# Statistical Averaging

Chemical Conformations From a chemical point of view, biomolecular systems are characterized by different conformations - a term, which simultaneously describes both distinguishable geometric configurations and the associated chemical functionality . In a conformation, the large scale geometric structure of the molecule is understood to be conserved, whereas on smaller scales the system may well rotate, oscillate or fluctuate. For a conformation to be an object of chemical interest, the duration of stay within that comformation should be long enough stable conformation) or, equivalently, it should make a significant contribution to any (statistical) averages. Conformational changes are therefore rare events, which will show up only in long-term simulations. [c.104]

Equilibration and Statistical Averaging [c.98]

The Statistical Averaging Period [c.318]

When some property of a system is measured experimentally, the result is an average for all of the molecules with their respective energies. This observed quantity is a statistical average, called a weighted average. It corresponds to the result obtained by determining that property for every possible energy state of the system, T(F), and multiplying by the probability of finding the system in that energy state, ti (F). This weighted average must be normalized by a partition function Q, where [c.13]

There are several other, equivalent ways to obtain a statistical average. One of these is to use a time average. In this formulation, a calculation is designed to simulate the motion of molecules. At every step in the simulation, the property is computed for one molecule and averaged over all the time steps equally. This is equivalent to the weighted average because the molecule will be in more probable energy states a larger percentage of the time. The accuracy of this result depends on the number of time steps and the ability of the simulation to correctly describe how the real system will behave. [c.15]

It is important to realize that many important processes, such as retention times in a given chromatographic column, are not just a simple aspect of a molecule. These are actually statistical averages of all possible interactions of that molecule and another. These sorts of processes can only be modeled on a molecular level by obtaining many results and then using a statistical distribution of those results. In some cases, group additivities or QSPR methods may be substituted. [c.110]

In the chapter on reaction rates, it was pointed out that the perfect description of a reaction would be a statistical average of all possible paths rather than just the minimum energy path. Furthermore, femtosecond spectroscopy experiments show that molecules vibrate in many dilferent directions until an energetically accessible reaction path is found. In order to examine these ideas computationally, the entire potential energy surface (PES) or an approximation to it must be computed. A PES is either a table of data or an analytic function, which gives the energy for any location of the nuclei comprising a chemical system. [c.173]

The rotational isomeric state (RIS) model assumes that conformational angles can take only certain values. It can be used to generate trial conformations, for which energies can be computed using molecular mechanics. This assumption is physically reasonable while allowing statistical averages to be computed easily. This model is used to derive simple analytic equations that predict polymer properties based on a few values, such as the preferred angle [c.308]

Equilibration and Statistical Averaging [c.98]

For purposes of exploring fluctuations and determining the convergence of these statistical averages the root mean square (RMS) deviation in x is also computed [c.312]

When using molecular dynamics to study equilibrium properties like enthalpy (the average energy), etc., you want the average over a trajectory to be equivalent to an ensemble average. This means the system must be in equilibrium the initial conditions have been forgotten and you are sampling from a set ofphase space configurations representative of the macroscopic equilibrium state. You should not begin sampling for the purpose of collecting statistical averages until this equilibration is performed. The lack of any long term drift is one indication of possible equilibration. To achieve equilibration to a temperature T, it may be necessary to rescale velocities through the use of the constant temperature bath algorithm, (the use of a heating phase with a small temperature step and overall temperature change such as one degree or less) or by re-initializing periodically. Equilibration requires the temperature to fluctuate about the requisite value T. [c.316]

HyperChem includes a number of time periods associated with a trajectory. These include the basic time step in the integration of Newton s equations plus various multiples of this associated with collecting data, the forming of statistical averages, etc. The fundamental time period is Atj s At, the integration time step stt in the Molecular Dynamics dialog box. [c.318]

The Statistical Averaging Period [c.318]

Emissions rates for a specific source can be measured directiy by inserting sampling probes into the stack or vent and this has been done for most large point sources. It would be an impossible task to do for every source in an area inventory, however. Instead, emission factors, based on measurements from similar sources or engineering mass-balance calculations, are appHed to most sources. An emission factor is a statistical average or quantitative estimate of the amount of a pollutant emitted from a specific source type as a function of the amount of raw material processed, product produced, or fuel consumed. Emission factors for most sources have been compiled (2). Emission factors for motor vehicles are determined as a function of vehicle model year, speed, temperature, etc. The vehicles are operated using various driving patterns on a chassis dynamometer. Dynamometer-based emissions data are used in EPA s MOBILE 4 model (3) to calculate total fleet emissions for a given roadway system. [c.366]

