Classical probability statistics

Classical probability statistics are inadequate for the treatment of small numbers of observations, and techniques developed only within recent years are necessary to avoid large errors in the estimates of error. For a finite (and usually small) number of observations of a quantity, one obtains data that show a certain amount of spread. The true mean fx and the true spread of the hypothetical infinite population of measurements are what one wishes to have. Usually the best that we can actually have with a finite number Adof measurements are estimates of jx and a. These are, respectively, the mean x and the spread... 

Marcus uses the Born-Oppenheimer approximation to separate electronic and nuclear motions, the only exception being at S in the case of nonadiabatic reactions. Classical equilibrium statistical mechanics is used to calculate the probability of arriving at the activated complex only vibrational quantum effects are treated approximately. The result is... 

Quantum mechanics, however, is different from the other applications in that the states themselves require a concept of probability for their physical interpretation. I shall call this intrinsic probability or quantum probability to distinguish it from the above classical or statistical probability. Intrinsic probability is not covered by the definition in 1.1 and cannot be regarded as an ensemble.510... 

Quantum mechanical, classical and statistical probabilities agree, on average, reasonably well with the experimental results [133] shown in Fig. 37 (vibrational distributions of NO were also measured by Harrison et al. [310]). In the experiment a high population of the state n o = 1 is found already 100 cm above its threshold. Moreover, the measured probabilities show some indications of fluctuations. Because of the limited number of data points, the inevitable incoherent averaging over several overall rotational states of NO2 and the averaging over the various possible electronic states of the 0 and NO products, these fluctuations are less pronounced than in the quantum mechanical calculations on a single adiabatic PES and for J = 0. 

Many Statistics students have suffered immensely over the years by having to solve classical probability problems. Marilyn vos Savant (1997) stumped many readers with the following classical probability problem ... 

This section summarizes the classical, equilibrium, statistical mechanics of many-particle systems, where the particles are described by their positions, q, and momenta, p. The section begins with a review of the definition of entropy and a derivation of the Boltzmann distribution and discusses the effects of fluctuations about the most probable state of a system. Some worked examples are presented to illustrate the thermodynamics of the nearly ideal gas and the Gaussian probability distribution for fluctuations. 

For a more explicit discussion of such a representation, we shall be guided by the physical view of the light beam as a statistical ensemble of wave trains emitted by the macroscopic source at some time and with a spectral distribution function which are both random quantities characterized by classical probability distribution functions. The individual wave packets have state vectors similar to (57) ... 

The statistical energy properties of the exciting light pulses are represented by the classical probability function, introduced in Eq. (88), which describes the distribution of the mean photon energies around the value E with the energy width AEi g ... 

Generally, for light sources other than mode-locked or single-mode lasers, the statistical features must be taken into account as well the mean energies of the photon wave packets are regarded as random variables distributed according to a classical probability function of a characteristic width A j e the energy profile of these non-transform-limited or chaotic pulses has then a width of the order... 

In the classical free statistics, the number of the functional groups on the surface of a tree-like cluster is of the same order of that of the groups inside the cluster, so that a simple thermodynamic limit without surface term is impossible to take. The equilibrium statistical mechanics for the polycondensation was refined by Yan [14] to treat surface correction in such finite systems. He found the same result as Ziff and Stell. Thus the treatment of the postgel regime is not unique. The rigorous treatment of the problem requires at least one additional parameter defining relative probability of occurrence of infra- and intermolecular reactions in the gel. 

The prefactor in terms of h is used to expUdtly show the correspondence of Zt with the corresponding PF in the quantum statistical mechanics in the classical limit h O. Despite the classical limit requirement h O.we are not allowed to set h = 0 in the final result, but keep its actual nonzero value. Accordingly, some problems remain such as Wigner s distribution function not being a classical probability distribution, which we do not discuss any further but refer the reader to the Uterature [117]. Keeping h at its nonzero value avoids infinities as we will see below but in no way implies that we are dealing with quantum effects. In particular, it does not imply that the entropy is nonnegative, as we have discussed elsewhere [75]. We... 

