Boltzmann tail

Inclusion in the vme of v-t terms only produces a Boltzmann distribution at the gas temperature Tg. The inclusion of e-v, v-v and v-t terms produces the distributions of Fig. 8 e-v processes tend to establish the non equilibrium vibrational temperature 8 j > Tg. v-v processes, which have rates several orders of magnitude larger than v-t ones, at low vibrational quantum numbers, tend to create a Treanor distribution up to approximately v = v. Above a given level, v-t processes dominate the v-v ones and determine a Boltzmann tail characterized by a temperature approaching Tg. The plateau, which extends from approximately v = vi up to the onset of the Boltzmann tail, is connected to the near resonant v-v terms. The dependence of kd on pressure (Fig. 9) can be interpreted along these lines by examining the normalized distributions of Fig. 10. The same arguments apply to the data of Fig. 11. 

Figure 7 shows the behaviour of as a function of Tg at two different E/N values. The small increase of k j with decreasing gas temperature should be compared with the strong dependence of k on Tg. This behaviour is due to the fact that vibrational levels responsible for dissociation in JVE are those far from the v-t deactivation region (i.e. far from the Boltzmann tail) while, in PVM, only levels belonging to the tail can dissociate. The strong dependence of k on Tg is therefore attributed to the sensitivity of the Boltzmann tail of the N distribution to the gas temperature. 

In molecular nitrogen large e-v and v-v rates are associated with unusually low v-t rates. This represents a crossing of conditions favourable to the PVM, as discussed in 2.3.1. This situation is immediately reflected in the N distributions of Fig. 19. These distributions consist of a Treanor distribution followed by a long plateau extending up to the dissociation limit. The Boltzmann tail has practically disappeared, as consequence of the low v-t rates, and the influence of Tg on the distribution is limited. The behaviour of the NP distributions is reflected on the k values reported... 

In Fig. 22a the arrows mark the onset of a Boltzmann tail for 1% and 10% of atomic nitrogen. The corresponding t> s have been calculated according to Ref.8). The occurrence of non-Boltzmann distributions of the type illustrated in Figs. 19 and 22 has been demonstrated experimentally. 

Figure 32 shows the complete N distribution at different times. Three portions of the N distribution are present, the Treanor, the plateau and the Boltzmann tail. The plateau is now controlled by v-v exchanges because the contribution coming from the recombination process is negligible as a consequence of the low concentration of oxygen atoms. An estimated contribution of the recombination process is also shown in Fig. 32 at a time at which the oxygen atom concentration is practically stationary. 

Figure 39 shows typical N distributions at different times for ne = 1014 cm-3. Also reported in this figure is the temporal evolution of the atoms. It should be noted that N distributions do not display a Boltzmann tail. In fact with ne = 1014 cm-3,... 

The values reported in the table are approximate averages deduced from data reported by the authors. The distributions also can be described by T = 1000-1800 K and 7 = 2500 500 K with truncation of the Boltzmann tail. 

Fig. 2. (a) Energy, E, versus wave vector, k, for free particle-like conduction band and valence band electrons (b) the corresponding density of available electron states, DOS, where Ep is Fermi energy (c) the Fermi-Dirac distribution, ie, the probabiUty P(E) that a state is occupied, where Kis the Boltzmann constant and Tis absolute temperature ia Kelvin. The tails of this distribution are exponential. The product of P(E) and DOS yields the energy distribution... 

The abundance of light elements at high altitude leads to a finite flux of these substances escaping the Earth s gravitational field. This results from a combination of a very long mean free path and a few particles having the requisite escape velocity due to the high-velocity "tail" of the Boltzmann velocity distribution. 

Tlj = V Mijcrji/eij, where mu is the mass. In addition, Boltzmann s constant was set equal to unity. The pair potential was cut and shifted at Rcut = 2.5. No tail correction was used. The shift to make the potential zero at the cutoff is necessary for consistency between the Monte Carlo and the molecular dynamics aspects of the computations. 

If U0 and U1 were the functions of a sufficient number of identically distributed random variables, then AU would be Gaussian distributed, which is a consequence of the central limit theorem. In practice, the probability distribution Pq (AU) deviates somewhat from the ideal Gaussian case, but still has a Gaussian-like shape. The integrand in (2.12), which is obtained by multiplying this probability distribution by the Boltzmann factor exp (-[3AU), is shifted to the left, as shown in Fig. 2.1. This indicates that the value of the integral in (2.12) depends on the low-energy tail of the distribution - see Fig. 2.1. 

Fig. 2.1. Po(AU), the Boltzmann factor exp (—/3AU) and their product, which is the integrand in (2.12). The Iovj-AU tail of the integrand, marked with stripes is poorly sampled with Po(AU) and, therefore, is known with low statistical accuracy. However, it provides an important contribution to the integral...

The rate constant is measured in units of moles dnr3 sec /(moles dnr3)", where n = a + b. Time may also be in minutes or hours. It should be noted that in case where the reaction is slow enough, the thermal equilibrium will be maintained due to constant collisions between the molecules and k remains constant at a given temperature. However, if the reaction is very fast the tail part of the Maxwell-Boltzmann distribution will be depleted so rapidly that thermal equilibrium will not be re-established. In such cases rate constant will not truly be constant and it should be called a rate coefficient. 

One end of each surface-active molecule in a monolayer is anchored firmly to the Uquid surface by the attraction of the polar head group for the aqueous subphase, while the hydrophobic portion is displaced easily from it. If the molecules are separated widely as in a gaseous monolayer, the simple two-dimensional gas law is approached, namely, irA = kT, where k is the Boltzmann constant. The hydro-phobic chains are free to assume almost any orientation above the surface and may sweep out circles with radii as long as their tails by rotating around their point of attachment at the head group. However, intermolecular translational movements are restricted to the two-dimensional interfacial plane because the hydrophilic head groups cannot leave the aqueous surface. 

In an attempt to model the spectral functions of rare gas mixtures, Fig. 3.2, it was noted that a Gaussian function with exponential tails approximates the measurements reasonably well [75], about as well as the Lorentzian core with exponential tails. Two free parameters were chosen such that at the mending point a continuous function and a continuous derivative resulted the negative frequency wing was again chosen as that same curve, multiplied by the Boltzmann factor, to satisfy Eq. 3.18. Subsequent work retained the combination of a Lorentzian with an exponential wing and made use of a desymmetrization function [320],... 

We show the Hydrogen ionization fraction in figure 10.1. We find a very rapid transition from X 1 to X 0 at z 1,100, corresponding to T(z) = 0.3eV. This is considerably lower than our naive expectation of 13.6eV, due to the small prefactors on the right hand side of Eq. 10.7, themselves due to the very small value of nB/n-y = 2.7 x 10 8(flBh2) there are many more photons than baryons, and so even the small fraction in the high-energy tail of the Boltzmann distribution are sufficient to keep the Universe ionized. 

Figure 3-9. Boltzmann energy distributions at 10°C (solid line) and 30°C (dashed line). The inset is a continuation of the right-hand portion of the graph with the scale of the abscissa (E) unchanged and that of the ordinate [ ( )] expanded by 104. The difference between the two curves is extremely small, except at high energies although very few molecules are in this "high-energy tail, there are many more such molecules at the higher temperature.

The Boltzmann factor, preferentially weights the low-energy tail of the... 

We briefly comment on some other treatments. One of the oldest precursors comes from Singer ), who applied lattice theory but assumed all segments to be restricted to the train layer. Frisch and Slmha ) presented a model accounting for loops and tails in addition to trains, using random-walk statistics with a Boltzmann factor for train segments. However, their statistical treatment is incorrect... 

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