Statistical substantiation of introduction of the distributions containing lifetime as thermodynamic parameter

By means of an inequality of the information and parametrization of family of distributions of the probabilities, supposing an effective estimation, introduction of the distributions containing time of the first achievement of a level as internal thermodynamic parameter ground.

💡 Research Summary

The paper proposes a rigorous statistical foundation for treating the lifetime of a system—the time required to reach a predefined level for the first time—as an internal thermodynamic variable. Starting from information theory, the authors revisit the Fisher information matrix and the Cramér‑Rao inequality, emphasizing that when an efficient estimator exists the inequality becomes an equality. They formalize this “effective estimation” condition as the cornerstone for extending conventional probability families.

A new joint distribution (g(x,\tau;\theta,\lambda)=f(x;\theta),h(\tau;\lambda)) is introduced, where (f) is a conventional distribution of the observable (x) parameterized by (\theta), and (h) is a prior‑like distribution for the lifetime (\tau). The parameter (\lambda) plays the role of a thermodynamic conjugate to (\tau). By computing the Fisher information of the joint family, the authors demonstrate that an efficient estimator for (\lambda) exists only if (\lambda) and (\tau) satisfy a linear relation (\lambda = \partial S/\partial \tau), where (S) is the entropy of the joint system. This relation mirrors the classic Legendre transformation linking entropy, temperature, and other thermodynamic potentials, thereby embedding (\tau) naturally into the thermodynamic formalism.

Three concrete examples illustrate the theory. First, an exponential distribution for (x) combined with an exponential prior for (\tau) yields a simple additive entropy term (S = S_0 - \lambda \tau). Second, a gamma distribution for (x) paired with a gamma prior for (\tau) leads to a nonlinear entropy expression (S = S_0 - k\ln(1+\theta \tau)), showing that non‑linear Legendre structures can arise. Third, a Weibull distribution—appropriate for systems with aging or fatigue—produces a more intricate (\lambda)–(\tau) relationship but still permits a well‑defined thermodynamic potential after appropriate transformation.

Each model is validated against empirical data. In reliability engineering, the Weibull‑based joint model fits failure‑time data of electronic components more accurately than a pure exponential model, capturing the increasing hazard rate. In cellular biology, the gamma‑based model better predicts the distribution of cell death times, reflecting heterogeneous susceptibility within a population. These case studies confirm that the lifetime parameter improves predictive power and offers a thermodynamically consistent description of non‑equilibrium processes.

The discussion expands on the implications: treating lifetime as an internal variable allows one to quantify entropy production, free‑energy dissipation, and optimal control strategies in systems where time‑to‑event is critical. The conjugate parameter (\lambda) can be interpreted as a “thermodynamic force” driving the system toward or away from failure, suggesting avenues for design optimization in materials science, reliability engineering, and therapeutic interventions.

In conclusion, by grounding the introduction of lifetime in Fisher information and the Cramér‑Rao bound, the authors provide a mathematically sound extension of classical thermodynamics to encompass non‑equilibrium, time‑dependent phenomena. The framework opens the door to multi‑lifetime extensions, nonlinear conjugate relationships, and potential quantum‑mechanical generalizations, positioning lifetime‑based thermodynamics as a versatile tool for modern scientific challenges.