Boltzmann distribution modification

Many interesting and useful concepts follow from classical statistical con side rations (eg, the Boltzmann distribution law) and their later modifications to take into account quantum mechanical effects (Bose-Einstein and Fermi-Dirac statistics). These concepts are quite beyond the scope of the present article, and the reader should consult Refs 14 16. A brief excursion into this area is appropriate, however. A very useful concept is the so-called partition function, Z, which is defined as ... 

Modification of this model to get the potential function is obtained considering the Fermi-Dirac distribution function for the electron density and the Boltzmann distribution for the ionic density. This was done by Stewart and Pyatt [58] to get the energy levels and the spectroscopic properties of several atoms under various plasma conditions. Here the electron density was given by... 

At each temperature there will be a thermodynamic equilibrium between the two forms. At the absolute zero, the molecules must be located at the lowest energy level, i.e. J = o, so that only the para modification is stable. At higher temperatures molecules may exist in both odd and even energy levels and in consequence both forms exist. The thermodynamic equilibrium between the two forms is governed by the Boltzmann distribution law. According to this law, the fraction of molecules JVj of the total number jV in the rotational state J is given by ... 

Bose-Einstein distribution - A modification of the Boltzmann distribution which applies to a system of particles that are bosons. The number of particles of energy E is proportional to [e<, where g is a normalization constant, k is the... 

In the previous discussion, a great deal of time has been spent on approaches to surface modification of the clay nanoparticles in order to render the particles to be more compatible with the polymer of interest. This approach mainly concentrates on the enthalpic portion of the Gibb s free energy of intercalation-exfoliation. In order to realize the maximum benefit from a nanocomposite, the exfoliated state is the ultimate goal, since this will present the maximum interfacial interaction between the nanoparticle and the polymer. In reality, a completely exfoliated system probably does not exist, but a Boltzmann distribution of energy states is more likely which invokes some of the entropic terms. In the following... 

Max Planck, utilizing his quantum theory postulates and modifications of the Boltzmann statistical procedure, established the theoretical formula for the spectral distribution curves of a black body ... 

Snider is best known for his paper reporting what is now referred to as the Waldmann-Snider equation.34 (L. Waldmann independently derived the same result via an alternative method.) The novelty of this equation is that it takes into account the consequences of the superposition of quantum wavefunctions. For example, while the usual Boltzmann equation describes the collisionally induced decay of the rotational state probability distribution of a spin system to equilibrium, the modifications allow the effects of magnetic field precession to be simultaneously taken into account. Snider has used this equation to explain a variety of effects including the Senftleben-Beenakker effect (i.e., is, the magnetic and electric field dependence of gas transport coefficients), gas phase NMR relaxation, and gas phase muon spin relaxation.35... 

Falkenhagen considered a modified distribution function which took account of the finite size of the ions by recognising that the total space available to the ions is less than the total volume of the solution. This implicitly means that (1 - - Ka) is not approximated to unity. A modified Poisson-Boltzmann equation was thus used in the derivation of the relaxation effect, but the solution of this modified Poisson-Boltzmann equation was approximated to the first two terms. These modifications gave higher order terms in Cacmai of the type which had been empirically observed. 

Implicit solvation models developed for condensed phases represent the solvent by a continuous electric field, and are based on the Poisson equation, which is valid when a surrounding dielectric medium responds linearly to the charge distribution of the solute. The Poisson equation is actually a special case of the Poisson-Boltzmann (PB) equation PB electrostatics applies when electrolytes are present in solution, while the Poisson equation applies when no ions are present. Solving the Poisson equation for an arbitrary equation requires numerical methods, and many researchers have developed an alternative way to approximate the Poisson equation that can be solved analytically, known as the Generalized Born (GB) approach. The most common implicit models used for small molecules are the Conductor-like Screening Model (COSMO) [96,97], the Dielectric Polarized Continuum Model (DPCM) [98], the Conductor-like modification to the Polarized Continuum Model (CPCM) [99], the Integral Equation Formalism implementation of PCM (lEF-PCM) [100] PB models and the GB SMx models of Cramer and Truhlar [52,57,101,102]. The newest Miimesota solvation models are the SMD (universal Solvation Model based on solute electron Density [57]) and the SMLVE method, which combines the surface and volume polarization for electrostatic interactions model (SVPE) [103-105] with semiempirical terms that account for local electrostatics [106]. Further details on these methods can be found in Chapter 11 of reference 52. 

In the case of electrode-electrolyte solution interfaces, the Poisson-Boltzmann equation has been modified for integrating many effects as, for example, finite ion size, concentration dependence of the solvent, ion polarizability, and so on. More often, this modification consists in the introduction of one or several supplementary terms to the energetic contribution in the distribution, which leads to modified Poisson-Boltzmann (MPB) nonlinear differential equations [52],... 

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