Statistical Fluctuations

A reactive species in liquid solution is subject to pemianent random collisions with solvent molecules that lead to statistical fluctuations of position, momentum and internal energy of the solute. The situation can be described by a reaction coordinate X coupled to a huge number of solvent bath modes. If there is a reaction... 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

Noise. So fat, as indicated at the beginning of this section on semiconductor statistics, equihbtium statistics have been considered. Actually, there ate fluctuations about equihbtium values, AN = N— < N >. For electrons, the mean-square fluctuation is given by < ANf >=< N > 1- ) where (Ai(D)) is the Fermi-Dirac distribution. This mean-square fluctuation has a maximum of one-fourth when E = E-. These statistical fluctuations act as electrical noise and limit minimum signal levels. 

The maximum statistical fluctuation of 10 atoms is the same in both homogeneous and heterogeneous nucleation. If Q is the volume occupied by one atom in the nucleus then we can easily see that... 

This chapter provided a common basis for understanding the assigning of numerical values to "risk." in the context of probability as the behavior of an ensemble of plants. Predictions of short-term behavior are subject to statistical fluctuations and may be very misleading. Qualitative... 

The emission of thermal electrons is subject to statistical fluctuations (lead shot effect). It is influenced by the current strength, the number of electric charges hberated and the frequency of the radiation [55] (see [40] for further details). 

Now, assume that we are getting closer to the critical point of our transition, i.e., to the point of the second-order transition. In the case of a uniform system the critical region can be described by the divergent correlation length of statistical fluctuations [138]... 

All the above scaling relations have one common origin in the behavior of the correlation length of statistical fluctuations, in a finite system [140,141]. Namely, the most specific feature of the second-order transition is the divergence of at the transition point, as is described by Eq. (22). In the finite system, the development of long-wavelength fluctuations is suppressed by the system size limitation can be, at the most, of the same order as L. Taking this into account, we find from Eqs. (22) and (26) that... 

In the next paper [160], Villain discussed the model in which the local impurities are to some extent treated in the same fashion as in the random field Ising model, and concluded, in agreement with earlier predictions for RFIM [165], that the commensurate, ordered phase is always unstable, so that the C-IC transition is destroyed by impurities as well. The argument of Villain, though presented only for the special case of 7 = 0, suggests that at finite temperatures the effects of impurities should be even stronger, due to the presence of strong statistical fluctuations in two-dimensional systems which further destabilize the commensurate phase. 

The relaxation time r of the mean length, = 2A Loo, gives a measure of the microscopic breaking rate k. In Fig. 16 the relaxation of the average length (L) with time after a quench from initial temperature Lq = 1.0 to a series of lower temperatures (those shown on the plot are = 0.35,0.37, and 0.40) is compared to the analytical result, Eq. (24). Despite some statistical fluctuations at late times after the quench it is evident from Fig. 16 that predictions (Eq. (24)) and measurements practically coincide. In the inset is also shown the reverse L-jump from Tq = 0.35 to = 1.00. Clearly, the relaxation in this case is much ( 20 times) faster and is also well reproduced by the non-exponential law, Eq. (24). In the absence of laboratory investigations so far, this appears the only unambiguous confirmation for the nonlinear relaxation of GM after a T-quench. 

T = 0.5,0.6,0.7,0.8, and 0.9. Despite some statistical fluctuations at late times after the T-jump, it is evident from Fig. 19 that the different curves collapse onto a single one if time is scaled by a single. As for the system of rate equations, (26), we again find = (I.SSLqo) where the power 5 is determined with an accuracy of 2%. An interpolation formula for the scaling function /(jc — = (0.215 + 8jc) appears to account well... 

These equations can also be used for approximate calculations of F c . The simplest approximation is MFA that neglects all statistical fluctuations and correlations. Therefore, the MFA expression Fmfa given by Eq. (13) corresponds to the omission of the last term in Eq. (16). 

The background is so large that statistical fluctuations in Nb may vitiate a determination of the element sought. 

In the fifth subdivision, which becomes important when trace determinations are pushed to the limit of detection, Nb is comparable with Nt. Statistical fluctuations in both counts may therefore lead to (1) the... 

When m < mc, product assemblages are unstable and thermodynamic considerations predict that the product (B) will tend to revert to the reactant unless the local statistical fluctuations of energy are sufficient to achieve... 

Third, a further simplification of the Boltzmann equation is the use of the two-term spherical harmonic expansion [231 ] for the EEDF (also known as the Lorentz approximation), both in the calculations and in the analysis in the literature of experimental data. This two-term approximation has also been used by Kurachi and Nakamura [212] to determine the cross section for vibrational excitation of SiHj (see Table II). Due to the magnitude of the vibrational cross section at certain electron energies relative to the elastic cross sections and the steep dependence of the vibrational cross section, the use of this two-term approximation is of variable accuracy [240]. A Monte Carlo calculation is in principle more accurate, because in such a model the spatial and temporal behavior of the EEDF can be included. However, a Monte Carlo calculation has its own problems, such as the large computational effort needed to reduce statistical fluctuations. 

The second procedure is to use a small number of large particles in the beginning of the simulation. The results of that are used as the initial conditions for a simulation with a large number of small particles. Thus, in the final results, the statistical fluctuations are reduced. Also, results from a previous simulation can be used as initial conditions for a new simulation with changed parameters, which saves a large amount of time during parameter scans. 

As previously discussed, we expect the scaling to hold if the polydisper-sity, P, remains constant with respect to time. For the well-mixed system the polydispersity reaches about 2 when the average cluster size is approximately 10 particles, and statistically fluctuates about 2 until the mean field approximation and the scaling break down, when the number of clusters remaining in the system is about 100 or so. The polydispersity of the size distribution in the poorly mixed system never reaches a steady value. The ratio which is constant if the scaling holds and mass is conserved,... 

Another attempt to explain the homochirality of biomolecules is based on autocatalysis. The great advantage of asymmetric catalysis is that the catalyst and the chiral product are identical and thus do not need to be separated (Buschmann et al., 2000). The racemic mixture must have been affected by a weak perturbation in order that autocatalysis, which acts as an amplifier of enantioselectivity, could have led to only one of the two enantiomeric forms. This perturbation could have been due to the slight energy difference of the enantiomers referred to above, or to statistical fluctuations. 

Generally, inaccuracies can also be expected at low photon counts (N < 100). Besides comparatively large statistical fluctuations, also a bias in lifetime is introduced by the data fitting procedure [37],... 

For Hamiltonians invariant under rotational and time-reversal transformations the corresponding ensemble of matrices is called the Gaussian orthogonal ensemble (GOE). It was established that GOE describes the statistical fluctuation properties of a quantum system whose classical analog is completely chaotic. 

In true thermospray, charging of the droplets is due to the presence of a buffer in the mobile phase. Both positively and negatively charged droplets are formed due to the statistical fluctuation in anion and cation density occurring when the liquid stream is disrupted. As with the interfaces previously described, involatile buffers are not recommended as blocking of the capillary is more likely to occur if temperature control is not carefully monitored and for this reason ammonium acetate is often used. 

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