Particle statistics, accuracy

Flow cytometry (FCM) is a high-precision technique for rapid analysis and sorting of cells and particles. In theory, it can be used to measure any cell component, provided that a fluorescent tracer is available that reacts specifically and stoichiometrically with that constituent. The technique provides statistical accuracy, reproducibility, and sensitivity. 

The extraction of more complex particle size distributions from PCS data (which is not part of the commonly performed particle size characterization of solid lipid nanoparticles) remains a challenging task, even though several corresponding mathematical models and software for commercial instruments are available. This type of analysis requires the user to have a high degree of experience and the data to have high statistical accuracy. In many cases, data obtained in routine measurements, as are often performed for particle size characterization, are not an adequate basis for a reliable particle size distribution analysis. 

There remains the problem of statistical accuracy, since the total count in the 239Pu peak, assuming a dosage of 0.008 Bqh m-3 and a counter efficiency of 32%, is less than 3 cpm. A statistical problem of another kind lies in the fact that the D.A.C. of 239Pu in air is equivalent to one Pu02 particle of diameter 1.6 jum Per m3 °f air. At a sampling rate of 301 min-1, one such particle would be sampled in half an hour. 

Next, Fig. 5 shows a typical center of mass trajectory for the case of a full chaotic internal phase space. The eyecatching new feature is that the motion is no more restricted to some bounded volume of phase space. The trajectory of the CM motion of the hydrogen atom in the plane perpendicular to the magnetic field now closely resembles the random motion of a Brownian particle. In fact, the underlying equation of motion at Eq. (35) for the CM motion is a Langevin-type equation without friction. The corresponding stochastic Lan-gevin force is replaced by our intrinsic chaotic force — e B x r). A main characteristic of random Brownian motion is the diffusion law, i.e. the linear dependence of the travelled mean-square distance on time. We have plotted in Fig. 6 for our case of a chaotic force for 500 CM trajectories the mean-square distance as a function of time. Within statistical accuracy the plot shows a linear dependence. The mean square distance

of the CM after time t, therefore, obeys the diffusion equation... 

Table 4. Excess free energy per particle for the different bulk structures and the liquid state calculated via a thermodynamic intergration in the limit of infinite number of particles [102]. The reference state for the free energy calculation of the liquid was the hard-sphere fluid, and for the bulk solid structures we used an Einstein-crystal. In some cases we also used the hard-sphere system as a reference state for the sohd structures. We found that the solid free energies obtained via these two distinct routes agreed to within 0.005kgT, which corresponds to our estimate of the statistical error in this calculation. The statistical accuracy of the computed free energy of the liquid is estimated to be iO.Olfc T. In the table, the values in brackets indicate the volume fraction at which the excess free energy was calculated. The calculated excess free energies for the fee and the hep structures can be compared directly, as they were calculated at the same pressure, whereas the others are not. The fcc-hcp free energy difference is always smaller than (1 x Q kgT)...

The shear viscosity is a tensor quantity, with components T] y, t],cz, T)yx> Vyz> Vzx> Vzy If property of the whole sample rather than of individual atoms and so cannot be calculat< with the same accuracy as the self-diffusion coefficient. For a homogeneous fluid the cor ponents of the shear viscosity should all be equal and so the statistical error can be reducf by averaging over the six components. An estimate of the precision of the calculation c then be determined by evaluating the standard deviation of these components from tl average. Unfortunately, Equation (7.89) cannot be directly used in periodic systems, evi if the positions have been unfolded, because the unfolded distance between two particl may not correspond to the distance of the minimum image that is used to calculate the fore For this reason alternative approaches are required. 

In classical molecular dynamics, on the other hand, particles move according to the laws of classical mechanics over a PES that has been empirically parameterized. By means of their kinetic energy they can overcome energetic barriers and visit a much more extended portion of phase space. Tools from statistical mechanics can, moreover, be used to determine thermodynamic (e.g. relative free energies) and dynamic properties of the system from its temporal evolution. The quality of the results is, however, limited to the accuracy and reliability of the (empirically) parameterized PES. 

