# Calculations of System Aging through the Stochastic Entropy

## Abstract

**:**

## 1. Introduction

_{i}) is function of the probability P

_{i}of the state A

_{i}

_{i}) = H(P

_{i}) = log

_{e}(P

**)**

_{i}_{i}of the discrete-state stochastic system S. Assuming S in A

_{i}, it may be said in intuitive terms that when S often abandons this state, A

_{i}is somewhat reversible and H(A

_{i}) is 'low'. When the system does not evolve from the state Ai, we say that the state A

_{i}is irreversible, and the stochastic entropy H(A

_{i}) is 'high'.

_{i}) is summable

_{ig}is the entropy of the generic substate (or component) g.

## 2. Aging interpretations

_{f}or the recovery state A

_{r}

_{f}= 1 − P

_{r}

_{f}slops down. This trend implies that the entropy H

_{f}= H(P

_{f}) declines, namely the functioning state becomes reversible and the capability of working of S diminishes in the physical world. The stochastic entropy illustrates that aging consists both of increased failure risk and of low ability of working, and in such a way answer an open riddle. In fact physicians, sociologists and other researches in the biological realm understand aging as the worsening of the performance characteristics of a system (e.g. see biomarkers of aging [6]). At the other side mathematicians interpret aging as the dramatic increase of the hazard rate. These views seem irreconcilable; instead the notion of reversibility/ irreversibility unifies the ‘failure interpretation’ of aging and the ‘performance interpretation’.

## 3. Maturity

- 1) –
- The ability of working of the generic component k decreases at constant rate with the passage of time, as a result of processes inherent in the functional block.
- 2) –
- The more a component is bust and the more the overall system degenerates. The capability of good functioning of the entire system S slops down due to the continuous decay of the components.

- 1.
- The entropy of the generic component k in the state A
_{f}decreases linearly in function of the timeH_{fk}(t) = – λ_{k}t λ_{k}> 0 k=1,2..n_{k}is the time-constant of k. - 2.
- ${H}_{f}(t)=f({H}_{f1},{H}_{f2},...{H}_{fn})$ is a monotonically increasing function of H
_{fk}(k = 1,2,..,n).

_{f}(t) is exponential [7] during the system maturity

## 4. Aging

- 1) –
- The foregoing considerations imply that the entropy ${H}_{f}^{*}$ of S* increases in function of the entropy H
_{fk}of the generic component k. - 2) –
- A component is capable of stopping the entire system. If only one component gives away, the overall system may give away, and this means that if the entropy H
_{fk}of the generic component k reaches the minimum, then the reliability entropy ${H}_{f}^{*}$ of S reaches the minimum value. - 3) –
- The aging mechanism begins when the system is no longer mature. Intuitively we may say that the reciprocal damaging of components can start only when ${H}_{f}^{*}$ has reached a 'certain' level below zero, which is the maximum of ${H}_{f}^{*}$. We assume that aging starts when the reliability entropy is -1, in order to normalize the function ${H}_{f}^{*}$.

- 1.
- ${H}_{f}^{*}=f({H}_{f1},{H}_{f2},...{H}_{fn})$ is a monotonically increasing function of H
_{fk}(k = 1,2,..,n). - 2.
- If ${H}_{fk}=-\infty $, then ${H}_{f}^{*}=-\infty $
- 3.
- If $\left|{H}_{fk}\right|=1$ for all the generic components k, then $\left|{H}_{f}^{*}\right|=1$ (Axiom 3 is said 'normalization axiom').

## 5. Aging of linear systems

_{f}, and we get

## 6. Aging of compound systems

_{fk}

^{+}for the generic component k which causes n chains of inference. We use (10) to calculate each chain and (2) to calculate the overall result

^{+}, thus we calculate the aging function through summation and we prove the theorem is true

_{f}. This assumption also fits with the experience because the speed toward fatal failure of a degeneration chain is indifferent to the number of the chain steps. Thus it is reasonable to take the constant-time of every mechanisms be equal and to write

## 7. Conclusive Remarks

## References

- Gertsbakh, I.B. Statistical Reliability Theory; Marcel Dekker, 1988. [Google Scholar]
- Arking, R. Biology of Aging; Oxford University Press, 2006; (3
^{rd}Ed). [Google Scholar] - Bengtson, V.L. Schaie, K.W., Ed.; Handbook of Theories of Aging; Springer Publishing, 1999. [Google Scholar]
- Nikulin, M. Accelerated Life and Degradation Models in Reliability: State of Art. In Proc. Stochastic Models in Reliability, Safety, Security and Logistics (SMRSSL’05), BeerSheva, 2005; pp. 257–265.
- Rocchi, P. Boltzmann-like Entropy in Reliability Theory. Entropy
**2002**, 4, 142–150. [Google Scholar] [CrossRef] [Green Version] - Schulz, R. (Ed.) Encyclopedia of Aging. Springer Publishing, 2006; (4
^{th}Ed); p. 137. - Rocchi, P. The Stochastic Entropy and the Physical Behavior of the Systems. In Proc. Fifth Intl. Scientific School on Modelling and Analysis of Safety and Risk in Complex Systems, St. Petersurg, 2005; pp. 71–77.
- Rigdon, S.E.; Basu, A.P. Statistical Methods for the Reliability of Repairable Systems; Wiley, 2000. [Google Scholar]

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**MDPI and ACS Style**

Rocchi, P.
Calculations of System Aging through the Stochastic Entropy. *Entropy* **2006**, *8*, 134-142.
https://doi.org/10.3390/e8030134

**AMA Style**

Rocchi P.
Calculations of System Aging through the Stochastic Entropy. *Entropy*. 2006; 8(3):134-142.
https://doi.org/10.3390/e8030134

**Chicago/Turabian Style**

Rocchi, Paolo.
2006. "Calculations of System Aging through the Stochastic Entropy" *Entropy* 8, no. 3: 134-142.
https://doi.org/10.3390/e8030134