Maximum entropy formalism

Keck, J. C. (1978). Rate-controlled constrained equilibrium method for treating reactions in complex systems. In Maximum Entropy Formalism" (R. D. Levine and M. Tribus, eds). M.I.T. Press, Cambridge, MA. 

Bricogne, G. (1988) A Bayesian statistical theory ofthe phase problem. I. A multichannel maximum-entropy formalism for constructing generalized joint probability distributions of structure factors, Acta Cryst., A44, 517-545. 

It should be indicated that a probability density function has been derived on the basis of maximum entropy formalism for the prediction of droplet size distribution in a spray resulting from the breakup of a liquid sheet)432 The physics of the breakup process is described by simple conservation constraints for mass, momentum, surface energy, and kinetic energy. The predicted, most probable distribution, i.e., maximum entropy distribution, agrees very well with corresponding empirical distributions, particularly the Rosin-Rammler distribution. Although the maximum entropy distribution is considered as an ideal case, the approach used to derive it provides a framework for studying more complex distributions. 

Evans, R.B. "A New Approach for Deciding Upon Constraints in the Maximum Entropy Formalism," The Maximum Entropy Formalism. M.I.T. Press, Cmbridge, MA, 1978, pp.169-203. 

Jaynes, E. T., Tribus, M."The Maximum Entropy Formalism". MIT Univ. Press, Cambridge (1978). 

We can describe irreversibility by using the kinetic theory relationships in maximum entropy formalism, and obtain kinetic equations for both dilute and dense fluids. A derivation of the second law, which states that the entropy production must be positive in any irreversible process, appears within the framework of the kinetic theory. This is known as Boltzmann s H-theorem. Both conservation laws and transport coefficient expressions can be obtained via the generalized maximum entropy approach. Thermodynamic and kinetic approaches can be used to determine the values of transport coefficients in mixtures and in the experimental validation of Onsager s reciprocal relations. 

Comparison of the results of quantum and wave-packet 539 calculations for the photodissociation of CH3I with results obtained with a simplified semiclassical approach Maximum entropy formalism applied to MPD. Applica- 540 tion to alkyl iodides... 

A study has been described of the real-time multiphoton ionization detection of iodine atoms produced by i.r. multiphoton dissociation of perfluoroalkyl iodides,and the relative importance of different possible fragmentation pathways in multiphoton ionization fragmentation of some alkyl iodides has been determined by a maximum entropy formalism. ... 

The point about the technique of cluster impact is that it enables one to heat the cluster on a time scale short compared to that needed for expansion and ipso facto for evaporation. In this way one can prepare superheated clusters with enough energy for breaking most or all intermolecular bonds so that the cluster shatters into its constituents. The sharp transition from evaporation regime to the shattering one, shown in this section, indicate a fast translational thermalization of the extreme disequilibrium formed immediately after the impact with the surface. The need for only two constraints in the maximum entropy formalism (conservation of matter and total energy) to predict the experimental results on the fragment size... 

Fig. 35. Two characterizations of the facile formation of new bonds in superheated clusters containing N2/O2. Top panel The yield of NO, computed by the maximum entropy formalism, when a cluster of 30 N2 and 30 O2 molecules is superheated. The velocity scale is that which is needed to heat the cluster if there is no energy dissipation to the surface. The experimental impact velocity is necessarily equal or higher than this velocity. Bottom panel The yield of new bonds (as a fraction of the number of initial bonds) computed by a molecular dynamics simulation when a cluster of 7 N2 and 7 O2 molecules, embedded in 97 Ar atoms, impacts on a cold (30 K) surface, vs. the impact velocity, previously discussed in Sec. 3.4.

Fig. 36. The yield of N2O molecules computed by the maximum entropy formalism for a cluster of 30 N2 and 30 O2 molecules. The velocity scale is the same as in Fig. 35 top panel. Far fewer N2O molecules are found in the molecular dynamics simulation, see text.

The papers have been published by The M.I.T. Press under the title, "The Maximum Entropy Formalism," 1978. 

Keck, James C. Rate Controlled Constrained Equilibrium Method of Treating Reactions in Complex Systems Maximize the Entropy Subject to Constraints, proceedings of the conference on Maximum Entropy Formalism, M.I.T. Press, 1979. 

In Section III we introduce the maximum entropy formalism.37 Results such as (2.17) and others below can then be derived within a unified framework. The price of a more general approach is that it is more cumbersome and hence it is delayed to the end. 

R. D. Levine, The Theory and Practice of the Maximum Entropy Formalism in Maximum Entropy and Bayesian Methods in Applied Statistics (J. H. Justice, ed.). Cambridge University Press, Cambridge, 1986. 

