Repeated randomness assumption

Let q represent an observable quantity of a macroscopic system such as the circuit in Figure 1. Assuming that there are no other macroscopic observables, one can derive Eq. (15) from the equation of motion of all particles at the expense of a regrettable, but indispensable, repeated randomness assumption, similar to Boltzmann s Stosszahlansatz. 6 It then also follows that, provided q is an even variable, W has a symmetry property called detailed balancing. 6,7... 

A serious difficulty now appears. The quantum master equation (3.14), obtained by eliminating the bath, does not have the required form (5.6) and therefore results in a violation of the positivity of ps(/). Only by the additional approximation rc Tm was it possible to arrive at (3.19), which does have that form (see the Exercise). The origin of the difficulty is that (3.14) is based on our assumed initial state (3.4), which expresses that system and bath are initially uncorrelated. This cannot be true at later times because the interaction inevitably builds up correlations between them. Hence it is unjustified to use the same derivation for arriving at a differential equation in time without invoking a repeated randomness assumption, such as embodied in tc rm. ) At any rate it is physically absurd to think that the study of the behavior of a Brownian particle requires the knowledge of an initial state. 

This does not express Pj(t) in terms of P, (0) rather the full microscopic specification of the initial state is needed. This shortcoming can only be remedied by some drastic assumptions, akin to the repeated randomness assumption in III.2. 

The equation (7.8) is not yet a differential equation and the evolution of the Pj is not yet Markovian. As a final step the repeated randomness assumption has to be invoked. It assumes that it is possible to partition the time axis into intervals At such that... 

Conclusion. In classical statistical mechanics the evolution of a many-body system is described as a stochastic process. It reduces to a Markov process if one assumes coarse-graining of the phase space (and the repeated randomness assumption). Quantum mechanics gives rise to an additional fine-graining. However, these grains are so much smaller that they do not affect the classical derivation of the stochastic behavior. These statements have not been proved mathematically, but it is better to say something that is true although not proved, than to prove something that is not true. 

In all considered above models, the equilibrium morphology is chosen from the set of possible candidates, which makes these approaches unsuitable for discovery of new unknown structures. However, the SCFT equation can be solved in the real space without any assumptions about the phase symmetry [130], The box under the periodic boundary conditions in considered. The initial quest for uy(r) is produced by a random number generator. Equations (42)-(44) are used to produce density distributions T(r) and pressure field ,(r). The diffusion equations are numerically integrated to obtain q and for 0 < s < 1. The right-hand size of Eq. (47) is evaluated to obtain new density profiles. The volume fractions at the next iterations are obtained by a linear mixing of new and old solutions. The iterations are performed repeatedly until the free-energy change... 

In the detection of repeats using SMART an algorithm is used that derives similarity thresholds that are dependent on the number of repeats already found in a protein sequence (Andrade et al., 1999b). These thresholds are based on the assumption that suboptimal local alignment scores of a profile/HMM against a random sequence database are well described by an extreme value distribution (EVD). The result of this protocol is that acceptance thresholds for suboptimal alignments are lowered below the optimal scores of nonhomologous sequences. 

By varying the distance between nearest adsorption sites, rs, one can control the composition variation period of the synthesized copolymer. From the chemical correlators defined by Eq. 16, it is easy to find the average number of segments in the repeating chain sections, N, for different rs values. It is instructive to analyze the relation between N and rs. As expected, a power law N oc is observed. It is clear that exponent //. in this dependence should be between //, = 1 (for a completely stretched chain) and //. = v 1 with v 0.6 (for a random coil with excluded volume [75]). The calculation [95] yields yu 1.33 for N > 15. This supports the aforementioned assumption that the repeating chain sections are strongly stretched between the adsorption sites. The same conclusion can be drawn from the visual analysis of typical snapshots similar to that presented in Fig. 22. 

Statistical formulas are based on various mathematical distribution functions representing these frequency distributions. The most widely used of all continuous frequency distributions is the normal distribution, the common bellshaped curve. It has been found that the normal curve is the model of experimental errors for repeated measurements of the same thing. Assumption of a normal distribution is frequently and often indiscriminately made in experimental work because it is a convenient distribution on which many statistical procedures are based. However, some experimental situations subject to random error can yield data that are not adequately described by the normal distribution curve. 