As of this writing, the only practical approach to solving turbulent flow problems is to use statistically averaged equations governing mean flow quantities. These equations, which are usually referred to as the Reynolds equations of motion, are derived by Reynold s decomposition of the Navier-Stokes equations (18). The randomly changing variables are represented by a time mean and a fluctuating part [c.101]

Amphibole minerals, in general, are characterized by prismatic cleavage planes which intersect at an angle of about 55°. Hence, in the cmshing of massive, nonftbrous amphiboles, microscopic fragments are frequendy found having the appearance of asbestos fibers. However, the statistical average of their aspect ratio is considerably lower than that of the asbestiform amphiboles. [c.346]

In effect, this represents the root of a statistical average of the squares. The divisor quantity (n — 1) will be referred to as the degrees of freedom. [c.488]

Estimating Emissions from Sources Knowledge of the types and rates of emissions is fundamental to evaluation of any air pollution problem. A comprehensive material balance on the process can often assist in this assessment. Estimates of the rates at which pollutants are discharged from various processes can also be obtained by utilizing pubhshed emission factors. See Compilation of Air Pollution Emission Factors (AP-42), 4th ed., U.S. EPA, Research Triangle Park, North Carolina, September, 1985, with all succeeding supplements and the EPA Technology Transfer Networks CHIEF. The emission fac tor is a statistical average of the rate at which pollutants are emitted from the [c.2173]

The statistical average of a variable described by a probability distribution [c.76]

Under the conditions (1.1) the rate constant is determined by the statistically averaged reactive flux from the initial to the final state. [c.3]

The rate constant is the statistical average of the reactive flux from the initial to the final state [c.12]

Equation (2.2) defines the statistically averaged flux of particles with energy E = P /2m -f V Q) and P > 0 across the dividing surface with Q =0. The step function 6 E — Vq) is introduced because the classical passage is possible only at > Vq. In classically forbidden regions, E < Vq, the barrier transparency is exponentially small and given by the well known WKB expression (see, e.g., Landau and Lifshitz [1981]) [c.12]

The first type of interaction, associated with the overlap of wavefunctions localized at different centers in the initial and final states, determines the electron-transfer rate constant. The other two are crucial for vibronic relaxation of excited electronic states. The rate constant in the first order of the perturbation theory in the unaccounted interaction is described by the statistically averaged Fermi golden-rule formula [c.26]

Now we take up the calculation of the rate constant for the decay of a metastable state. In principle it can be done by statistically averaging E from (3.32), but there is a more elegant and general way which relates the rate constant with the imaginary part of free energy. Recalling (3.19) we write the rate constant as [c.43]

The term proportional to cTy after averaging goes to zero. It is easy to verify that exp( —

Minimum ignition energy hazards of different materials can be directly compared by collecting samples from the smaller of two sieves whose sizes differ by only a few microns. This eliminates most of the MIE variation due to size and shape, it also avoids disparities found in the literature due to the reporting of various statistical average diameters if the PSD is very narrow, all of the commonly used averages converge to a single value. As an extension of this technique, the powder can be fractionated into a series of particle ranges by cascade sieving, enabling MIE to be found for a series of average particle sizes. Provided the sieves differ only incrementally in size from 200 down to (say) 400 mesh, there is no need to consider the PSD for each size fraction. Eor large capacity production especially, the MIE might be [c.171]

Chemical reactions are processes in which reactants are transformed into products. In some processes, the change occurs directly and the complete description of the mechanism of the reaction present can be relatively obtained. In complex processes, the reactants undergo a series of step-like changes and each step constitutes a reaction in its own right. The overall mechanism is made up of contributions from all the reactions that are sometimes too complex to determine from the knowledge of the reactants and products alone. Chemical kinetics and reaction mechanism as reviewed in Chapter 1 can often provide a reasonable approach in determining the reaction rate equations. Chemical kinetics is concerned witli analyzing tlie dynamics of chemical reactions. The raw data of chemical kinetics are the measurements of the reactions rates. The end product explains these rates in terms of a complete reaction mechanism. Because a measured rate shows a statistical average state of the molecules taking part in the reaction, chemical kinetics does not provide information on the energetic state of the individual molecules. [c.109]