In this section we present a point of view that provides a convenient parametrization of the spectrum and apply it to discuss three aspects (i) the different frequency scales in the spectra and the interpretation in terms of the sequential exploration of phase space, (ii) the fluctuation of spectral intensities, and (iii) the extraction of the time crosscorrelation function from observed Raman excitation profiles. For the convenience of the reader we first summarize the practical results so that one can skip Sec. IV.A.1-3 and proceed directly to the applications in Sec. IV.B. The derivation of the parametrization, in Sec. IV.A.2, uses the maximal entropy formalism in the manner of (101) and further details can be found in (102,103). Two extensive reviews are (104,105). Section IV.A.l is a brief discussion of why we use the maximum entropy formalism. The discussion is brief and cannot replace the more detailed presentations available (e.g., (55,101,107,108)) in the literature. 

Any other dynamical information derivable using only the one-photon spectrum for a cw source will equally depend only on the transition intensities and hence invariably will lead to Gaussian uncorrelated fluctuations, of the type of Eq. (72), when used in a maximum entropy formalism. 

The final application we discuss is one where the maximum entropy formalism is used not only to fit the spectrum but also to extract new results. Specifically we discuss the determination of the time cross-correlation function, Cf, t) (Eq. (43)), which is the Fourier transform of the Raman scattering amplitude a/((Tu) (Eq. (44)) when what is measured is the Raman scattering cross section afi(m) a/((Tii) 2. The problem is that the experiment does not appear to determine the phase of the amplitude. The application proceeds in two stages (i) Representing the Raman spectrum as one of maximal entropy, using as constraints the Fourier transform of the observed spectrum. At the end of this stage one has a parametrization of a/,( nr) 2 whose accuracy can be determined by how well it fits the observed frequency dependence, (ii) The fact that the Raman spectrum can be written as a square modulus as in Eq. (97) implies that it can be uniquely factorized into a minimum phase function... 

Figure 18 Time cross-correlation functions for three Raman transitions in iodobenzene (from the ground state to the B excited electronic state with v = 1, 2, 3 quanta in the vu vibrational mode. (Left) Computed for a harmonic B state potential and convoluted with a 125-fs-wide window function. The spectrum is computed from this cross-correlation function. (Right) The time correlation function determined from the Raman frequency spectrum (the excitation profile) via the maximum entropy formalism, as discussed in the text, using nine Lagrange multipliers kr. (From Ref. (102).)...

One example is the maximum entropy formalism (MEF), which assiunes the final distribution is that which maximizes the entropy production. A detailed discussion is given in [24]. In general, the method is able to calculate the correct shape of the distribution. However, experimental results are needed to obtain the magnitude. At the current time, no method is predictive without some experimental input [4]. 

Keywords Characteristic drop diameter Cumulative volume fraction Discrete probability function (DPF) Drop size distribution Empirical drop size distribution Log-hyperbolic distribution Log-normal distribution Maximum entropy formalism (MEF) Nukiyama-Tanasawa distribution Number distribution function Probability density function (pdf) Representative diameter Root-normal distribution Rosin-Rammler distribution Upper limit distribution Volume distribution... 

Drop size distributions are typically described using raie of four methods empirical, maximum entropy formalism (MEF), discrete probability function (DPF) method, or stochastic. The empirical method was most popular before about the year 2000, when drop size distributions were usually determined by fitting spray data to predetermined mathematical functions. Problems arose when extrapolating to regimes outside the range of experimental data. Two analytical approaches were proposed to surmount this, MEF and DPF, as well as one numerical approach, the stochastic breakup model. 

Four methods of modeling drop size distributions, empirical, maximum entropy formalism (MEF), Discrete Probability Function (DPF), and stochastic were reviewed. Key conclusions are ... 

C. W. M. van der Geld, H. Vermeer Predictirai of Drop Size Distributions in Sprays Using the Maximum Entropy Formalism The Effect of Satellite FOTmation, Int. J. Multiphase Flow 20 (2). 363-381 (1994). 

J. Cousin, S. J. Yotm, C. Dumouchel Coupling of Classical Linear Theory and Maximum Entropy Formalism for Prediction of Drop Size Distribution in Sprays Applicatiini to Pressure-Swirl Atomizers, Atomizatirai Sprays 6, 601-622 (1996). 

C. Dumouchel A New Formulation of the Maximum Entropy Formalism to Model Liquid Spray Drop-Size Distribution, Part. Part Syst Char. 23,468-479 (2006). 

S. Boyaval, C. Dumouchel, Investigation on the drop size distribution of sprays produced by a high-pressure swirl injector Measurements and application of the maximum entropy formalism Part. Part. Syst. Charact. 18, 33-49 (2001). 

A reconstruction of the orientation distribution function using the Maximum Entropy Formalism (4) reveals that the chlorophyll chromophores are most likely to be oriented with their planes perpendicular to the bilayer plane, see fig.6. 

Levine, R.D. and Tribus, M. (1979) The Maximum Entropy Formalism, MIT Press, Cambridge. 

Often extended nonequilibrium thermodynamics with maximum-entropy formalism leads to more general expression for the entropy not limited to second order in fluxes. 

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