Plausibility argument

If we wish to know the number of (VpV)-collisions that actually take place in this small time interval, we need to know exactly where each particle is located and then follow the motion of all the particles from time tto time t+ bt. In fact, this is what is done in computer simulated molecular dynamics. We wish to avoid this exact specification of the particle trajectories, and instead carry out a plausible argument for the computation of r To do this, Boltzmann made the following assumption, called the Stosszahlansatz, which we encountered already in the calculation of the mean free path ... 

Though by no means a complete theory, this is at least a reasonable explanation of the Knudsen minimum, and it then remains to explain why the minimum is not observed for flow through porous media. Pollard and Present attributed this to the limited length/diameter ratio of the channels in a typical porous medium and gave a plausible argument in favor of this view. 

This state of affairs is in effect unchanged. But a plausibility argument that relates angular to linear momentum can be given to illuminate the meaning of Lz. 

Obviously an exponential mapping looks also good in the sense of the first criterion. One sees easily that J r) is independent of the choice of j, such that we may as well take 7 = 1. Of course, this is only a plausiblity argument and we need a rigorous criterion for the optimum mapping. We come back to this problem in the conclusions. 

In this section we will approach the question which is at the very heart of density functional theory can we possibly replace the complicated N-electron wave function with its dependence on 3N spatial plus N spin variables by a simpler quantity, such as the electron density After using plausibility arguments to demonstrate that this seems to be a sensible thing to do, we introduce two early realizations of this idea, the Thomas-Fermi model and Slater s approximation of Hartree-Fock exchange defining the X(/ method. The discussion in this chapter will prepare us for the next steps, where we will encounter physically sound reasons why the density is really all we need. 

The authors assume different and statistically independent mechanisms of electron-ion recombination in the quasi-free and trapped states. Thus P = w, where and wt are respectively the probabilities of escaping recombination in the quasi-free and trapped states. Based on some heuristic and not entirely plausible arguments, wq( is approximately equated to 1/2. The probability of finding a trapped electron at a distance between r and r + dr from the positive ion is given by (crtP dr/v) exp(-crPr/v), where P, is again the probability of finding an... 

We are now in a position which allows us to give a microscopic analysis of electrophoresis. However, the detailed calculations rapidly become extremely complicated and we shall often be obliged to replace strict mathematical proofs by physical plausibility arguments. 

For completeness, we show in Fig. 30 the functions gon(R) and gim(-R) for this simulation. In the next section we will discuss the little that can be inferred about these from diffraction data and plausibility arguments. 

We shall present here an intuitively plausible argument on how to construct the probabilities of the various events of our new system, having m independent sites, opened with respect to the ligand. Each site, say j, can be in one of two states empty, with probability Pj 0), and occupied, with probability P,(l). Since the system is at equilibrium with the ligand at some fixed chemical potential, it is reasonable to assume that the probability ratio Pj(l)/Pj(0) is proportional to two factors one that measures the affinity of the site to the ligand, which we denote by q., and the second that depends on the concentration C of the ligand in the reservoir, i.e.. 

The center manifold approach of Mercer and Roberts (see the article Mercer and Roberts, 1990 and the subsequent article by Rosencrans, 1997) allowed to calculate approximations at any order for the original Taylor s model. Even if the error estimate was not obtained, it gives a very plausible argument for the validity of the effective model. This approach was applied to reactive flows in the article by Balakotaiah and Chang (1995). A number of effective models for different Damkohler numbers were obtained. Some generalizations to reactive flows through porous media are in Mauri (1991) and the preliminary results on their mathematical justification are in Allaire and Raphael (2007). 

In classical mechanics, Newton s laws of motion determine the path or time evolution of a particle of mass, m. In quantum mechanics what is the corresponding equation that governs the time evolution of the wave function, F(r, t) Obviously this equation cannot be obtained from classical physics. However, it can be derived using a plausibility argument that is centred on the principle of wave-particle duality. Consider first the case of a free particle travelling in one dimension on which no forces act, that is, it moves in a region of constant potential, V. Then by the conservation of energy... 

From the view point of statistical mechanical theory there are little more than plausible arguments about the universality class of primitive models. The MSA is analytical and thus it yields classical exponents. Therefore, applications to calculate critical exponents or to prove the apparent mean-field criticality of ionic systems [230] are not meaningful [300]. By construction, other integral equation theories also are also not suitable for determining critical exponents [301], The ultimate goal of an RG analysis of the RPM or UPM has not yet been reached. 

PROBLEM 3.1.1. In one dimension "derive," or give a plausibility argument, for the Schrodinger equation by combining the one-dimensional classical wave equation... 

A long-lasting open question in the study of Actinomycetes is why this group of bacteria produce such highly diverged compounds as secondary metabolites. It is well known that some of the compounds exhibit antimicrobial activities, which leads to a plausible argument that the defensive function of these compounds is essential for the life of this kind of bacteria in the natural environment, although this simple idea does not fully explain the marked diversity of structure and activity of the secondary metabolites. 

R. A. Ogg, Jr., J. Chem. Phys. 21, 2079 (1953), has shown that NO and N2O6 catalyze the exchange of labeled O2 with NO2 (no exchange without catalysis) and has presented very plausible arguments for believing the catalysis to take place via the species O—N—O—O and O2N—O—O, respectively ... 

The evaluation of the constant is achieved through the following plausible argument. The probability that a hole has some radius must be unity (i.e., is certain by definition). Equation (5.30) expresses the probability of the radius of the hole lying between r and r + dr. Similarly, one can write down the probabilities of the radius being between r, and Tj + dr, between r2 and + dr, etc. If all these probabilities for r from zero to infinity are summed up (or integrated), then the sum must be unity, i.e.,... 

For 1 = 0 the monopole structure can be determined completely if Boo(h) is known. In fact, the phase problem of reduces to the determination of the sign of F(,(,. This is usually not too difficult a task for Bqo as plausible arguments can be made concerning the corresponding radial mass distribution v fr) of the resonant label atoms. Once the signs of the sinusoidal function BQQ(h) are known for each peak, the phases of Ao(,(h) can be determined directly by using the cross term. 

In Ex. 14.4 a plausibility argument was developedfrom the LLE equilibrium equations to demonstrate that positive deviations from ideal-solution behavior are conducive to liquid/Uquid phase splitting. 

In other words, do the easy splits first. A split is easy if the relative volatility between the two key species is large. The (plausible) argument is that the hard splits should be done when no other species are present. Since the marginal vapor flow computation neither supports nor rejects this heuristic, we might draw the conclusion that this heuristic is not valid for the problem as we have posed it above. But there is some justification for this heuristic when we consider the energy integration of columns (e.g., using the heat expelled from the condenser of one column as the heat input into the reboiler of another). 

We have shown that [Aik] does not change upon addition (or withdrawsil) of CO2. In the literature one finds the plausibility argument if CO2 dissolves in a natural water, the CO2 reacts with CO3 ions CO2 + CO3 -I- H2O = 2 HCOf. Thermodynamically there is the following equilibrium ... 

This is the Schrodinger equation for a particle moving in one dimension. The development provided here is not a derivation of this central equation of quantum mechanics rather, it is a plausibility argument based on the idea that the motions of particles can be described by a wave function with the wavelength of the particle being given by the de Broglie relation. 

The following plausibility argument illustrates the reasoning. There is a 6 given by... 

This is at first sight a plausible argument, but as it stands it has a number of glaring holes. 

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