The bulk associating fluids have been intensively studied using computer simulation [12-20]. In many cases simulation has been performed to verify theoretical predictions. The studies of Chapman and Zhang [17,18], in particular, have been concerned with a comparison of the results of a traditional Metropolis Monte Carlo simulation for dimerization of methanol with the prior predictions of the theory of Wertheim. It appeared, however, that in several applications the traditional Monte Carlo method does not provide reasonable statistical averages. At high degrees of association bonded configurations represent a small part of the configuration space. In the traditional method, the configurational space is randomly sampled, thus the small spatial volume of bonded configurations may be difficult to sample adequately in the framework of the common algorithm. In order to overcome this difficulty, Busch et al. [19] have proposed the association-biased Monte Carlo method. The canonical ensemble algorithm biases sampling to the regions of configuration space where the association or dissociation of particles is Ukely to occur. This is an efficient simulation technique for associating fluids in a wide range of densities. Unfortunately, the application of the method to nonuniform associating fluids requires a great deal of numerical effort. [c.169]

The interaction with the solvent is of similar importance as the intramolecuiar energy contributions and a correct representation of the solvent is therefore es.sential. If an explicit solvent description is chosen, averaging over many different solvent configurations is necessary in order to obtain converged statistical averages. Advantageous in this respect is describing the solvent as [c.67]

For purposes of exploring fluctuations and determining the convergence of these statistical averages the root mean square (RMSl deviation m x is also computed [c.312]

HyperChem in eludes a num ber of tim e periods associated with a trajectory. These include the basic time step in the integration of Newton s equations plus various imiltiples of this associated with collecting data, Ihc forining of statistical averages, etc. fhe fundamental time period is At]. At, the integration limestc p set in the Molecular Hyn am ics dialog box. [c.318]

Statistical mechanics states that the macroscopic values of certain quantities, like the energy, can be obtained by ensemble averaging over a very large number of possible states of the microscopic system. In many realms of chemistry, these statistical averages are what computational chemistry requires for a direct comparison with experiment. Afundamental principle of statistical mechanics, the Ergodic Hypothesis, states that it is possible to replace an ensemble average by a time average over the trajectory of the microscopic system. Molecular dynamics thus allows you to compute a time average over a trajectory that, in principle, represents a macroscopic average value. These time averages are fundamental to the use of molecular dynamics. [c.311]

Among the most successful of the Hquid chromatographic reversed-phase chiral stationary phases have been the cyclodextrin-based phases, introduced by Armstrong (78,79) and commercially available through Advanced Separation Technologies, Inc. or AUtech Associates. The most commonly used cyclodextrin in hplc is the P-cyclodextrin. In the bonded phases, the cyclodextrins are thought to be tethered to the silica substrate through one or two spacer ligands (Fig. 7). The mechanism thought to be responsible for the chiral selectivity observed with these phases is based on the formation of an inclusion complex between the hydrophobic moiety of the chiral analyte and the hydrophobic interior of the cyclodextrin cavity (Fig. 8). Preferential complexation between one optical isomer and the cyclodextrin through stereospecific interactions with the secondary hydroxyls which line the mouth of the cyclodextrin cavity results in the enantiomeric separation. Unlike the Pirkle-type phases, enantiospeciftc interactions between the analyte and the cyclodextrin are not the result of a single, well-defined association, but more of a statistical averaging of all the potential interactions with each interaction weighted by its energy or strength of interaction (80). [c.64]

In conclusion note that for a sufficiently dense energy spectrum the caustic segments have been shown [Benderskii et al. 1992b] to disappear after statistical averaging, which brings one back to the instanton and, for the present model, leads to eqs. (2.80a, b). [c.74]

The basic equations of filtration cannot always be used without introducing corresponding corrections. This arises from the fact that these equations describe the filtration process partially for ideal conditions when the influence of distorting factors is eliminated. Among these factors are the instability of the cake resistance during operation and the variable resistance of the filter medium, as well as the settling characteristics of solids. In these relationships, it is necessary to use statistically averaged values of both resistances and to introduce corrections to account for particle settling and other factors. In selecting filtration methods and evaluating constants in the process equations, the principles of similarity modeling are relied on heavily. [c.80]

See pages that mention the term

**Statistical Averaging**:

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