Boltzmann thermal distribution

If the bath is kept at thennal equilibrium, the system should approach the same thermal equilibrium at long time. In practical situations we often address this distribution in the representation defined by the system eigenstates, in which case the statement holds rigorously in the limit of zero coupling. Detailed-balance relationships such as Eq. (10.161) indeed imply that a Boltzmann thermal distribution is a stationary (da/dt = 0) solution of the Redfield equation. 

With use of the microcanonical reaction mechanism and the nonadiabatic transition probability P E) from the left to right in Figure 12.1 at a given translation energy E, the thermal rate k can be obtained by an average with the Boltzmann thermal distribution as a weight function ... 

As stated earlier, within C(t) there is also an equilibrium average over translational motion of the molecules. For a gas-phase sample undergoing random collisions and at thermal equilibrium, this average is characterized by the well known Maxwell-Boltzmann velocity distribution ... 

Control of sonochemical reactions is subject to the same limitation that any thermal process has the Boltzmann energy distribution means that the energy per individual molecule wiU vary widely. One does have easy control, however, over the energetics of cavitation through the parameters of acoustic intensity, temperature, ambient gas, and solvent choice. The thermal conductivity of the ambient gas (eg, a variable He/Ar atmosphere) and the overaU solvent vapor pressure provide easy methods for the experimental control of the peak temperatures generated during the cavitational coUapse. 

The relative population ratio FJFi was slightly higher than expected from a 300 K thermal distribution (e.g. 2.1 vs 1.8). Of particular note, in comparison to a simple Boltzmann distribution, there was a substantial absence of population in the F2(J < S.S) levels from that expected based on a thermal (300 K) distribution. Approximately 1% of the desorbed molecules were vibrationally excited. 

Consideration of the energy level diagram of Fig. 1 leads to the conclusion that there must be a mechanism by which the Boltzmann thermal equilibrium distribution can be maintained during resonance absorption. For, if there were no such mechanism, upon application of a field Hi at the resonance frequency initial absorption of energy would occur, but would... 

As discussed by Fuchs and Sutugin (1970, 1971) and Motz and Wise (1960), in this continuous regime, distortion of the Boltzmann velocity distribution in the region close to the surface occurs if there is rapid uptake. In effect, the normal thermal velocity distribution is distorted so that the effective speed toward the surface is higher. In the case of a surface where the uptake occurs on every collision, the net speed toward the surface is effectively doubled. This adds an additional term to the rate of transfer of the gas to the surface, which when normalized using Eq. (PP), gives an additional resistance term of -1 /2. The overall normalized conductance is therefore given by... 

To estimate the broadening of spot size by the thermal velocity of the ions, let us assume that the image gas atoms immediately before ionization have a Maxwell-Boltzmann velocity distribution with an effective temperature T which is very close to the tip temperature Tt. If n(y) dy represents the number of ions arriving at the screen between y and y + dy, we have... 

Solution The Maxwell - Boltzmann velocity distribution for the random motion of a thermally equilibrated neutron gas is... 

Rotational Population Distributions. As expected, there is an overpopulation in the A state level and an underpopulation in the X state level connected by the laser, compared to a thermal distribution (see Fig. 1). Higher rotational levels (N 6) in the A-state are described by a Boltzmann distribution with T 940°K, well less than the gas temperature and reflecting the energy dependence of the or. The high-N levels of the X-state are described by a Boltzmann distribution with very high T (3200°K) but this may be an artifact of the model, due principally to the assumed... 

Thus, even in the presence of pumping, of energy Sk, from the thermal bath into the modes k, a Planck-type distribution of photons over the modes k, and hence a Boltzmann-type distribution of molecular systems over vibrational states (v, V2- -vj,..vk,..vz) results from linear systems. 

Here v is the (scalar) velocity, f v) is the normalized three-dimensional velocity distribution as determined by the molecular dynamics simulation at a given point in time. f u) is the three dimensional Maxwell-Boltzmann velocity distribution with a temperature determined by the condition that / (i/) has the same mean energy as the velocity distribution obtained in the simulation after a long propagation time. At thermal equilibrium DS = 0 and otherwise it is positive. The larger is DS, the more extreme is the deviation from equilibrium. The results for the entropy deficiency are shown in... 

Applications of RRKM theory often focus on the high pressure limit of the dissociation rate coefficient. The presence of multiple collisions prior to reaction generally maintains a Boltzmann population distribution. Correspondingly, the thermal rate coefficient is then expressed as a Boltzmann average over the energy and angular momentum resolved rate coefficient, which reduces to... 

The important point that arises from the Rodger-Sceats reduction is that the dynamics can take place on an effective potential P defined by Eq. (2.19). The origin of the potential can be traced back to the requirement that at long times the system must achieve a thermal distribution which is consistent with a Boltzmann distribution on the full potential and the use of P simply ensures that the partition functions of the system will be given correctly. This will be very important in applications to chemical reactions, because the partition function plays an important role in determining Arrhenius activation parameters. The dependence of P on the reaction coordinate simply accounts for this effect. For example, the reactant and transition state configurations are defined by the minima and maxima of P at qi and gj, and the Boltzmann factor for activation is... 

Using the Maxwell-Boltzmann energy distribution (Glasstone and Sesonske, 1981), it may be shown that the average speed of perfectly thermalized neutrons is given by... 

In the Westcott formulation the energy distribution (E) is treated as a Maxwell-Boltzmann energy distribution 0 f( of thermalized neutrons on which is superimposed an epithermal distribution 0 -( of nonthermalized neutrons, so that... 

In our work the SF has a twofold application. First, the formula (254) is used for description of thefar-IR spectrum in the high-frequency approximation. Such a spectrum accounts for resonance or, more often, quasi-resonance interaction of dipoles with radiation. An absorption peak arises when the radiation frequency co is near the mean thermal frequency determined by the Boltzmann energy distributions. 

Simple collision theories neglect the internal quantum state dependence of ct. The rate constant as a function of temperature T results as a thermal average over the Maxwell-Boltzmann velocity distribution p E ... 

---