Extremum Principles

2019 ◽  
pp. 184-194
Author(s):  
Robert H. Swendsen
Keyword(s):  
Free Energy ◽  
Maximum Principle ◽  
Total Energy ◽  
Composite System ◽  
Minimum Principle ◽  
Maximum Principles ◽  
Energy Minimum ◽  
Total Entropy ◽  
Entropy Maximum ◽  
Internal Constraint

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.

scholarly journals The Equation of State

1993 ◽  
Vol 137 ◽  
pp. 208-221
Author(s):  
Werner Däppen
Keyword(s):  
Free Energy ◽  
Equation Of State ◽  
Traditional Method ◽  
Minimum Principle ◽  
Energy Minimum ◽  
Fundamental Particles ◽  
Diagnostic Potential ◽  
Basic Approaches ◽  
Stellar Envelopes ◽  
Made In

AbstractThere are two basic approaches to the equation of state for stellar envelopes and interiors. The traditional method chooses the so-called “chemical picture”, in which the notion of atoms is maintained despite the plasma environment. A mixture of atoms, molecules, ions, electrons and nuclei is considered, and the occurring ionization and dissociation reactions (thus the name chemical picture) are treated according to the entropy-maximum (or free-energy-minimum) principle. The alternative method is based on the so-called “physical picture”, where only fundamental particles (electrons, nuclei) explicitly enter. Through the means of activity expansions, the problems of plasma physics and statistical mechanics are treated simultaneously and on the same footing. For helio- and asteroseismology, an accurate and precise equation of state is essential. Progress towards a better equation of state can be made in several ways: purely theoretical efforts, checks with experiments, including astrophysical data, and comparisons between different theoretical formalisms. Comparisons are useful to assess the domain of temperature and density where the theoretical complications matter, and to determine the diagnostic potential of astrophysical observables for equation of state issues.

scholarly journals The adsorption of vapours on mercury. I. Non-polar substances

1946 ◽  
Vol 187 (1008) ◽  
pp. 53-73 ◽  
Keyword(s):  
Free Energy ◽  
Total Energy ◽  
Langmuir Equation ◽  
Phase Changes ◽  
Two Dimensional ◽  
Surface Phase ◽  
Standard State ◽  
Spreading Pressure ◽  
Benzene Toluene ◽  
Entropy Of Adsorption

Reversible results for the adsorption of benzene, toluene and n -heptane vapours on mercury have been obtained. The films were found to be gaseous and obeyed the Volmer eqution F ( A - b ) = kT , where F = spreading pressure, A =area per molecule and b = co-area. The possibility that the films might be immobile was considered and the Langmuir equation was applied but found unsatisfactory. A standard state for the surface phase was defined and the free energy, total energy and entropy of adsorption evaluated. The heat of adsorption was shown to increase with the amount on the surface. A number of phase changes were found to occur after the completion of monolayer adsorp­tion, the most striking being interpreted as the change over from ‘flat’ to ‘vertical’ adsorp­tion of the toluene molecules. Others were thought to be either two-dimensional condensation or adsorption of a second layer.

Theoretical Thermodynamic Analysis and Phase Analysis of AZ91D/Flyash Composites

2014 ◽  
Vol 788 ◽  
pp. 604-607
Author(s):  
Hong Chao Chu ◽  
Si Rong Yu ◽  
Cui Xiang Wang ◽  
Qi Lou
Keyword(s):  
Free Energy ◽  
Gibbs Free Energy ◽  
Minimum Principle ◽  
Solution Treatment ◽  
Entropy Change ◽  
Treatment Temperature ◽  
Free Energy Change ◽  
Energy Change ◽  
Solution Treatment Temperature ◽  
Gibbs Free Energy Change

The thermodynamic calculation is valuable for judging the feasibility of a reaction. In the present paper, the enthalpy change (∆HR), entropy change (∆SR) and Gibbs free energy change (∆GR) among various components in AZ91D Mg alloy-Cenosphere composites (FAC/AZ91D) were calculated. Through the calculation, we obtained the relationships between the Gibbs free energy changes and temperatures. The difficulty degree of every potential reaction could be directly reflected by the correlation curve between the temperature and the Gibbs free energy change. The analysis result provided the theoretical basis for the reaction temperature and the solution treatment temperature of the FAC/AZ91D system. At the same time, the analysis based on the minimum principle of the reaction free energy revealed the final components (MgO, Mg2Si and MgAl2O4), which was partially similar to the result of XRD analysis (MgO, Mg2Si and Mg17Al12).

