Entropy concavity

This is evidently positive (since a2 < 0, the second entropy being concave down). 

Adamantane (47), however, only offers weak van der Waals interactions to the inner concave surface of the host assembly. This interaction, together with the enthalpic gains provided by the formation of 16 hydrogen bonds, is enough to overcome the entropy penalty for bringing together four monomers and one guest. 

The method of coordinatewise optimization was proposed for simultaneous choice of flow rates and pressure losses on the closed redundant schemes (Merenkov and Khasilev, 1985 Merenkov et al., 1992 Sumaro-kov, 1976). According to this method motion to the minimum point of the economic functional F(x, Pbr) is performed alternately along the concave (F(x)) and convex (F(Pbr)) directions. The convex problem is solved by the dynamic programming method and the concave one reduces to calculation of flow distribution. The pressure losses in this case are optimized on the tree obtained as a result of assumed flow shutoff at the end points of some branches. The concave problem is solved on the basis of 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... 

Theorem 1 (Concavity). The average entropy Enta(9) is a concave function of the set of probability distributions for 9. 

Proof. The result follows from the fact that Shannon entropy is concave in the space of probability distribution (DeGroot, 1970), and the average entropy is a convex combination of Shannon entropies. 

Extension and stiffening of the backbone because of the tight adsorption of the side chains are connected with a significant loss of entropy. A certain decrease in the entropy by bending of the backbone without changing the degree of adsorption of the side chains can, however, be enabled by an uneven distribution of the side chains between the two sides of the backbone. On the concave side of the bend, the space available per side chain is increased compared... 

It has been shown that concavity of the entropy is a consequence of the second part of the second principle and extensivity. Then the energy results to be convex, on the assumption that it be a monotonic function of the entropy [9, 10]. A relation of the concavity of the entropy to thermodynamic stability has been established [11]. 

We start with a formal derivation of the equivalency of the minimum principle of energy and maximum principle of entropy and illustrate subsequently the principle with a few examples in back-breaking work. A more detailed treatment based on the concavity of entropy, dealing with thermodynamic potentials, can be found in the literature [16]. 

Prestipino, S., Giaquinta, P.V. The concavity of entropy and extremum principles in thermodynamics. J. Stat. Phys. 111(1-2), 479 93 (2003). URL http //www.springerlink.com/ content/n411802313285077/fuUtext.pdf... 

GENERIC tries to formulate a general time evolution equation by which the time evolution (derivative) of a state variable (which can be, e.g., mass density or fraction, momentum, energy) is determined by two potentials the total energy of the system and a dissipation function. Just the latter one introduces the irreversibility (and, in this way, the thermodynamics ) into consideration and description of the system behavior. The dissipation function or potential is a function of derivatives (with respect to the state variables) of a quantity which should have the physical meaning of the entropy of the system and this latter function is minimum at zero state variables, is zero at zero entropy derivatives just mentioned and a concave function. The general evolution equation can be reformulated by means of Poisson brackets. To apply the GENERIC formalism first one has to select suitable state variables for the problem or system which is to be modeled. The next step is to formulate... 

FIGU RE 3.2 The signature dependence of U on volume and entropy. U versus V and U versus S mandate concave-upward curves. The curves remain above any and all tangent lines that can be drawn. The dotted curves are examples of functions strictly forbidden for stable systems. 

I have proposed [10] that the decreased entropy of the water adjacent to the hydrophobic groups arises from a particular type of H... bonding orientation enforced as a result of the stereochemical fit to the surrounded hydrophobic groups the hydrophobically enforced water structures - being concave with respect to the convexity of the hydrophobic groups - displays toward bulk-water a convex face of externally oriented hydrogens with their positive polarity characteristics see Figure 1. 

Figure 10.2 Schematic form of generic (total, cofigurational, or lattice communal) entropy functions as continuous and concoire functions of E for a fixed volume V for various possible states, and the resulting continuous and concave Helmholtz free energies in the inset. The communal entropy is defined in (10.8). At present, we need notwor aboutthedistinotion between the total and configurational entropies except to note that both of these and the communal entropy on a lattice have similar features. Note the presence of an energy gap...

All we require of the extension of Scomm.dis ( ) in the gap is that the resulting entropy be continuous and concave. 

In most computations, the analytic form of the communal entropy Scomm,dis( ) will be known over the range E > Eq, so the extension is unique. Otherwise, the extended portion of the entropy can be arbitrary provided it does maintain continuity and concavity. This arbitrariness is irrelevant as its values over the range [ o> Tr] will not determine the physics of the system. This will become apparent in Section 10.12. 

We clearly see an energy gap at the lower end of the curve at K gK > 0. For the square lattice, we find that gx ci 0.227 [52], where SFH,dis(g) vanishes with 2i finite (nonzero) slope, as was the case for the communal entropy in Figure 10.2, even though the analytical form of spH.dis (g) derived by Flory is over the full entire range [0,1 ] of g. This entropy is also concave over the entire range, including the gap. As the point H in... 

It should be stressed that the proof does not use the vanishing of Sdis ( Tk) Thus, the equality Fdis(Teq) = o is also valid if Sdis (Tk) > 0. The proofalso does not depend on the entropy slope at r. From the concavity of Sais ( ), it should be obvious that slope at k is larger than 1/Teq. 

Figure 10.11 The free energies and entropies from the two FP solutions. The entropy crisis occurs at Tk, below which Sn (dashed cyan) becomes negative, but the corresponding free energy F) (cyan) remains concave despite an unphysical communal entropy. This explains their labeling as unphysical in the figure. The...

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