Entropy maximization

At very high temperatures, however, the excited state will also be occupied. Entropy maximization requires that both levels be equally populated. The high-temperature limit of the partition function is... 

Bricogne, G. (1993) Direct phase determination by entropy maximization and likelihood ranking status report and perspectives, Acta Cryst., D49, 37-60. 

W Dong, T Baird, JR Fryer, CJ Gilmore, DD McNicol, G Bricogne, DJ Smith, MA O Keefe, S Hovmoller. Electron microscope at 1 A resolution by entropy maximization and likelihood ranking. Nature 355 605-609, 1992. 

The algorithm of entropy maximization is nonlinear, and must therefore be applied iteratively. It is possible to solve for both X(n + 1) and pf n + 1), starting from X(n) and p/n) at each iteration it. The starting values are A(0) % 0, and p/0) equal to the prior density. Achieving convergence involves a two-step process, in which first the y2 — IV constraint is satisfied, and subsequently the entropy S is maximized. In a variation of the method, X is kept fixed at a small value adequate to ensure convergence (Sakata and Sato 1990). 

Collins DM (1982) Electron density images from imperfect data by iterative entropy maximization. Nature 298 49-51... 

The corresponding formulation was made by von Neumann2 for quantum mechanics. This entropy-maximizing (or information-minimizing) principle is the most direct path to the canonical distribution and thus to the whole equilibrium theory. It is understood that the extremalizing is conditional, i.e., certain expected values, such as that of the Hamiltonian, are fixed. 

DUALITY OF FUNDAMENTAL EQUATIONS ENTROPY MAXIMIZATION VERSUS ENERGY MINIMIZATION... 

However, these potentials do not yet express the second law in the form most convenient for chemical applications. Open laboratory vessels exposed to the temperature and pressure of the surroundings are subject neither to constraints of isolation (as required for entropy maximization) nor to adiabatic constant-volume conditions (as required for energy minimization). Hence, we seek alternative thermodynamic potentials that express the criteria for equilibrium under more general conditions. 

The Gibbs conditions of total entropy maximization under isolation conditions are (following the example of Section 5.2)... 

In essence, we wish to follow the Gibbs entropy maximization procedure of Section 5.2 backward in time. Specifically, we seek to characterize the final stage of equilibration when effective local equilibrium has been achieved in each small cell n, but cell intensities are not yet equalized throughout the system. To make direct contact with Section 5.2, we shall temporarily revert to the entropy representation (Section 5.3), which generates a scalar product and metric geometry that is conformally equivalent to the (/-based metric... 

In equilibrium thermodynamics, entropy maximization for a system with fixed internal energy determines equilibrium. Entropy increase plays a large role in irreversible thermodynamics. If each of the reference cells were an isolated system, the right-hand side of Eq. 2.4 could only increase in a kinetic process. However, because energy, heat, and mass may flow between cells during kinetic processes, they cannot be treated as isolated systems, and application of the second law must be generalized to the system of interacting cells. 

Hence, the extremum L(x) is the point of minimum. Thus, the problem of entropy maximization is transformed into the problem of heat minimization and the Kirchhoff and Prigogine theorems result from the extension of the second law to the passive isothermal circuits. The graphical interpretation of problem (21) is given in Figure 3b. 

Note that the coordinatewise optimization method has already found numerous practical applications to optimization of heat, oil, water, and gas supply systems (Merenkov and Khasilev, 1985 Merenkov et al., 1992 Sumarokov, 1976). As a matter of fact, in the algorithms used for applied problems the flow distribution was calculated not on the base of entropy maximization, but with the help of the closed system of equations of the first and second Kirchhoff laws. However, because of equivalence of approaches that are based on the principle of conservation and equilibrium (extremality) the Kirchhoff equations can be strictly replaced by thermodynamic relations. And the extreme thermodynamic approach in many cases should be preferable owing to the known low sensitivity of the extremal methods to variation of the space of variables. 

Notice that this information approach to Levy statistics is even more direct than the nonextensive thermodynamic approach. As shown in Ref. 52, the adoption of the method of entropy maximization, with the Shannon entropy replaced by the Tsallis entropy [53], does not yield directly the Levy distribution, but a probability density function n(x) whereby reiterated application of the convolution generates the stable Levy distribution. 

Entropy Maximization and the Dishonest Die In this problem we will examine the way in which maximum entropy can be used to confront a problem in incomplete information. The basic idea is that we are told that the mean value that emerges after many rolls of a die is 4.5. On this information, we are urged to make a best guess as to the probability of rolling a 1, 2, 3,4, 5 and 6. Apply the principle of maximum entropy in order to deduce the probability p n) by applying the constraint 4.5 = np n). 

Next consider the right-hand chain. There the maximally diverse partitions (for given entropy) maximize the number of rows. If there were n-q elements (boxes) in the first row, then the corresponding maximally diverse partition would have q rows of one box each. 

Xiang, S., Carter Jr., C.W., Bricogne, G. and Gilmore, C.J. (1993) Entropy maximization constrained by solvent flatness a new method for macromolecular phase extension and map improvement, Acta Cryst, D49, 193-212. 

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