The Boltzmann relation

The potential of mean force is a useful analytical tool that results in an effective potential that reflects the average effect of all the other degrees of freedom on the dynamic variable of interest. Equation (2) indicates that given a potential function it is possible to calculate the probabihty for all states of the system (the Boltzmann relationship). The potential of mean force procedure works in the reverse direction. Given an observed distribution of values (from the trajectory), the corresponding effective potential function can be derived. The first step in this procedure is to organize the observed values of the dynamic variable, A, into a distribution function p(A). From this distribution the effective potential or potential of mean force, W(A), is calculated from the Boltzmann relation ... 

According to the Boltzmann relation, the entropy change AS for the process of compression is given by... 

This is immedicately recognizable as the Boltzmann relation. Moreover, since r 0 + r c = 1, then... 

The wave equation is built from V E cx pext/ - Because electrostatic double-layer equations are easier to think about in terms of potentials rather than electric fields E = -V0, we set up the problem of ionic-charge-fluctuation forces in terms of potentials. Charges pext come from the potential 0 through the Boltzmann relation... 

Let us derive an expression for the entropy of mixing of the large particles with a fluid containing small particles. The procedure employed is similar to that of Ref. [17]. The entropy of mixing is given by the Boltzmann relation... 

Substituting Eq. (A.4) into the Boltzmann relation, and using the Stirling approximation for the factorials, one obtains... 

It is worth remembering that we are still working with the one-electron picture, and that we have applied the Boltzmann relation in order to approximate Fermi and quasi-Fermi distribution functions, assuming the quasi-free electron and hole densities of states in the bands. 

The work gained in bringing a hydrogen ion from the interior of a solution to the surface is —c . Hence, by the Boltzmann relation,... 

Saturation leads to equalization in the populations of the energy levels, contrary to the Boltzmann distribution. On the other hand, a number of NMR techniques can be employed to increase the population difference well beyond that given by the Boltzmann distribution. In some instances it is convenient to retain the formalism of the Boltzmann relation by defining a spin temperature Ts that satisfies Eq. 2.19 for a given ratio n /na. For times much less than T, it is meaningful to have Ts = T, the macroscopic temperature of the sample, because the spin system and lattice do not interact in this time frame.Viewed in this way, saturation... 

Because of assumption 1, however, we need not find an explicit expression for U but can concentrate on the entropy expression. For this we shall again invoke the Boltzmann relation [Appendix 2, equation (g)], as we did for the isolated chain ... 

A common method to determine voltage sensitivity is shown in Figure 5A, in which the data for the larger channel from one experiment are fitted to a form of the Boltzmann relation that can be plotted as a linear function of... 

Figure 5. Voltage dependence of P0 for the larger (130-pS) connexin-32 channels. A, Fit to ln( P0/Pj) as a function of voltage. B, Fit to P0 as a function of voltage. The solid line is calculated with the Boltzmann relation (SSE — 0.007) and the dashed line is an independent fit (SSE = 0.001).

Suppose that the spin populations are now disturbed so that some additional nuclei in state a are promoted to state b. While the system may no longer be in thermal equilibrium with the lattice, we can still describe the populations by the Boltzmann relation provided that we change the definition of T to keep the relationship correct. Since the ratio is now... 

In NMR, we say that the spins are saturated and this corresponds to the situation immediately after a n/2 pulse. From the Boltzmann relation, Tg= and what s more Tg=- is a valid... 

There are certain consequences of the small size of the energy gap involved in a nuclear-spin transition. The Boltzmann relation gives the populations of nuclear spins in the upper energy state N and in the lower energy state Nj) in terms of the energy gap A between them ... 

Furthermore, the Boltzmann relation that describes the probability and stability of a given cluster can be used. From the theory of Nyrkova and Semenov this gives ... 

The Boltzmann relation provides the bridge from a statistical mechanical description of molecular structure to experimentally determined thermodynamic quantities. It is an elegant yet simple statement of entropy (5) in terms of thermodynamic probability (W, the number of a priori equally probable states accessible to the system) (Eyring et al, 1964), i.e.,... 

At equilibrium, ions are distributed according to the Boltzmann relation so that p x) can be written as... 

The Boltzmann relation provides a statistical interpretation of the entropy. The greater the number of accessible states, the less our knowledge of the system and the more randomness or disorder in the system. This entropy is a measure of disorder. The tendency of the entropy to increase reflects the tendency of thermodynamic systems to increase in disorder just as an initially ordered deck of cards increases in disorder during a game of cards. 

In semiconductors the electrical conductivity is proportional to the electron density in the conduction band. With the band gap A g this density is, according to the Boltzmann relation... 

The most probable state of a system is the state that has the greatest number of ways, Y, of being realized. The entropy S is related to this number of complexions Y by the Boltzmann relation ... 

Resonance absorption can only be detected if the spin levels involved differ in population. For a very large number N of protons in thermal equilibrium. the distribution of the spin 1/2 nuclei between the ground (a) slate and the excited (/() state is given by the Boltzmann relation ... 

To get the translational entropy, apply the Boltzmann relation and Stirling s approximation, Equation (4.26), to Equation (27.3),... 

The observable effect of an orienting potential upon the ESR spectrum is the anisotropic distribution of orientations produced by the potential. The equilibrium distribution of orientations obeys the Boltzmann relation. 

---