Classical statistics compared with

Robust system identification and estimation has been an important area of research since the 1990s in order to get more advanced and robust identification and estimation schemes, but it is still in its initial stages compared with the classical identification and estimation methods (Wu and Cinar, 1996). With the classical approach we assume that the measurement errors follow a certain statistical distribution, and all statistical inferences are based on that distribution. However, departures from all ideal distributions, such as outliers, can invalidate these inferences. In robust statistics, rather than assuming an ideal distribution, we construct an estimator that will give unbiased results in the presence of this ideal distribution, but will be insensitive to deviation from ideality to a certain degree (Alburquerque and Biegler, 1996). 

It is often impossible to obtain the quantized energies of a complicated system and therefore the partition function. Fortunately, a classical mechanical description will often suffice. Classical statistical mechanics is valid at sufficiently high temperatures. The classical treatment can be derived as a limiting case of the quantum version for cases where energy differences become small compared with ksT. 

In this chapter we will mostly focus on the application of molecular dynamics simulation technique to understand solvation process in polymers. The organization of this chapter is as follow. In the first few sections the thermodynamics and statistical mechanics of solvation are introduced. In this regards, Flory s theory of polymer solutions has been compared with the classical solution methods for interpretation of experimental data. Very dilute solution of gases in polymers and the methods of calculation of chemical potentials, and hence calculation of Henry s law constants and sorption isotherms of gases in polymers are discussed in Section 11.6.1. The solution of polymers in solvents, solvent effect on equilibrium and dynamics of polymer-size change in solutions, and the solvation structures are described, with the main emphasis on molecular dynamics simulation method to obtain understanding of solvation of nonpolar polymers in nonpolar solvents and that of polar polymers in polar solvents, in Section 11.6.2. Finally, the dynamics of solvation with a short review of the experimental, theoretical, and simulation methods are explained in Section 11.7. 

If g.t the number of available states, is very large compared with tip so that gj = Be kT + a a = oor 1 for the three cases above) is very large compared to 1, then the difference between these three forms disappears, that is to say the classical statistics holds. At otherwise the same conditions, that is to say, with the same value of z kT, this will be the case for electrons only at extremely high temperatures, on account of the very small mass m in fact B is always very much smaller for electrons than for atoms. Thus the classical statistical laws, such as the law of equipartition from which the above-mentioned extra specific heat was derived, do not hold for electrons this will only be the case at very much higher temperatures. Thus i° K for H2 molecules corresponds to a temperature of 224000° K for electrons, both with s = kT. 

As shown above, classical unimolecular reaction rate theory is based upon our knowledge of the qualitative nature of the classical dynamics. For example, it is essential to examine the rate of energy transport between different DOFs compared with the rate of crossing the intermolecular separatrix. This is also the case if one attempts to develop a quantum statistical theory of unimolecular reaction rate to replace exact quantum dynamics calculations that are usually too demanding, such as the quantum wave packet dynamics approach, the flux-flux autocorrelation formalism, and others. As such, understanding quantum dynamics in classically chaotic systems in general and quantization effects on chaotic transport in particular is extremely important. 

The first step is similar to the classical sensing method the value of a physical quantity in the sensing medium is measured, for example the output voltage of a chemical sensor. Then the microscopic spontaneous fluctuations of these measurements are strongly amplified (typically 1-100 million times) and the statistical properties of these fluctuations are analyzed. These fluctuations are due to the dynamically changing molecular-level interactions between the odor molecules and the sensing media, thus they contain the chemical signature of the odor. The results are compared with a statistical pattern database to identify the odor. 

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