Expansion of probability density

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. 

For computer simulations, (5.35) leads to accurate estimates of free energies. It is also the basis for higher-order cumulant expansions [20] and applications of Bennett s optimal estimator [21-23], We note that Jarzynski s identity (5.8) follows from (5.35) simply by integration over w because the probability densities are normalized to 1 ... 

Abstract For the case of small matter effects V perturbation theory using e = 2V E/ Am2 as the expansion parameter. We derive simple and physically transparent formulas for the oscillation probabilities in the lowest order in e which are valid for an arbitrary density profile. They can be applied for the solar and supernova neutrinos propagating in matter of the Earth. Using these formulas we study features of averaging of the oscillation effects over the neutrino energy. Sensitivity of these effects to remote (from a detector), d > PE/AE, structures of the density profile is suppressed. 

In "concluding remarks", Lutzky stated that the calculation of C-J T with the help of LSZK equations, assuming cv=0.3cal/g (approx average value for deton products), gave results which were too low at high densities (See Table 2). The reason for this is not known - probably it is due to incompleteness of LSZK theory. In any case, it is believed that in all applications where the ealen of T is not needed, and only an (e, p, v) equation of state is required (such as the calculation of the non-reactive, isentropic expansion of detonation products by means of hydrodynamic computer codes), the LSZK equation of state, in particular ... 

Conclusion. The expansion in atc, which was developed in 3 and 4 for linear equations, can be adapted to nonlinear equations by the simple device of introducing the corresponding Liouville equation. Moreover, this results in an equation for the entire probability density P(w, t) rather than just the average . In fact, the same device can be applied to linear equations as well, if one wants to find the entire distribution, as in the case of (5.14) or to inhomogeneous equations. 

Charge transfer and metal d-d bands are probably buried in the n- n bands. The n- n absorption bands shift bathochromically with electron density increase or expansion of the n system. The effect of 7,8-dehydrogenation is larger than that of 2,3-unsaturation. 

It is noted that the right-hand side of Eq. (10.20) is just the series expansion of an exponential function. Therefore the overall residence time distribution probability density function in the SCISR is obtained to be... 

Table 1. Basic Quantities in Analyses of CW Laser Scattering for Probability Density Function. In Eq. 1 within the table, F(J) is the photon count distribution obtained over a large number of consecutive short periods. For example, F(3) expresses the fraction of periods during which three photons are detected. The PDF, P(x), characterizes the statistical behavior of a fluctuating concentration. Eq. 1 describes the relationship between Fj and P(x) provided that the effects of dead time and detector imperfections such as multiple pulsing can be neglected. In order to simplify notation, the concentration is expressed in terms of the equivalent average number of counts per period, x. The normalized factorial moments and zero moments of the PDF can be shown to be equal by substitution of Eq.l into Eq.2. The relationship between central and zero moments is established by expansion of (x-a)m in Eq.(4). The trial PDF [Eq.(5)] is composed of a sum of k discrete concentration components of amplitude Ak at density xk. [The functions 5 (x-xk) are delta functions.]...

All the aforesaid also applies to the classical limit of the quantum density operator p, namely to the probability density p(0, tensor operators to which the spherical functions Ykq 0, different ways of expansion are used by different authors, both in the quantum approach and in the classical limit. This complicates considerably comparison of the results obtained by these authors, including experimental data. 

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