Morante S, Rossi GC, Testa M (2006) The stress tensor of a molecular system an exercise in statistical mechanics. J Chem Phys 125 034101 66. Nelson DF, Lax M (1976) Asymmetric total stress tensor. Phys Rev B 13 1770-1776 Das A (1978) Stress tensor in a class of gauge theraies. Phys Rev D 18 2065-2067 Cohen L (1979) Local kinetic energy in quantum mechanics. J Chem Phys 70 788-789 Cohen L (1984) Representable local kinetic tmergy. J Chem Phys 80 4277-4279 Cohen L (1996) Local values in quantum mechanics. Phys Lett A 212 315-319 Ayers PW, Parr RG, Nagy A (2002) Local kinetic tmergy and local temperature in the density-functional theory of electronic structure. Int J Quantum Chem 90 309-326 Cohen L (1966) Generalized phase-space distribution functions. J Math Phys 7 781-786 Cohen L (1966) Can quantum mechanics be formulated as classical probability theory. Philos Sci 33 317-322... 

The statistical probability converges to the classical probability when the number of trials is infinite. If the number of tri s is small, then the value of the statistical probability fluctuates. We show later in this chapter that the magnitude of fluctuations in the value of P(Ei) is inversely proportional to /iv. 

The Restart check box can be used in ctiii junction with the explicit editing of a IIIX file to assign completely user-specified initial velocities. This may be useful in classical trajectory analysis of chemical reactions where the initial velocities and directions of the reactants are varied to statistically determine the probability of reaction occurring, or n ot, in the process of calculating a rate con -Stan t. 

The controversy (for a lucid discussion refer to Mann, Shefer and Singpurwala, 1976) between "Bayesians" and "classicists" has nothing to do with precedence, for Bayes preceded much of classical statistics. The argument hinges on a) what prior knowledge is acceptable, and b) the treatment of probabilities as random variables themselves. 

In the introduction to this section, two differences between "classical" and Bayes statistics were mentioned. One of these was the Bayes treatment of failure rate and demand probttbility as random variables. This subsection provides a simple illustration of a Bayes treatment for calculating the confidence interval for demand probability. The direct approach taken here uses the binomial distribution (equation 2.4-7) for the probability density function (pdf). If p is the probability of failure on demand, then the confidence nr that p is less than p is given by equation 2.6-30. 

There is thus assumed to be a one-to-one correspondence between the most probable distribution and the thermodynamic state. The equilibrium ensemble corresponding to any given thermodynamic state is then used to compute averages over the ensemble of other (not necessarily thermodynamic) properties of the systems represented in the ensemble. The first step in developing this theory is thus a suitable definition of the probability of a distribution in a collection of systems. In classical statistics we are familiar with the fact that the logarithm of the probability of a distribution w[n is — J(n) w n) In w n, and that the classical expression for entropy in the ensemble is20... 

The probability of an ensemble distribution in classical statistics is maximized under the condition of given total energy in the ensemble, to yield the familiar Boltzmann distribution ... 

It is shown in classical statistics that the probability of the grand ensemble is a maximum, under the restriction to given average energy and given avenge population per system, when the distribution is chosen to be... 

In physical chemistry the most important application of the probability arguments developed above is in the area of statistical mechanics, and in particular, in statistical thermodynamics. This subject supplies the basic connection between a microscopic model of a system and its macroscopic description. The latter point of view is of course based on the results of experimental measurements (necessarily carried out in each experiment on a very large number of particle ) which provide the basis of classical thermodynamics. With the aid of a simple example, an effort now be made to establish a connection between the microscopic and macroscopic points of view. 

Percolation theory describes [32] the random growth of molecular clusters on a d-dimensional lattice. It was suggested to possibly give a better description of gelation than the classical statistical methods (which in fact are equivalent to percolation on a Bethe lattice or Caley tree, Fig. 7a) since the mean-field assumptions (unlimited mobility and accessibility of all groups) are avoided [16,33]. In contrast, immobility of all clusters is implied, which is unrealistic because of the translational diffusion of small clusters. An important fundamental feature of percolation is the existence of a critical value pc of p (bond formation probability in random bond percolation) beyond which the probability of finding a percolating cluster, i.e. a cluster which spans the whole sample, is non-zero. 

If the dependence of nA and nB on q is taken into account in the calculation of the statistical operators for heavy particles, we obtain the improved Condon approximation (ICA) which differs from Eq. (17) only by the change of p and p°f to p, and pf, respectively. In the classical limit for p, and p/ the expression for the transition probability takes the form... 

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