the functioning state Af and the recovery entropy Hr of the recovery state Ar  whose meanings can be described as follows. When the functioning state is irreversible, the system S works steadily. In particular, the more Af is irreversible, the more Hf is high and S is reliable. On the other hand, when Hf is low, S often abandons Af and switches to Ar since it fails and we say that S is unreliable. The recovery entropy calculates the irreversibility of the recovery state, this implies that the more Hr is high, the more Ar is stable and in practice S is hard to be repaired and/or cured in the world. In sum Hr expresses the aptitude of S to work or to live without failures; the entropy Hr illustrates the disposition of S toward reparation or restoration to health.   Universal experience brings evidence how the components of the functioning state Af degenerate by time passing and at last impede the correct functioning to S. Thus we assume that the entropy Hfg of the generic component g of the functioning state Af decreases linearly as time goes by; and from this assumption a theorem demonstrates that the hazard rate (or mortality rate) of S is constant with time. When the system is rather old, an endangered part of S can harm to close components and starts a cascade effect while the machine proceeds to run. The cascade effect accelerates the evolution of S toward definitive stop. Now we face two alternative models of system. If the system is linear, one can prove that the hazard rate is power of time. If the system is a mesh, the hazard rate is exponential of time. One can map the reliability entropy Hf with the recovery entropy Hr using the reparability function     This function demonstrates four basic properties of repairable systems. In conclusion, fundamental laws tested in the reliability domain can be deduced from precise assumptions using the Boltzmann-like entropy. The theorems provide deep insights on how systems degenerate. The assumptions make clear the causes of the system failures which instead cannot be justified using usual statistical inference. The present frame seems to be a promising approach for developing a deductive theory of aging integrating mathematical methods with engineering notions and specific biological knowledge.
Keywords: Reliability theory; boltzmann-like entropy