The reconstruction functionals may be understood as substantially renormalized many-body perturbation expansions. When exact lower RDMs are employed in the functionals, contributions from all orders of perturbation theory are contained in the reconstructed RDMs. As mentioned previously, the reconstruction exactly accounts for configurations in which at least one particle is statistically isolated from the others. Since we know the unconnected p-RDM exactly, all of the error arises from our imprecise knowledge of the connected p-RDM. The connected nature of the connected p-RDM will allow us to estimate the size of its error. For a Hamiltonian with no more than two-particle interactions, the connected p-RDM will have its first nonvanishing term in the (p — 1) order of many-body perturbation theory (MBPT) with a Hartree-Fock reference. This assertion may be understood by noticing that the minimum number of pairwise potentials V required to connectp particles completely is (p — 1). It follows from this that as the number of particles p in the reconstmcted RDM increases, the accuracy of the functional approximation improves. The reconstmction formula in Table I for the 2-RDM is equivalent to the Hartree-Fock approximation since it assumes that the two particles are statistically independent. Correlation corrections first appear in the 3-RDM functional, which with A = 0 is correct through first order of MBPT, and the 4-RDM functional with A = 0 is correct through second order of MBPT. 

Once the analytical method is validated for accuracy at the laboratory scale, it can be used to obtain extensive information on process performance (blend homogeneity, granulation particle size distribution, and moisture content) under various conditions (blender speed, mixing time, drying air temperature, humidity, volume, etc.). Statistical models can then be used to relate the observable variables to other performance attributes (e.g., tablet hardness, content uniformity, and dissolution) in order to determine ranges of measured values that are predictive of acceptable performance. 

Summing up, note that the direct statistical (computer) simulation does not demonstrate serious errors of the superposition approximation for equal reactant concentrations. Divergence begins first of all for unequal concentrations for not very large reaction depths T 2 it is almost negligible, at T > 2 and especially asymptotically (as t —> oo) it becomes important, but the complete quantitative analysis cannot be done due to unreliable statistics of results. In this Section we have restricted ourselves to the A + B —> 0 reaction without particle generation. Testing of the superposition approximation accuracy for the case of particle creation will be done in Chapter 7. 

The uncertainty principle shows that the classical trajectory of a particle, with a precisely determined position and momentum, is really an illusion. It is a very good approximation, however, for macroscopic bodies. Consider a particle with mass I Xg, and position known to an accuracy of 1 pm. Equation 2.41 shows that the uncertainty in momentum is at least 5 x 10 29 kg m s-1, corresponding to a velocity of 5 x 10 JO m s l. This is totally negligible for any practical purpose, and it illustrates that in the macroscopic world, even with very light objects, the uncertainty principle is irrelevant. If we wanted to, we could describe these objects by wave packets and use the quantum theory, but classical mechanics gives essentially the same answer, and is much easier. At the atomic and molecular level, however, especially with electrons, which are very light, we must abandon the idea of a classical trajectory. The statistical predictions provided by Bom s interpretation of the wavefunction are the best that can be obtained. 

For the attrition experiments approximately 25 g of polymer particles were used. After three, six and nine consecutive impacts with a velocity of 40 m/s, the attrition rate A was determined as the relative loss of mass (see Eq. (1)). This procedure was necessary since the attrition rates for some polymer classes were not measurable with satisfying accuracy for lower impact numbers. Since the attrition process is highly statistical, each experiment was repeated three times. This also holds for the other experimental setups. In the following diagrams, the median values of these three repeated experiments are plotted with the standard deviations as error bars. For all experiments, the attrition rate A was calculated according to Eq. (1), where M denotes the initial particle mass. Ma is the mass of particles at the initial size after the attrition experiment. 

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