Nonphysical solutions

To prevent the optimization procedure from discovering trivial, or nonphysical solutions, the yield must be optimized with respect to a set of constraints. These constraints can take many forms, including details of the experimental apparatus and the physical system [23-30]. 

In addition to physically realizable multiple solutions, there are also nonphysical steady states available. These solutions can be problematic, since they usually appear superficially to be valid solutions. In fact, simply by inspection, it is usually impossible to discern if a solution is physical or not. Nonphysical solutions, however, while available and computable in steady state, are unstable in their transient response. That is, a physically valid... 

From these examples it is apparent that one needs to be cautious when using steady-state methods and continuation procedures near turning points. While the solutions may converge rapidly and even appear to be physically reasonable, there can be significant errors. Fortunately, a relatively simple time-stepping procedure can be used to identify the nonphysical solutions. Beginning from any of the solutions that are shown in Fig. 15.9 as shaded diamonds, a transient stirred-reactor model can be solved. If the initial solution (i.e., initial condition for the transient problem) is nonphysical, the transient procedure will march toward the physical solution. If the initial condition is the physical solution, the transient computational will remain stationary at the correct solution. 

Figure 15-5 gives the shape of an isotherm calculated with Equation 15-14 at a temperature below the critical temperature. Points a through f are equivalent to points a through f on Figures 15-2 and 15-3. Points e are the values of z-factor that would be measured experimentally. Point f is a nonphysical solution. 

The boundary conditions to be used with the Burnett equations have also been determined for a BGK model by Sone, and for more general models by de Wit using variational methods, but in fact this set of boundary conditions is not complete. The Burnett and higher-order hydrodynamic equations have nonphysical solutions showing spatial variations on the length scale of the mean free path. One would like to have boundary conditions that could be used to reject these unphysical solutions. However, the available set of boundary conditions is not sufficient for that. Instead one must postulate that the rapidly varying solutions are absent and then use the available boundary conditions to determine the remaining hydrodynamic solution. 

Force field calculations often truncate the non bonded potential energy of a molecular system at some finite distance. Truncation (nonbonded cutoff) saves computing resources. Also, periodic boxes and boundary conditions require it. However, this approximation is too crude for some calculations. For example, a molecular dynamic simulation with an abruptly truncated potential produces anomalous and nonphysical behavior. One symptom is that the solute (for example, a protein) cools and the solvent (water) heats rapidly. The temperatures of system components then slowly converge until the system appears to be in equilibrium, but it is not. 

Continuum models remove the difficulties associated with the statistical sampling of phase space, but they do so at the cost of losing molecular-level detail. In most continuum models, dynamical properties associated with the solvent and with solute-solvent interactions are replaced by equilibrium averages. Furthermore, the choice of where the primary subsystem ends and the dielectric continuum begins , i.e., the boundary and the shape of the cavity containing the primary subsystem, is ambiguous (since such a boundary is intrinsically nonphysical). Typically this boundary is placed on some sort of van der Waals envelope of either the solute or the solute plus a few key solvent molecules. 

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

Multiple solutions to equations occur whenever they have sufficient nonlinearity. A familiar example is equHihrium composition calculations for other than A B. The reaction composition in the reaction A i B yields a cubic polynomial that has three roots, although all but one give nonphysical concentrations because thermodynamic equilihrium (the solution for a reactor with f —> co or T — co) is unique. 

Schell (1965) recognized that the major deficiency of the Wiener inverse filter is the nonphysical nature of the partially negative solutions that it is prone to generate. He sought to extrapolate the band-limited transform O(co) by seeking a nonnegative physical solution 6 + (x) through minimization of... 

Thus, the author considered forcing the solution to lie within the physical bounds. Besides eliminating the objectionable nonphysical result, this approach, it was hoped, would also improve the accuracy of the solution within the physical bounds. Because of the limited performance evidenced in previous literature available on deconvolution, the author was unprepared for the magnitude of the improvement that resulted. 

With the present definition of r, however, an overcorrection that would normally disappear gradually through ensuing iterations results in a value of d(k)(x) that vanishes for all subsequent iterations. This behavior occurs because further corrections to that value are prohibited. To use the method, the investigator is compelled to take small values for r0. Even in this case, erroneously nonphysical values of o(k) that have been forced to zero are never allowed to return to the finite range that might better represent the true spectrum o(x). This form of the method therefore demands excessive computation and yields a solution that, although physically realizable, is not the best achievable estimate. 

For a basic deconvolution problem involving band-limited data, the trial solution d(0) may be the inverse- or Wiener-filtered estimate y(x) (x) i(x). Application of a typical constraint may involve chopping off the nonphysical parts. Transforming then reveals frequency components beyond the cutoff, which are retained. The new values within the bandpass are discarded and replaced by the previously obtained filtered estimate. The resulting function, comprising the filtered estimate and the new superresolving frequencies, is then inverse transformed, and so forth. 

Fig. 15.9 Steady-state solutions for the benzene mole fraction from the simulation of benzene oxidation near a turning point in a perfectly stirred reactor. Depending on the starting estimates, a number of spurious (nonphysical) solultions may be encountered. The true solution is indicated by the filled circles, while the shaded diamonds indicate (sometimes spurious) solutions that are computed through various continuation sequences.

We focus only on the plus solution, the minus sign giving nonphysical results. Using Eq. (8), the kinetic energy of B relative to C is given as... 

Obviously, such interpetation led to disregard advanced solutions as nonphysical. For instance, Ritz [30] and Tetrode [31] considered that the mathematical existence of advanced solutions was a major weakness of Maxwell s equations. An attempt to provide a physical basis for advanced potentials is due to Lewis, who proposed focusing on the process of propagation from an emitter to an absorber far away from the emitter [32]. This concept also appears in the work of Wheeler and Feynman [33]. However, such model constitutes another form of causality violation. Lewis [32, p. 25] himself stated I shall not attempt to conceal the conflict between these views and common sense. ... 

Several details with respect to implementation of Equations [22] and [23] deserve further discussion. Whereas the approximation of the solute residing in a spherical cavity is clearly of limited utility, since most molecules are not approximately spherical in shape, there is also the issue of the choice of the cavity radius, a. Obvious approaches include (1) recognizing that the spherical cavity approximation is arbitrary and thus treating a as a free parameter to be chosen by empirical rules, and (2) choosing a so that the cavity encompasses either the solvent-accessible van der Waals surface of the solute or the same volume. Wong et al. have advocated a quantum mechanical approach like the last method wherein the van der Waals surface is replaced by an isodensity surface. Because g depends on the third power of a, the calculations are quite sensitive to the radius choice, and some nonphysical results have been reported in the literature when insufficient care was taken in assigning a value to a. Implementations that replace the cavity sphere with an ellipsoid have also appeared. 

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