Statistics Boltzmann

Nearly ten years ago, Tsallis proposed a possible generalization of Gibbs-Boltzmann statistical mechanics. [1] He built his intriguing theory on a reexpression of the Gibbs-Shannon entropy S = —k Jp r) np r)dr written... 

When g = 1 the extensivity of the entropy can be used to derive the Boltzmann entropy equation 5 = fc In W in the microcanonical ensemble. When g 1, it is the odd property that the generalization of the entropy Sq is not extensive that leads to the peculiar form of the probability distribution. The non-extensivity of Sq has led to speculation that Tsallis statistics may be applicable to gravitational systems where interaction length scales comparable to the system size violate the assumptions underlying Gibbs-Boltzmann statistics. [4]... 

In the limit 5 -+ 1 of Gibbs-Boltzmann statistics, using the fact that limi o[l — a(l — 6 )] / = we find that... 

T. Huber and A.E. Torda, Protein fold recognition without Boltzmann statistics or explicit physical basis, submitted to Protein Sci. (1997). 

Examining transition state theory, one notes that the assumptions of Maxwell-Boltzmann statistics are not completely correct because some of the molecules reaching the activation energy will react, lose excess vibrational energy, and not be able to go back to reactants. Also, some molecules that have reacted may go back to reactants again. 

As long as the system is in thermal equilibrium the individual molecules in a sample are distributed among the 15 accessible states according to Boltzmann statistics ... 

In case of the doped semiconductor of -type under consideration the situation gets simple due to large thickness of SCR if compared to the free path length of the carriers. Therefore, substituting expression (1.42) into (1.41) under condition of applicability of the Boltzmann statistics for the free electrons and holes leads to expression... 

Thermal computation which comprises algorithms inspired by multiparticle systems like BROWNian motion (BROWNian search, Monte Carlo search) on the one hand and simulated annealing which is inspired by Boltzmanns statistics on the other hand. 

The Boltzmann statistical weight for each rotamer is multiplied by (1 + aPt(cos 8)), with a being an adjustable parameter, before a particular rotamer is selected. Values of a, between zero and unity, will give rise to an order parameter for segment orientations. Average of Pt(cos 8) of 0.0 to 0.5. 

The probability that an atom will successfully move can be estimated by using Maxwell-Boltzmann statistics. The probability p that an atom will move from one position of minimum energy to an adjacent position is... 

The ionic densities must in turn depend on the potential x). We choose (j) po) = 0 as our reference, and apply Boltzmann statistics ... 

According to the calorimetrists view, if we consider an electron as a particle that takes part in chemical reactions as any other chemical element, then it is fair to agree on Af7/°(c-,g) = 0, at any temperature. Moreover, because we regard the electron as a regular monoatomic perfect gas, Boltzmann statistics or the equipartition principle [1] imply that C° = Cy + R = 5R/2 (three translational degrees of freedom) or X = — //(, )c = 298.15 x C" =... 

Electrode reactions can be classified into two groups one in which an electron transfer takes place across the electrode interface, such as ferric-ferrous redox reaction (Fet, + e = Fe ) and the other in which an ion transfer takes place across the electrode interface, such as iron dissolution-deposition reaction (Fe M = FeVq). Since electrons are Fermi particles in contrast to ions that obey the Boltzmann statistics as described in Chap. 1, the reaction kinetics of the two groups differ in their electrode reactions. 

We consider a simple reaction from reactant R to product P in Eqn. 7-8 these R and P are the particles of Boltzmann statistics ... 

Debye and Huckel applied the Boltzmann statistical distribution law and the Poisson equation for electrostatics in the model above (1,6, 10). In the calculations using the model above they considered one particular ion (the reference ion, or central ion) with... 

In the classical high-temperature limit, kBT hv, where kB is the Boltzmann constant, and hv is the spacing of the quantum-mechanical harmonic oscillator energy levels. If this condition is fulfilled, the energy levels may be considered as continuous, and Boltzmann statistics apply. The corresponding distribution is... 

---