Boltzmann forms

One may also show that MPC dynamics satisfies an H theorem and that any initial velocity distribution will relax to the Maxwell-Boltzmann distribution [11]. Figure 2 shows simulation results for the velocity distribution function that confirm this result. In the simulation, the particles were initially uniformly distributed in the volume and had the same speed v = 1 but different random directions. After a relatively short transient the distribution function adopts the Maxwell-Boltzmann form shown in the figure. 

Chemical processes, in contrast, are processes that are not limited by rates of energy transfer. In thermal processes, chemical reactions occur under conditions in which the statistical distribution of molecular energies obey the Maxwell-Boltzmann form, i.e., the fraction of species that have an energy E or larger is proportional to e p(—E/RT). In other words, the rates of intermolecular collisions are rapid enough that all the species become thermalized with respect to the bulk gas mixture (Golden and Larson, 1984 Benson, 1976). 

If we write the partition function in the standard Boltzmann form Z = Yn exp (—( ), the pressure is then... 

Fukutani et al. [8] also observed NO desorption from on-top species at X = 193 nm. The results are very similar to those obtained by Buntin et al. at X = 532 and 355 nm. The decay of the desorption yield on this surface at X = 532 nm gives a desorption cross section of 1 x 10-22 cm2 [6]. A fit to the TOF spectrum by the non-Boltzmann form gives Tt = 910 K. Rotational energy distributions of... 

The rotational energy distribution of desorbed NO from on-top species on Pt(l 1 1) at A. = 192 nm is observed to have a non-Boltzmann form, as shown in Fig. 14. Furthermore, the population in the two spin-orbit states is substantially inverted, since the population ratio of 2 = 1/2 and 3/2 is 1 2.2 in low J region. For desorption of hep hollow species at = 193 nm, on the other hand, the population ratio... 

The production of MO in ground and electronically excited states by the reactions of M + 02, NO, N20, N02, 03 and S02 has been studied by laser-induced fluorescence and visible chemiluminescence methods. The reactions M + 02 have been particularly well studied. For Y + 02 and Sc + 02, emission is observed [389] from the A2Tl and A 2A states of MO and the branching ratios for population of these states are statistical. The MO vibrational population distribution in the A2Yli/2 state is statistical, whilst the distributions for the other states show inversion. The MO (A) rotational distributions can be fitted by Boltzmann forms with temperatures which differ from the vibrational temperatures [392]. For... 

Figure 2 translates the charge carrier formation reaction into an energy level diagram for various systems. In fact these levels refer to standard chemical potentials or (in the case of the Fermi-levels ) to full chemical potentials (see e.g. Refs.3,35). As long as —in pure materials— the gap remains large compared to RT, the Boltzmann-form of the chemical potential of the respective charge carrier (defect) is valid,... 

As anticipated above, the Boltzmann-form of the chemical potential results. If n in Eq. (25) is identified with nd, the effective... 

Here x = p (ri)1 a/2, (S = piy = C>/y/2IkBT is the inertial effects parameter (large (S corresponds to small inertial effects and vice versa), and we have noted that all the c, 0(0) vanish with the exception n - 0, namely, Cq °(0) = ,/3, where , = pF/(kBT). The last equality follows from the linearized initial (at t = 0) distribution function, which has the Maxwell-Boltzmann form... 

To proceed further we note that Onsager and Machlup s [4] results imply the following Boltzmann form, similar to Eq. (A.46c), for the distribution of initial velocities y... 

We begin by noting that at some time t, all the deep states for which E > E =kT In coot have been accumulating electrons without reemitting them, since their mean time for thermal emission t = Wq cxp(E/kT) is much longer than t. It follows that the distribution of electrons in these states must parallel the density of states. Similarly, the electrons in the shallow states for which EKE have on average experienced many thermal-emission - recapture events. The end result of this statistical scrambling for the shallow states (EKE ) is well known in solid-state physics—a Boltzmann distribution results. Thus the distribution function for the electrons in the shallow states must have the Boltzmann form exp( /fc7 )iV,( ). The proportionality constant can be calculated from the requirement that the distribution of electrons in the shallow localized states be continuous with the distribution in the deep states. If the (constant) fractional occupancy of the deep states isf, then for continuity the population in the shallow states (E K E ) must be ftxv>[iE-E )/kT]N,(E). 

This form of the entropy is more general than the Boltzmann form (6.1) it was introduced in Gibbs treatise [147]. 

Returning to our original problem, we would like to prove that for any initial distribution /(r, v, t) and for boundary interactions that satisfy the thermostat condition, Eq. (40a), with a constant wall temperature at all points, the gas will eventually reach a state of total equilibrium in which the distribution function has the Maxwell-Boltzmann form given by Eq. (38) with the temperature equal to the wall temperature. 

The Boltzmann form of the reactor equations. The foregoing formulation of reactor theory puts the entire substance of the theory into the all-powerful and rather mysterious kernel H, or the derived kernels—only shghtly less powerful and mysterious, K, Kq, and G. Actually the kernel G, at least, can be determined experimentally if the medium is uniform, isotropic, and infinitely large. In that case, all the kernels, besides being essentially positive, are displacement operators in However, if the reactor is... 

By means of (10) it is a simple matter to obtain the Boltzmann form of (1). Differentiation of (1) with respect to t gives... 

We see that the velocity autocorrelation function of a free Brownian particle falls exponentially with time. The time required for an arbitrary initial velocity distribution of the particle to settle down to the Maxwell-Boltzmann form corresponding to the temperature of the bath (thermaliza-tion of velocities) is on the order of... 

With this assumption at hand, it is possible to obtain the evolution of the translational, rotational and vibrational contributions to the energy of the fullerene fragments, as well as the emitted dimers. In particular, from the respective distributions the corresponding temperatures can be inferred by appropriate fltting onto Boltzmann forms. Figure 4.8 shows the variations of the three temperatures of carbon dimers emitted successively from after a... 

When the primary facets of ice are defect free, the relevant step generation mechanism is the formation of two-dimensional nuclei. The process is driven by thermally activated fluctuations of the liquid phase that create two-dimensional heterophase clusters with solid-like structure. The most probable cluster geometry is tiiat of a pillbox, for which the edge-to-surface free energy ratio will favor spreading at a given drive. In analogy with three-dimensions, the nucleation frequency /per unit facet area. 4 is of a Maxwell-Boltzmann form, / exp - n cr /Ap kbT), where [Pg.47]

Please note that higher concentrations demand corrections with respect to the Boltzmann-form, already via statistical effects, that is corrections with respect to the configurational entropy. Cf. page 214ff. 

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