Potential of maximum entropy

Figure 7.10 Potential of maximum entropy (PME) of a Pt(lll) electrode modified by Bi, Pb, Se, and S deposition in 1 mM HCIO4 + 0.1 M KCIO4 solution, as a function of adatom coverage. The dashed, zero-slope line corresponds to the averaged reference PME value of unmodified Pt(lll). The cartoons show the schematic interpretation for the effect of the adatoms at high coverage on the potential transients. (Reprinted with permission from Garcia-Araez et al. [2008].)...

Climent V, Coles BA, Compton RG. 2002b. Coulostatic potential transients induced by laser heating of a Pt(lll) single-crystal electrode in aqueous acid solutions. Rate of hydrogen adsorption and potential of maximum entropy. J Phys Chem B 106 5988-5996. 

Climent V, Garcia-Araez N, Compton RG, Feliu JM. 2006. Effect of deposited bismuth on the potential of maximum entropy of Pt(lll) single-crystal electrodes. J Phys Chem B 110 21092-21100. 

V. WATER REORIENTATION ON SINGLE-CRYSTAL ELECTRODES FROM NANOSECOND LASER-PULSED EXPERIMENTS. POTENTIAL OF MAXIMUM ENTROPY OF DOUBLE-LAYER FORMATION... 

Instead of continuous modulation, single thermal pulses have been imposed by laser beams. This can be seen as a continuation and an expansion of the temperature-jump technique which had been introduced to study kinetics of ionic processes [36]. The method has found application preferably with single-crystal electrodes [65-70]. Many fundamental quantities have been determined, among them the potential of zero charge (Fpzc) of Au(l 11) [65], the potential of maximum entropy [66, 70], the process of hydrogen adsorption at platinum surfaces [67, 68] and the entropy of double-layer formation [69]. This quantity also has been... 

This can be inserted in equation (02.2.3) to give tlie orientational distribution function, and tlius into equation (02.2.6) to deteniiine the orientational order parameters. These are deteniiined self-consistently by variation of tlie interaction strength iin equation (c2.2.7). As pointed out by de Gemies and Frost [20] it is possible to obtain tlie Maier-Saupe potential from a simple variational, maximum entropy metliod based on tlie lowest-order anisotropic distribution function consistent witli a nematic phase. 

The Maximum Entropy (abbreviated MaxEnt) method has been used in the field of accurate charge density studies for some time now (see Section 2.2) it has the potential to overcome some of the limitations of traditional multipolar modelling, but great care must be taken not to apply it outside the range of validity of its own foundations. 

The second law of thermodynamics states that an isolated system in equilibrium has maximum entropy. This is the basis for a variational principle often used in determining the equilibrium state of a system. When the system contains several elements which are allowed to exchange mass with each other, the variational principle yields the condition that all elements must have equal chemical potential once equilibrium is established. 

Recently, Attard [30] proposed a different approach which provides a variational formulation of the electrostatic potential in dielectric continua. His formulation of the free energy functional starts from Equation (1.77), which he justifies using a maximum entropy argument. He defines a fictitious surface charge, s, located on the cavity boundary. The charge s, which produces an electric field /, contributes together with the solute... 

More detailed information can be obtained from noise data analyzed in the frequency domain. Both -> Fourier transformation (FFT) and the Maximum Entropy Method (MEM) have been used to obtain the power spectral density (PSD) of the current and potential noise data [iv]. An advantage of the MEM is that it gives smooth curves, rather than the noisy spectra obtained with the Fourier transform. Taking the square root of the ratio of the PSD of the potential noise to that of the current noise generates the noise impedance spectrum, ZN(f), equivalent to the impedance spectrum obtained by conventional - electrochemical impedance spectroscopy (EIS) for the same frequency bandwidth. The noise impedance can be interpreted using methods common to EIS. A critical comparison of the FFT and MEM methods has been published [iv]. 

Entropy at constant energy and density of Brownian particles has a maximum, is uniform in velocity space, and equal to the thermodynamic potential of a Brownian ideal gas, so we have... 

The success of the maximum entropy procedure to predict the shattering of clusters encourages us to use it in more complicated systems, where very little is known about the potential energy surface. In the next section the results from both molecular dynamics simulations and information theory analysis for clusters made up of N2 and O2 molecules are presented. 

Therefore, novel techniques potentially applicable to solving crystal structures are under continuous testing and development. A recent collective monograph on the structure determination from powder diffraction data provides an excellent discussion of the problem and introduces different approaches that may be used in its solution. In this chapter, unconventional structure solution methods are only briefly reviewed most of them are still controversial and do not always work well with different kinds of compounds and data, although solutions of several complex structures have been demonstrated. Summarized below are the genetic algorithm, maximum entropy, maximum likelihood, and simulated annealing methods. 

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