Plancks Formulation

In Nemst s statement of the third law, no comment is made on the value of the entropy of a substance at 0 K, although it follows from his hypothesis that all pure crystalline substances must have the same entropy at OK. Planck [2] extended Nemst s assumption by adding the postulate that the value of the entropy of a pure solid or a pure liquid approaches zero at 0 K  

The assumption of any hnite constant for the entropy of all pure solids and liquids at OK leads to Nernst s theorem [Equation (11.3)] for these substances. 

Equation (11.4) provides a convenient value for that constant. Planck s statement asserts that 5qk is zero only for pure solids and pure liquids, whereas Nernst assumed that his theorem was applicable to all condensed phases, including solutions. According to Planck, solutions at 0 K have a positive entropy equal to the entropy of mixing. (The entropy of mixing is discussed in Chapters 10 and 14). 

---

We have introduced the Fokker-Planck equation as a special kind of M-equation. Its main use, however, is as an approximate description for any Markov process Y(t) whose individual jumps are small. In this sense the linear Fokker-Planck equation was used by Rayleigh 0, Einstein, Smoluchowskin), and Fokker, for special cases. Subsequently Planck formulated the general nonlinear Fokker-Planck equation from an arbitrary M-equation assuming only that the jumps are small. Finally Kolmogorov8 provided a mathematical derivation by going to the limit of infinitely small jumps. 

After addressing the Langevin and fractional Fokker-Planck formulations of Levy flight processes in some more detail, we will show that in the presence of steeper than harmonic external potentials, the situation changes drastically The forced Levy process no longer leads to an Levy stable density but instead to a multimodal PDF with steeper asymptotics than any Levy stable density. 

The dynamics of transition between the states have to be considered independently, which is most conveniently described by a Fokker—Planck formulation. 

The first part of Carnot s theorem is another variant of the Thomson-Planck formulation of the II. Principle of Thermodynamics. 

The Kelvin-Planck formulation of the second law of thermodynamics states "It is impossible to construct a heat engine operating in a cycle that absorbs heat from a reservoir and performs an equal amount of work." Redraw Figure 3.1 to represent such an impossible engine. What would its efficiency be Why is such an engine impossible ... 

Radiation is a process that is different from both conduction and convection, because the substances exchanging heat need not be touching and can even be separated by a vacuum. A law formulated by German physicist Max Planck in... 

Historical Background.—Relativistic quantum mechanics had its beginning in 1900 with Planck s formulation of the law of black body radiation. Perhaps its inception should be attributed more accurately to Einstein (1905) who ascribed to electromagnetic radiation a corpuscular character the photons. He endowed the photons with an energy and momentum hv and hv/c, respectively, if the frequency of the radiation is v. These assignments of energy and momentum for these zero rest mass particles were consistent with the postulates of relativity. It is to be noted that zero rest mass particles can only be understood within the framework of relativistic dynamics. 

The visualization of light as an assembly of photons moving with light velocity dates back to Isaac Newton and was formulated quantitatively by Max Planck and Albert Einstein. Formula [1] below connects basic physical values ... 

In 1913, the Danish physicist Niels Bohr (1885-1962) sees problems in Rutherford s model and refines it to suggest that electrons exist only in specific states. He uses Planck s constant, formulated by German physicist Max Planck, to explain the stability that these states confer on atoms. In 1919, Rutherford—now director of the Cavendish Lab at Cambridge—discovers the... 

Application of elementary conservation laws leads to formulation of a general expression for which is often denoted as the Nernst-Planck equation ... 

Equation (10) is known as the Nernst-Planck equation. This equation can be given in all kinds of formulations. Another common one is ... 

In order to go beyond the simple description of mixing contained in the IEM model, it is possible to formulate a Fokker-Planck equation for scalar mixing that includes the effects of differential diffusion (Fox 1999).83 Originally, the FP model was developed as an extension of the IEM model for a single scalar (Fox 1992). At high Reynolds numbers,84 the conditional scalar Laplacian can be related to the conditional scalar dissipation rate by (Pope 2000)... 

For these and other phenomena, thermal and work quantities, although controUing factors, are only of indirect interest. Accordingly, a more refined formulation of thermodynamic principles was established, particularly by Gibbs [6] and, later, independently by Planck [7], that emphasized the nature and use of several special functions or potentials to describe the state of a system. These functions have proved convenient and powerful in prescribing the rules that govern chemical and physical transitions. Therefore, in a sense, the name energetics is more descriptive than is... 

