Extremum Principles

This chapter derives the energy minimum principle from the entropy maximum principle. It postulates and consider the consequences of extensivity. From this are further derived minimum principles for the Helmholtz free energy, enthalpy, and Gibbs free energy. Because of its importance in engineering, exergy is also introduced, and the exergy minimum principle is justified. Analogously to these minimum principles, maximum principles can be derived for the Massieu functions from the entropy maximum principle. For the analysis of the entropy maximum principle, we isolated a composite system and released an internal constraint. Since the composite system was isolated, its total energy remained constant. The composite system went to the most probable macroscopic state after release of the internal constraint, and the total entropy went to its maximum.

Classical Gases: Ideal and Otherwise

As preparation for the derivation of the entropy for systems with interacting particles, the position and momentum variables are treated simultaneously, in this chapter, for the ideal gas. Releasing a constraint on the exchange of volume between two systems leads to an entropy maximum, just as the release of an energy- or particle-number constraint. This same principle is shown to be true for asymmetric pistons, which allow the total volume to change. The entropy of systems with interacting particles is then derived. The Second Law of Thermodynamics is established for general systems. Finally, the Zeroth Law of Thermodynamics is derived.

Hybrid-Optional Effectiveness Functions Entropy Conditional Extremization Doctrine Contributions into Engineering Systems Reliability Assessments

In this publication a Doctrine for the Conditional Extremization of the Hybrid-Optional Effectiveness Functions Entropy is discussed as a tool for the Reliability Assessments of Engineering Systems. Traditionally, most of the problems having been dealt with in this area relate with the probabilistic problem settings. Regularly, the optimal solutions are obtained through the probability extremizations. It is shown a possibility of the optimal solutions “derivation”, with the help of a model implementing a variational principle which takes into account objectively existing parameters and components of the Markovian process. The presence of an extremum of the objective state probability is observed and determined on the basis of the proposed Doctrine with taking into account the measure of uncertainty of the hybrid-optional effectiveness functions in the view of their entropy. Such approach resembles the well known Jaynes’ Entropy Maximum Principle from theoretical statistical physics adopted in subjective analysis of active systems as the subjective entropy maximum principle postulating the subjective entropy conditional optimization. The developed herewith Doctrine implies objective characteristics of the process rather than subjective individual’s preferences or choices, as well as the states probabilities maximums are being found without solving a system of ordinary linear differential equations of the first order by Erlang corresponding to the graph of the process. Conducted numerical simulation for the proposed mathematical models is illustrated with the plotted diagrams.

Максимум энтропии в масштабно-инвариантных процессах с 1/f-=SUP=-alpha-=/SUP=--спектром мощности: влияние анизотропии белого шума

Large value fluctuations are modeled by a system of nonlinear stochastic equations describing the interacting phase transitions. Under the action of anisotropic white noise, random processes are formed with the 1/f^alpha dependence of the power spectra on frequency at values of the exponent from 0.7 to 1.7. It is shown that fluctuations with 1/f^alpha power spectra in the studied range of changes correspond to the entropy maximum, which indicates the stability of processes with 1/f^alpha power spectra at different values of the exponent alpha.