Boltzmann statistical fluctuations

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. 

Uk U k Up Velocity vector of phase k Fluctuating velocity vector of phase k Velocity vector of particles W mb Total number of possible arrangements for a certain set of in the corrected Maxwell-Boltzmann statistics... 

This equation can be obtained as follows. Since the order parameter is conserved, it obeys a conservation law, 90/9/ - - V J = 0. The diffusion current J is given by J = — where fx is the chemical potential difference between oil and water. With the relation / = which follows from standard thermodynamics, we arrive at the first part of Eq. (49). The second part, a random (Gaussian) noise source, is necessary to describe thermal fluctuations. It can be shown that the order parameter field has the correct Boltzmann statistics if the noise has the correlations... 

To preserve Boltzmann statistics in the canonical ensemble, the relationship between the weight fnnctions, (r) = [w (r)] results in the following fluctuation-dissipation relation (Espafiol and Warren 1995) ... 

This quite general concept is used to specify any shift (in time and space) between the instantaneous value of magnetic moments and dieir mean value given by flie molecular field approximation and the Boltzmann statistics. Spin fluctuations can be individual and/or collective. In the latter case processes such as spin waves or paramagnons are considered and many models have been developed to interprete experiments with these concepts. 

The mean-squared fluctuations in thermal equilibrium can be calculated as for any pair of energy conjugated variables, by applying Boltzmann statistics. The probability distribution for (j)k is... 

Particles in a magnetic fluid are always attracted towards the direction of an applied magnetic gradient. This process competes with the diffusion of the particles due to thermal fluctuation. According to Boltzmann statistics, the maximum diameter of a particle in which thermal fluctuation overcomes the aggregation force of the magnetic field is expressed by the following equation ... 

Boltzmann statistics, the difference in population of the Zeeman levels is therefore extremely small. The absorption of photons of energy h would rapidly equalize these populations (bringing the spin temperature to infinity) if no other process than Einstein s spontaneous emission would contribute to restore the equilibrium Boltzmann distribution. Indeed, the natural life time at these frequencies is of the order of 10 s, while in practice the equalization of the spin temperature with the lattice temperature requires time of the order of the second in liquids. The relaxation mechanisms which act in these circumstances are linked to the fluctuation of the magnetic field (or quadrupolar coupling) inside of the sample, more precisely to the Fourier component of this fluctuation at the NMR frequency. The relaxation so far described correspond to the recovery of the z component of the magnetisation which measures the actual difference of population of the Zeeman levels. Therefore it is called longitudinal or spin lattice relaxation time T. ... 

It is most remarkable that the entropy production in a nonequilibrium steady state is directly related to the time asymmetry in the dynamical randomness of nonequilibrium fluctuations. The entropy production turns out to be the difference in the amounts of temporal disorder between the backward and forward paths or histories. In nonequilibrium steady states, the temporal disorder of the time reversals is larger than the temporal disorder h of the paths themselves. This is expressed by the principle of temporal ordering, according to which the typical paths are more ordered than their corresponding time reversals in nonequilibrium steady states. This principle is proved with nonequilibrium statistical mechanics and is a corollary of the second law of thermodynamics. Temporal ordering is possible out of equilibrium because of the increase of spatial disorder. There is thus no contradiction with Boltzmann s interpretation of the second law. Contrary to Boltzmann s interpretation, which deals with disorder in space at a fixed time, the principle of temporal ordering is concerned by order or disorder along the time axis, in the sequence of pictures of the nonequilibrium process filmed as a movie. The emphasis of the dynamical aspects is a recent trend that finds its roots in Shannon s information theory and modem dynamical systems theory. This can explain why we had to wait the last decade before these dynamical aspects of the second law were discovered. 

Not to be forgotten is the assumption that neither the presence of the electrolyte nor the interface itself changes the dielectric medium properties of the aqueous phase. It is assumed to behave as a dielectric continuum with a constant relative dielectric permittivity equal to the value of the bulk phase. The electrolyte is presumed to be made up of point charges, i.e. ions with no size, and responds to the presence of the charged interface in a competitive way described by statistical mechanics. Counterions are drawn to the surface by electrostatic attraction while thermal fluctuations tend to disperse them into solution, surface co-ions are repelled electrostatically and also tend to be dispersed by thermal motion, but are attracted to the accumulated cluster of counterions found near the surface. The end result of this electrical-thermodynamic conflict is an ion distribution which can be represented (approximately) by a Boltzmann distribution dependent on the average electrostatic potential at an arbitrary point multiplied by the valency of individual species, v/. 

Here, vmech is the mechanically effective chain density specified, e.g., in [168], Ac 0.67 [170] is a microstructure factor which describes the fluctuations of network junctions, Na the Avogadro number, p mass density, Ms and Zs molar mass and length of a statistic segment, respectively, kB the Boltzmann constant, and T absolute temperature. 

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