An essential step in the Caratheodory formulation of the second law of thermodynamics is a proof of the following statement Two adiabatics (such as a and b in Fig. 6.12) cannot intersect. F rove that a and b cannot intersect. (Suggestion Assume a and b do intersect at the temperature Ti, and show that this assumption permits you to violate the Kelvin-Planck statement of the second law.)... 

We will adopt this statement as the working form of the third law of thermodynamics. This statement is the most convenient formulation for making calculations of changes in the Gibbs function or the Planck function. Nevertheless, more elegant formulations have been suggested based on statistical thermodynamic theory [5]. 

The connection between the classical and quantum formulations of the transport coefficients has been studied by applying the WKB method to the quantum formulation of the kinetic theory (B16, B17). In this way it was shown that at high temperatures the quantum formulas for the transport coefficients may be written as a power series in Planck s constant h. When the classical limit is taken (h approaches zero), then the classical formulas of Chapman and Enskog are obtained. 

Generalized local Darcy s model of Teorell s oscillations (PDEs) [12]. In this section we formulate and study a local analogue of Teorell s model discussed previously. The main difference between the model to be discussed and the original one is the replacement of the ad hoc resistance relaxation equation (6.1.5) or (6.2.5) by a set of one-dimensional Nernst-Planck equations for locally electro-neutral convective electro-diffusion of ions across the filter (membrane). This filter is viewed as a homogenized aqueous porous medium, lacking any fixed charge and characterized... 

This book treats a selection of topics in electro-diffusion—a nonlinear transport process whose essence is diffusion of charged particles, combined with their migration in a self-consistent electric field. Basic equations of electro-diffusion were formulated about 100 years ago by Nernst and Planck in the ionic context [1]—[3]. Sixty years later Van Roosbroeck applied these equations to treat the transport of holes and electrons in semiconductors [4]. Correspondingly, major applications of the theory of electro-diffusion still lie in the realms of chemical and electrical engineering, related to ion separation and semiconductor device technology. Some aspects of electrodiffusion are relevant for electrophysiology. 

The Fokker-Planck method was set forth in a series of papers by Kirkwood and collaborators.3 After taking into account a certain error in the original formulation of this method,4 the theory may be regarded as complete, in the sense that it provides a well-defined method of calculation. (There are reasons, however, for questioning the correctness of the model, i.e., point sources of friction in a hydrodynamic continuum, for which the theory was constructed. These reasons will be discussed in another place.)... 

In Kirkwood s original formulation of the Fokker-Planck theory, he took into account the possibility that various constraints might apply, e.g., constant bond length between adjacent beads. This led to the introduction of a chain space of lower dimensionality than the full 3A-dimen-sional configuration space of the entire chain and it led to a complicated machinery of Riemannian geometry, with covariant and contravariant tensors, etc. 

So far the Fokker-Planck approximation has only been formulated for cases where there is no boundary, or where the boundary is too far away to bother about it. The question now is how a boundary with certain physical properties is to be translated into a boundary condition for the differential equation. In the case of a reflecting boundary the answer is clear the probability flow (1.3) has to vanish, as in (3.6),... 

Since a number of particles involved in any reaction event are small, a change in concentration is of the order of 1 /V. Therefore, we can use for the system with complete particle mixing the asymptotic expansion in this small parameter 1 /V. The corresponding van Kampen [73, 74] procedure (see also [27, 75]) permits us to formulate simple rules for deriving the Fokker-Planck or stochastic differential equations, asymptotically equivalent to the initial master equation (2.2.37). It allows us also to obtain coefficients Gij in the stochastic differential equation (2.2.2) thus liquidating their uncertainty and strengthening the relation between the deterministic description of motion and density fluctuations. 

For systems that exhibit slow anomalous transport, the incorporation of external fields is in complete analogy to the existing Brownian framework which itself is included in the fractional formulation for the limit a —> 1 The FFPE (19) combines the linear competition of drift and diffusion of the classical Fokker-Planck equation with the prevalence of a new relaxation pattern. As we are going to show, also the solution methods for fractional equations are similar to the known methods from standard partial differential equations. However, the temporal behavior of systems ruled by fractional dynamics mirrors the self-similar nature of its nonlocal formulation, manifested in the Mittag-Leffler pattern dominating the system equilibration. 

The formulation of the Fokker-Planck equation is due to Fokker s and Planck s independent works on the description of the Brownian motion of particles [17, 18]. Commonly, an N variables equation of the type... 

---