Theoretical probability identifies the possible outcomes of a statistical experiment, and uses theoretical arguments to predict the probability of each. Many applications in chemistry take this form. In atomic and molecular structure problems, the general principles of quantum mechanics predict the probability functions. In other cases the theoretical predictions are based on assumptions about the chemical or physical behavior of a system. In all cases, the validity of these predictions must be tested by comparison with laboratory measurements of the behavior of the same random variable. A full determination of experimental probability, and the mean values that come from it, must be obtained and compared with the theoretical predictions. A theoretical prediction of probability can never be tested or interpreted with a single measurement. A large number of repeated measurements is necessary to reveal the true statistical behavior. 

The underlying assumption in statistical analysis is that the experimental error is not merely repeated in each measurement, otherwise there would be no gain in multiple observations. For example, when the pure chemical we use as a standard is contaminated (say, with water of crystallization), so that its purity is less than 100%, no amount of chemical calibration with that standard will show the existence of such a bias, even though all conclusions drawn from the measurements will contain consequent, determinate or systematic errors. Systematic errors act uni-directionally, so that their effects do not average out no matter how many repeat measurements are made. Statistics does not deal with systematic errors, but only with their counterparts, indeterminate or random errors. This important limitation of what statistics does, and what it does not, is often overlooked, but should be kept in mind. Unfortunately, the sum-total of all systematic errors is often larger than that of the random ones, in which case statistical error estimates can be very misleading if misinterpreted in terms of the presumed reliability of the answer. The insurance companies know it well, and use exclusion clauses for, say, preexisting illnesses, for war, or for unspecified acts of God , all of which act uni-directionally to increase the covered risk. 

In this and subsequent spreadsheet exercises, we will use the method of least-squares to fit data to a function rather than to repeat measurements. This is based on several assumptions (1) that, except for the effect of random fluctuations, the experimental data can indeed be described by a particular function (say, a straight line, a hyperbola, a circle, etc.), that (2) the random fluctuations are predominantly in the dependent parameter, which we will here call y, so that random fluctuations in the independent parameterxcanbeneglected, and (3) that those random fluctuations can be described by a single Gaussian distribution. 

It must be stresse that all conclusions we make about our determination are based on the assumption that we have obtained a random sample from our material. If we repeat the process of choosing an object at random from a population n times, the values xi,...Xn of a random variable x so obtained will be a random sample. 

Standard error. A measure of the variability of a statistic from sample to sample. (As opposed to standard deviation which is a measure of variability for original observations.) Since repeated samples are not usually obtained, standard errors cannot be calculated directly. They can be calculated on the basis of a single sample using an appropriate model and the assumptions it requires. For example, on the assumption that a sample has been obtained by simple random sampling, the standard error of the mean will equal the standard deviation divided by the square root of the sample size. This particular assumption, which implies that variability between samples can be simply estimated with the help of variability within samples, is rarely valid, however, and this formula is often inappropriately used in clinical trials. In fact, it is a standard error to use it. 

Intra-laboratory RSD (or repeatability standard deviation, as defined in ISO 5725-2) of 0.5 the interlaboratory standard deviation, and the previously obtained mean laboratory values, were input into the random-number generator to obtain four replicate laboratory values for samples at the three concentrations. The assumption of intra-laboratory standard deviation... 

Suppose we apply a constant dead-weight load to a force-measurement system and keep the load applied for several hours. If the output O is monitored over tins period of time, it is often likely that the output value will fluctuate about an expected value for the output. For example, if the load is 100 N, for which the expected output value of the sensing element should be 1 V, it is likely that over time the output wiU assume values such as 1.01, 0.98, 0.99, 0.98, 1.02, etc. This effect is termed as a lack of repeatability. Repeatability is the ability of a system to give the same output for the same input, when this input is repeatedly applied to it. The most common causes for lack of repeatability are random variations in the measurement system elements and their environment. By making reasonable assumptions about the fluctuations of the various inputs, including environment-related ones, it is possible to analytically characterize the fluctuations expected in the output. 

For the confirmation of this assumption we have made an attempt to describe curves a(t), shown in Figure 10.5 and 10.6, in frameworks of the scaling approaches for the reactions of low-molecular substance [33]. Let us consider the reaction in which particles P, of a chemical substance diffuse in the medium containing the random located static nonsaturated traps T. By the contact of a particle P with a trap T the particle disappears. Nonsaturation of a trap means that the reaction P + T T can repeat itself an infinite number of times. It is usually considered that the concentration of particles and traps is large or the reaction occurs at intensive stirring, the process can be considered as the classical reaction of the... 

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