Entropy conditions

A mathematical proof of the inverse relationship starts from the maximum entropy conditions... 

The theoretical conformational analysis of a molecule, whatever the quantum technique used, provides quantities related to the free molecule at 0°K and within ideal standard entropy conditions. It follows that such results must be compared with experimental results obtained in conditions as close as possible to these. Obviously, any study in the gas phase will be preferable to corresponding ones performed on liquid or solid states. The most suitable experimental approaches will thus be electron diffraction and microwave spectroscopy. 

Joint Entropy Conditional Entropy Mutual Entropy... 

This quantity averages the Shannon entropies conditional on the Gamma and lognormal models, with weights given by their posterior probabilities. In Appendix B, we show that the average entropy is a concave function on the space of probability distributions which is monotone under contractive maps (Sebastiani and... 

When the conjunctive thermodynamically forbidden reaction (in a system with two stoichiometric stepwise reactions) results in a decrease in the system entropy, condition (1.22) determines the top limit rate of the conjugated reaction ... 

Eq. (A.16), however, is just the matrix form of the maximum entropy condition of Eq. (A.l) for. .., A. Thus, Onsager s theory reduces to classical thermodynamics in the statics limit. 

Chemical reactions are possible at simultaneously constant pressure and temperature. Reaction feasibility cannot be determined by the entropy condition. 

Here, we will review basic properties of low-order wave equations that admit shocks, demonstrate that correct entropy conditions follow as direct consequences of high-order derivative terms, and show how artificial viscosity and upstream differencing can lead to errors in modeling important physical quantities and also in describing shock front speed. 

Entropy conditions. Once the high-order model is agreed upon, for example, Equation 13-7 or 13-8, the complete physical description of the problem is self-contained. That is, the entropy conditions one pulls from hats in thermodynamics can be obtained from integration by parts. Let us consider Equation 13-7. For simplicity, we move with the shock speed, so that... 

The existence of truncation errors in finite difference approximations to differential equations is discussed in numerical analysis texts with respect to round-off error and computational instabilities (Roache, 1972 Richtmyer and Morton, 1957), but Lantz (1971) was among the first to address the form of the truncation error as it related to diffusion. Lantz considered a linear, convective, parabolic equation similar to 9u/9t + U 9u/9x = e S u/Sx and differenced it in several ways. He showed that the effective diffusion coefficient was not 8, as one might have suggested analytically, but 8 + 0(Ax, At) (so that the actual diffusion term appearing in computed solutions is the modified coefficient times c2u/9x2) where the 0(Ax,At) truncation errors, being functions of u(x,t), are comparable in magnitude to 8. Because this artificial diffusion necessarily differs from the actual physical model, one would expect that the entropy conditions characteristic of the computed results could likely be fictitious. 

Another approach is also possible we can indeed consider the problem in a mathematical framework suitable for the discontinuous solutions we will look for a solution c(x,t) in the sense of distributions ("weak solutions"), which must be, on the basis of chemical equilibrium, directly that given in the preceding section. It can be shown that, in the new framework, the solution of the equation is no longer unique. Mathematicians use to add a so-called "entropy condition" that makes the selection of the solution with physical meaning [ll]. In the following sections, we will see the correspondence between the approach of mathematicians and that we can follow in thermodynamics. As may seem obvious for a phenomenon that involves strong gradients, the diffusion will be considered as the perturbative phenomenon used for the choice of the "physical" solution [ l]. 

The preceding results allow us to discuss the question of the entropy bound to the shocks. In the entropy condition (9) obtained after (7), let us notice that, in absence of diffusion, the first member is equal to zero when there is no shock. The inequality has interest only for shocks it then appears infinite gradients 9c/9x and, though D goes to zero, we do not know the convergence of the last term in Eq. (7). The only thing we know is its sign. 

The entropy condition (9) also result tions -the condition... 

Remark 5.7 Theorem 5.6 can be proven in greater generality, notably for general bounded disorder variables iv (see Section 5.7). The proof turns out to be less technical under the relative entropy condition in Theorem 5.6. We spell out this condition explicitly here for a < we require... 

Theorem 6.9 Consider the copolymer model, with uj satisfying the standard assumptions. If either there exists C > 0 such that P (]wi > C) = 0 or if uJi is a continuous random variable satisfying the entropy condition of Theorem 5.6, then for every A > 0 there exists c(A) > 0 such that... 

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