Coarse variables

Metadynamics defines coarse-grained variables which are assumed to be slow coordinates of the system. Those coordinates are similar to the order parameters considered earlier in this chapter. The coarse variables are evolved independently following a steepest-descent equation. In the case of a single variable, Laio and Parrinello [34] use... 

All the calculations described below were done with the program MOIL [28]. Moil is a freely available molecular simulation package with a focus on reaction path and rate calculations. Structures selected from the trajectory described in the previous paragraph were fed into MOIL. They were refined into a detailed SDP from a helix to a coil conformation with the action formnlation of Olender and Elber [29]. The SDP was computed in a rednced snbspace of coarse variables, namely the positions of the C -s of the peptide. We are making the plausible assumption that side chain motions equilibrate more rapidly than backbone degrees of freedom. 

As in cases of bulk water and salt solution, all the functions are discretized. The range tl must be longer when ions are included in water. Typical values of N and 6r in the pure water case are 512 and 0.02d (salt concentration is IM, for example, N is 1024 and N must be even larger as the salt concentration becomes lower. We iterate on r)ab(ri) (there are 2m and 4m distinct pairs in the pure water and salt-solution cases, respectively). The coarse variables, r)ab(ri) for ri < Dab Dab vl), are decomposed into the coarse parts and the fine parts using the projective representation described in 1.1(b). r/ab(ri) for r > Hat are treated as the fine variables. The coarse parts are converged in the inner N-R loop and the fine parts and the fine variables are successively updated in accordance with the Picard method in the outer loop. The values of ibt and ((t = l,...,2mor 4m)can all be different. 

As explained above, the longest time scales accessible by MD simulations are limited by the short time steps dictated by the femtosecond time scale of the fastest atomic motions, hi the CMD method of Hummer and Kevrekidis this limitation is overcome by considering the time evolution of the system in terms of a few coarse variables [18,92]. These M variables q = q (r),q2(r),.. - each of which... 

The second class, indeterminate or random errors, is brought about by the effects of uncontrolled variables. Truly random errors are as likely to cause high as low results, and a small random error is much more probable than a large one. By making the observation coarse enough, random errors would cease to exist. Every observation would give the same result, but the result would be less precise than the average of a number of finer observations with random scatter. 

Soft-wheat flours are sold for general family use, as biscuit or cake flours, and for the commercial production of crackers, pretzels, cakes, cookies, and pastry. The protein in soft wheat flour mns from 7 to 10%. There are differences in appearance, texture, and absorption capacity between hard- and soft-wheat flour subjected to the same milling procedures. Hard-wheat flour falls into separate particles if shaken in the hand whereas, soft-wheat flour tends to clump and hold its shape if pressed together. Hard-wheat flour feels slightly coarse and granular when mbbed between the fingers soft-wheat flour feels soft and smooth. Hard-wheat flour absorbs more Hquid than does soft-wheat flour. Consequently, many recipes recommend a variable measure of either flour or Hquid to achieve a desired consistency. 

The characteristics of a powder that determine its apparent density are rather complex, but some general statements with respect to powder variables and their effect on the density of the loose powder can be made. (/) The smaller the particles, the greater the specific surface area of the powder. This increases the friction between the particles and lowers the apparent density but enhances the rate of sintering. (2) Powders having very irregular-shaped particles are usually characterized by a lower apparent density than more regular or spherical ones. This is shown in Table 4 for three different types of copper powders having identical particle size distribution but different particle shape. These data illustrate the decisive influence of particle shape on apparent density. (J) In any mixture of coarse and fine powder particles, an optimum mixture results in maximum apparent density. This optimum mixture is reached when the fine particles fill the voids between the coarse particles. 

The following variables can affect wall friction values of a bulk soHd. (/) Pressure as the pressure acting normal to the wall increases, the coefficient of sliding friction often decreases. (2) Moisture content as moisture increases, many bulk soHds become more frictional. (3) Particle size and shape typically, fine materials are somewhat more frictional than coarse materials. Angular particles tend to dig into a wall surface, thereby creating more friction. (4) Temperature for many materials, higher temperatures cause particles to become more frictional. (5) Time of storage at rest if allowed to remain in contact with a wall surface, many soHds experience an increase in friction between the particles and the wall surface. (6) Wall surface smoother wall surfaces are typically less frictional. Corrosion of the surface obviously can affect the abiUty of the material to sHde on it. 

AU processed material is screened to return the coarse fraction for a second pass through the system. Process feed rates are matched to operating variables such as rpm speed and internal clearances, thus minimizing the level of excess fines (—200 mesh (<0.075 mm mm)). At one installation (3) the foUowing product size gradation of total smaller than mesh size (cumulative minus) was obtained ... 

The major variable in setting entrainment (E, weight of liquid entrained per weight of vapor) is vapor velocity. As velocity is increased, the dependence of E on velocity steepens. In the lowest velocity regime, E is proportional to velocity. At values of E of about 0.001 (around 10 percent of flood), there is a shift to a region where the dependence is with (velocity) ". Near flood, the dependence rises to approximately (velocity). In this regime, the kinetic energy of the vapor dominates, and the bulk of the dispersion on the plate is often in the form of a coarse spray. 

The mechanical properties of Watts deposits from normal, purified solutions depend upon the solution formulation, pH, current density and solution temperature. These parameters are deliberately varied in industrial practice in order to select at will particular values of deposit hardness, strength, ductility and internal stress. Solution pH has little effect on deposit properties over the range pH 1 0-5-0, but with further increase to pH 5 -5, hardness, strength and internal stress increase sharply and ductility falls. With the pH held at 3-0, the production of soft, ductile deposits with minimum internal stress is favoured by solution temperatures of 50-60°C and a current density of 3-8 A/dm in a solution with 25% of the nickel ions provided by nickel chloride. Such deposits have a coarse-grained structure, whereas the harder and stronger deposits produced under other conditions have a finer grain size. A comprehensive study of the relationships between plating variables and deposit properties was made by the American Electroplaters Society and the results for Watts and other solutions reported... 

Although other descriptions are possible, the mathematical concept that matches more closely the intuitive notion of smoothness is the frequency content of the function. Smooth functions are sluggish and coarse and characterized by very gradual changes on the value of the output as we scan the input space. This, in a Fourier analysis of the function, corresponds to high content of low frequencies. Furthermore, we expect the frequency content of the approximating function to vary with the position in the input space. Many functions contain high-frequency features dispersed in the input space that are very important to capture. The tool used to describe the function will have to support local features of multiple resolutions (variable frequencies) within the input space. 

Assuming that the coarse velocity can be regarded as an intensive variable, this shows that the second entropy is extensive in the time interval. The time extensivity of the second entropy was originally obtained by certain Markov and integration arguments that are essentially equivalent to those used here [2]. The symmetric matrix a 2 controls the strength of the fluctuations of the coarse velocity about its most likely value. That the symmetric part of the transport matrix controls the fluctuations has been noted previously (see Section 2.6 of Ref. 35, and also Ref. 82). 

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