Probability distribution functions for

It is instructive to see this in temis of the canonical ensemble probability distribution function for the energy, NVT - Referring to equation B3.3.1 and equation (B3.3.2I. it is relatively easy to see that... 

In either case, first-order or continuous, it is usefiil to consider the probability distribution function for variables averaged over a spatial block of side L this may be the complete simulation box (in which case we... 

Under specific circumstances, alternative forms for ks j have been proposed like the parabolic or the truncated Gaussian probability distribution function for example [154]. 

The histogram reweighting methodology for multicomponent systems [52-54] closely follows the one-component version described above. The probability distribution function for observing Ni particles of component 1 and No particles of component 2 with configurational energy in the vicinity of E for a GCMC simulation at imposed chemical potentials /. i and //,2, respectively, at inverse temperature ft in a box of volume V is... 

While some younger physicists, like Born s student Werner Heisenberg and Ralph Fowler s student P. A. M. Dirac, were not sanguine about Bom s interpretation, it pleased chemists like Lewis, who earlier had been willing to think about the "average" position of an electron in its orbit, so as to reconcile Bohr s dynamic atom with Lewis s static atom. For Pauling, it was a natural step to take Y2 to be the probability distribution function for an electron s position in space.32... 

Figure 8. The structure of hydrated Na and CP ions at the water/Pt(IOO) interface (dotted lines) compared with the structure in bulk water (solid lines). In the two top panels are the oxygen ion radial distribution functions, and in the two bottom panels are the probability distribution functions for the angle between the water dipole and the oxygen-ion vector for water molecules in the first hydration shell. (Data adapted from Ref. 100.)...

Rhodes, P. R. 1975. A probability distribution function for turbulent flows. In Turbulent mixing in nonreactive and reactive mixing. Ed. S. N. B. Murthy. New York, NY Plenum Press. 235-41. 

The parallel-replica method [5] is perhaps the least glamorous of the AMD methods, but is, in many cases, the most powerful. It is also the most accurate AMD method, assuming only first-order kinetics (exponential decay) i.e., for any trajectory that has been in a state long enough to have lost its memory of how it entered the state (longer than the correlation time icorr, the time after which the system is effectively sampling a stationary distribution restricted to the current state), the probability distribution function for the time of the next escape from that state is given by... 

Fig. 3.6 The probability distribution functions for a harmonic oscillator in the n = 0 and n =10 levels, each plotted at the height corresponding to its energy, with the curve showing the potential energy function. The points where the energy equals the potential energy are the classical turning points, corresponding to the maximum possible displacement of a classical particle with the same energy.

It can be established by the following reasoning. If n = %, each particle contains at most one free radical. Growing chains in the latex particles can thus either grow or be terminated instantaneously by entrant free radicals. These mutually exclusive kinetic events immediately prescribe the Flory most probable distribution function for the growing chains (12) this is an exponential distribution function with a polydispersity index of 2.00 (13). 

The probability distribution function for the fixed end-to-end distance R of macromolecule can be written down on either ground. In the simplest case, it is the Gaussian distribution... 

A probability distribution function for a continuous random variable, denoted by fix), describes how the frequency of repeated measurements is distributed over the range of observed values for the measurement. When considering the probability distribution of a continuous random variable, we can imagine that a set of such measurements will lie within a specific interval. The area under the curve of a graph of a probability distribution for a selected interval gives the probability that a measurement will take on a value in that interval. 

The point of the scheme is to consider the probability distribution function for the variable 4> defined by Eq. (17-62). In the solution system, the probability distribution P < >) is given by... 

Fig. 3.20. Histogram of the probability distribution function for the proportion of main plasma ELM energy loss (AWelm) that reaches the divertor target (AWe/m) for ASDEX Upgrade discharges [33,44]...

The test of a fit can be put into the form of the following question At a specified level of confidence, is the value of 5(i) consistent with the assumption that the residuals are representative of a normal distribution as they would be if the model were correct and the weights properly assigned Here we make use of the fact that in principle the minimized quantity known as should conform to the so-called chi square distribution. The probability distribution function for with v = (m - n) degrees of freedom, is ... 

If now we assume that the molecules move completely at random and independently of each other and that the system is at equilibrium and isolated (no exchange of energy between the gas and its environment) then the total energy of the gas is simply the kinetic energy attributable to the random motion of the molecules. This total energy and the volume then fix completely the thermodynamic properties of the gas. If now we could know the probability distribution function for the molecular velocities (at equilibrium), that would determine uniciuely the properties of the system. 

Probability distribution function for the binding energy of a distinguished molecule of type a to the solution in the case that these two subsystems are decoupled, as indicated by the superscript zero. 

The characteristics of the water associated with the polar lipid bilayer surfaces are of particular interest because they become very important in processes of physiological significance such as membrane fusion, and they may play a role in the mechanisms of association of proteins and small molecules with lipid bilayers and biological membranes. A very small fraction of the lipid bilayer-associated water molecules are actually immobilized, and a larger fraction (about 30% of the total bilayer-associated water in multilamellar PC bilayers in the L phase) has a probability distribution function that is more or less coincident with the probability distribution function for the polar head group of the lipids. Nevertheless, the pressure, P, which must be exerted to remove the bilayer-associated water, is large and varies (from 0.5-500 N cm ) with the thickness of the inter-bilayer water space as ... 

Number of particles per unit volume of dispersion at time t in incremental range X, X -I- dx in phase space Rotational speed of impeller General function, Eq. (105) Impeller power Distribution function for kinetic energy, Eq. (37) Probability distribution function for local velocity, Eq. (44)... 

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