Boltzmanns results

If one accepts the ergodic hypothesis, then Eq. (34) becomes a purely mechanical theorem independent of any probability considerations. 

If one discards the ergodic hypothesis, or tries to retain it in a modified form,117 then for the time being one lacks any criterion to decide whether Eq. (34) is still valid or even whether it represents a somehow useful approximation. 

If we accept the validity of Eq. (34), we can conclude from it more than just a statement about average behavior. In fact, it also determines essentially the relative time intervals that the gas spends in the various distributions of state. 

This deeper formulation of the question was shelved in Boltzmann s investigations by the soon emerging formulation of the 17-theorem (1872). Boltzmann took up this problem again only to counter Loschmidt s Umkehreinwand (Section 7b) and to obtain a modified formulation of the 17-theorem. He tried to show that, if we consider a motion of unlimited duration, then the Maxwell-Boltzmann distribution very strongly dominates in time over all other distributions, and hence the tendency to approach this particular distribution is quite understandable. 

We will discuss all this in detail in Sections 13 and 14 before doing so, however, we want to discuss certain problems which will be of help later. 

---

A long time ago, Stem noted that the traditional assumption that the ions interact only with a mean electrical field (the Poisson - Boltzmann approach) leads to an ionic density in the vicinity of the interface that exceeds the available volume. A simple way to avoid this difficulty is to consider that the ions are hydrated, and therefore there are fewer positions available to them in the vicinity of charged surfaces [4.1]. When compared to the traditional Poisson-Boltzmann result, this correction leads to an increase in the repulsive force at... 

In all of the discussion above, comparisons have been made between various types of approximations, with the nonlinear Poisson-Boltzmann equation providing the standard with which to judge their validity. However, as already noted, the nonlinear Poisson-Boltzmann equation itself entails numerous approximations. In the language of liquid state theory, the Poisson-Boltzmann equation is a mean-field approximation in which all correlation between point ions in solution is neglected, and indeed the Poisson-Boltzmann results for sphere-sphere [48] and plate-plate [8,49] interactions have been derived as limiting cases of more rigorous approaches. For many years, researchers have examined the accuracy of the Poisson-Boltzmann theory using statistical mechanical methods, and it is... 

A and grid dimensions of 36 and 35. The resulting solvation energies are -6.63, —6.60, —6.56 and —6.69 kcal/mol, respectively. From these values we can conclude that we have calculated the Poisson-Boltzmann result for the electrostatic contribution to the solvation energy of acetic acid within about 0.1 kcal/mol. 

This completes the heuristic derivation of the Boltzmann transport equation. Now we trim to Boltzmaim s argument that his equation implies the Clausius fonn of the second law of thennodynamics, namely, that the entropy of an isolated system will increase as the result of any irreversible process taking place in the system. This result is referred to as Boltzmann s H-theorem. 

When this equation is applied to a system composed of a macromolecule immersed in an aqueous medium containing a dissolved electrolyte, the fixed partial charges of each atom of the macromolecule result in a charge density described by p, and the mobile charges of the dissolved electrolyte are described by /O , which i derived from a Boltzmann distribution of the ions and coions. 

Tine force field was then used to predict the results for fhe addition of the E and Z isomers c Ihe enol boronate of butanone (R = Me) to ethanol (R = Me). The relevant transitio. Iructures are shown in Figure 11.34. A Boltzmann distribution, calculated at the ten perature of the reaction (—78°C), predicted that the Z isomer would show almost complel syn selectivity syn anti = 99 1) and that the E isomer would be selective for the an product anti syn = 86 14). These results were in good agreement with the experunenti... 

In order to reduce the amount of computation time, some studies are conducted with a smaller number of solvent geometries, each optimized from a different starting geometry. The results can then be weighted by a Boltzmann distribution. This reduces computation time, but also can affect the accuracy of results. 

In a plasma, the constituent atoms, ions, and electrons are made to move faster by an electromagnetic field and not by application of heat externally or through combustion processes. Nevertheless, the result is the same as if the plasma had been heated externally the constituent atoms, ions, and electrons are made to move faster and faster, eventually reaching a distribution of kinetic energies that would be characteristic of the Boltzmann equation applied to a gas that had been... 

For most purposes only the Stokes-shifted Raman spectmm, which results from molecules in the ground electronic and vibrational states being excited, is measured and reported. Anti-Stokes spectra arise from molecules in vibrational excited states returning to the ground state. The relative intensities of the Stokes and anti-Stokes bands are proportional to the relative populations of the ground and excited vibrational states. These proportions are temperature-dependent and foUow a Boltzmann distribution. At room temperature, the anti-Stokes Stokes intensity ratio decreases by a factor of 10 with each 480 cm from the exciting frequency. Because of the weakness of the anti-Stokes spectmm (except at low frequency shift), the most important use of this spectmm is for optical temperature measurement (qv) using the Boltzmann distribution function. 

---