scholarly journals Maximum Entropy Production Theorem for Transitions between Enzyme Functional States and Its Applications

Entropy ◽  
2019 ◽  
Vol 21 (8) ◽  
pp. 743 ◽  
Author(s):  
Davor Juretić ◽  
Juraj Simunić ◽  
Željana Bonačić Lošić
Keyword(s):  
Free Energy ◽  
Entropy Production ◽  
Conformational Changes ◽  
Triosephosphate Isomerase ◽  
Biological Evolution ◽  
Maximal Entropy ◽  
Total Entropy ◽  
Functional States ◽  
Rate Limiting ◽  
Total Entropy Production

Transitions between enzyme functional states are often connected to conformational changes involving electron or proton transport and directional movements of a group of atoms. These microscopic fluxes, resulting in entropy production, are driven by non-equilibrium concentrations of substrates and products. Maximal entropy production exists for any chosen transition, but such a maximal transitional entropy production (MTEP) requirement does not ensure an increase of total entropy production, nor an increase in catalytic performance. We examine when total entropy production increases, together with an increase in the performance of an enzyme or bioenergetic system. The applications of the MTEP theorem for transitions between functional states are described for the triosephosphate isomerase, ATP synthase, for β-lactamases, and for the photochemical cycle of bacteriorhodopsin. The rate-limiting steps can be easily identified as those which are the most efficient in dissipating free-energy gradients and in performing catalysis. The last step in the catalytic cycle is usually associated with the highest free-energy dissipation involving proton nanocurents. This recovery rate-limiting step can be optimized for higher efficiency by using corresponding MTEP requirements. We conclude that biological evolution, leading to increased optimal catalytic efficiency, also accelerated the thermodynamic evolution, the synergistic relationship we named the evolution-coupling hypothesis.

scholarly journals The Maximum Principle for Variable-Order Fractional Diffusion Equations and the Estimates of Higher Variable-Order Fractional Derivatives

2020 ◽  
Vol 8 ◽  
Author(s):  
Guangming Xue ◽  
Funing Lin ◽  
Guangwang Su
Keyword(s):  
Maximum Principle ◽  
Fractional Derivatives ◽  
Arbitrary Point ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Variable Order ◽  
The Maximum Principle ◽  
Fractional Diffusion Equations

In this paper, the maximum principle of variable-order fractional diffusion equations and the estimates of fractional derivatives with higher variable order are investigated. Firstly, we deduce the fractional derivative of a function of higher variable order at an arbitrary point. We also give an estimate of the error. Some important inequalities for fractional derivatives of variable order at arbitrary points and extreme points are presented. Then, the maximum principles of Riesz-Caputo fractional differential equations in terms of the multi-term space-time variable order are proved. Finally, under the initial-boundary value conditions, it is verified via the proposed principle that the solutions are unique, and their continuous dependance holds.

Discrete maximum principles for nonlinear elliptic finite element problems on surfaces with boundary

2018 ◽  
Vol 40 (2) ◽  
pp. 1241-1265 ◽  
Author(s):  
János Karátson ◽  
Balázs Kovács ◽  
Sergey Korotov
Keyword(s):  
Finite Element ◽  
Maximum Principle ◽  
Elliptic Equations ◽  
Real Life ◽  
Maximum Principles ◽  
The Maximum Principle ◽  
Nonlinear Elliptic ◽  
Discrete Analogues ◽  
Second Order Elliptic Equations ◽  
The Given

AbstractThe maximum principle forms an important qualitative property of second-order elliptic equations; therefore, its discrete analogues, the so-called discrete maximum principles (DMPs), have drawn much attention owing to their role in reinforcing the qualitative reliability of the given numerical scheme. In this paper DMPs are established for nonlinear finite element problems on surfaces with boundary, corresponding to the classical pointwise maximum principles on Riemannian manifolds in the spirit of Pucci & Serrin (2007, The Maximum Principle. Springer). Various real-life examples illustrate the scope of the results.

Implications of the entropy maximum principle

1988 ◽  
Vol 56 (6) ◽  
pp. 560-561 ◽  
Author(s):  
E. Kazes ◽  
P. H. Cutler
Keyword(s):  
Maximum Principle ◽  
Entropy Maximum
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