Boltzmann equation definition

Both entropic and coulombic contributions are bounded from below and it can be verified that the second variation of is positive definite so that the above equations correspond to a minimum [27]. Using conditions in the bulk we can eliminate //, from the equations. Then we get the Boltzmann equation in which the electric potential verifies the Poisson equation by construction. Hence is equivalent within MFA to the... 

One will immediately recognize this as a form of the BOLTZMANN equation, or the GAUSSIAN LAW. We can modify this equation and put it into a form more suitable for our use by making the following definitions. 

The question to be discussed is whether saturation of the electric field (asserted by Proposition 2.1) implies saturation of the interparticle force of interaction. Consider for definiteness repulsion between two symmetrically charged particles in a symmetric electrolyte solution. In the onedimensional case (for parallel plates) the answer is known—the force of repulsion per unit area of the plates saturates. (This follows from a direct integration of the Poisson-Boltzmann equation carried out in numerous works, primarily in the colloid stability context, e.g., [9]. Recall that again in vacuum, dielectrics, or an ionic system with a linear screening, the appropriate force grows without bound with the charging of the particles.)... 

Now we return to Eq. (3.15). Using the definition of F we can eliminate the bound-state part F 2 and obtain the Boltzmann equation for the free particles in the form... 

Brpnsted theory, 23 Definition of Kb, 38 Lewis theory, 24 HSAB theory, 12 Base saturation (%), 163 Basic organic compounds, 356 Bicarbonate, 30-33 Biotite, 104, 108 Boltzmann equation, 143 Bonding, 6-12 Covalent, 7 Ionic, 7 Boron, 127 Buffer capacity, 86... 

To describe polydispersed multiphase systems the Boltzmann equation can be extended by including the dependency of the internal property coordinates such as the particle size and shape in the definition of the distribution function. In this way a statistical balance formulation can be obtained by means of a distribution function on the form p, r,v, c,t)d dr dv dc, defined as the probable number of particles with internal properties in the range about with a velocity range in property space dv about v, located in the spatial range dr about the position r, with a velocity range dc about c, at time t. The particular Boltzman type of equation is given by ... 

Entropy is a measure of the total number of microstates in a system. There have been two widely used definitions of entropy, which were suggested by Ludwig Boltzmann and J. Willard Gibbs. We ll just look at the one specified by Boltzmann, since it s a little more straightforward to understand. The equation for Boltzmann s definition of entropy is ... 

According to Katchalsky et al. [21] the resolution of the Poisson-Boltzmann equation in the presence of added salt leads to a definition of a minimum distance of approach, R, for a free counter-ion such that ... 

Once modifications to functions of this kind have been made, the Boltzmann superposition principle can no longer be assumed to apply, and there is no simple replacement for it. This marks a significant change in the level of difficulty when moving from linear to non-linear theory. In the linear case, the material behaviour is defined fully by single-step creep and stress relaxation, and the result of any other stress or strain history then can be calculated using the Boltzmann integral. In the non-linear case we have lost the Boltzmann equation, and it is not even clear what measurements are needed for a full definition of the material. 

We have repeatedly used the term hydrodynamic, and we now give it a more precise definition. By a hydrodynamic process we mean one for which the local thermodynamic variables, temperature, chemical potential (or density), and velocity, are determined by the past history of their boundary values. The normal solution to the Boltzmann equation, as well as its generalization obtained in the previous Sections, then clearly corresponds to a hydrodynamic process. The significance of the term hydro-dynamic may be clarified by the consideration of some processes of non-hydrodynamic type. A process of relaxation in momentum space in a spatially uniform gas is clearly non-hydrod5mamic, since the local thermod5mamic variables are not at all pertinent to its description. Another example is provided by processes in a Knudsen gas. Here there is an essential dependence on the particular form of the boundary forces. An insensitivity to the nature of the boundary forces is implied in the definition of a hydrod5mamic process, for which it is immaterial whether a thermal reservoir is constructed of, say, copper or aluminum, and... 

Argue from Boltzmann s definition for entropy that S can never have a negative value. Hint See equation 3.26.)... 

Now consider Boltzmann s definition of the disorder of a system, using the thermodynamic measure of entropy. Boltzmann s equation relates entropy to the number of possible arrangements of a system by ... 

The contenet of the third law of thermodynamics is summarized in Fig. 2.4. The third law is particularly easy to understand if one combines the macroscopic entropy definition of entropy with its statistical, microscopic interpretation through the Boltzmann equation, Eq. (11). The symbol k is the Boltzmaim constant, the gas constant R divided by Avogadro s number and W is the thermodynamic probability, representing the number of ways a system can be arranged on a microscopic level. One can state the third law, as proposed by Nernst and formulated by Lewis and Randall, as follows "If... 

Attempts to improve the theory by solving the Poisson-Boltzmann equation present other difficulties first pointed out by Onsager (1933) one consequence of this is that the pair distribution functions g (r) and g (r) calculated for unsymmetrically charged electrolytes (e.g., LaCl or CaCl2) are not equal as they should be from their definitions. Recently Outhwaite (1975) and others have devised modifications to the Poisson-Boltzmann equation which make the equations self-consistent and more accurate, but the labor involved in solving them and their restriction to the primitive model electrolyte are drawbacks to the formulation of a comprehensive theory along these lines. The Poisson-Boltzmann equation, however, has found wide applicability in the theory of polyelectrolytes, colloids, and the electrical double-layer. Mou (1981) has derived a Debye-Huckel-like theory for a system of ions and point dipoles the results are similar but for the presence of a... 

Applying the Enskog perturbation method, the properties of dilute gases which are only slightly deviating from equilibrium can be approximated. Only under these conditions will the flux vectors be about linear in the derivatives so that the formal definitions of the transport coefflcients apply. In this limit the distribution function is still nearly Maxwellian, and the Boltzmann equation can be solved by a perturbation method. The resulting solutions are then used to obtain approximate expressions for the heat and momentum fluxes and for the corresponding transport coefficients. 

For convenience, the Maxwell-Boltzmann equation (2.67) with the collision operator definition (2.69) is rewritten here defining the point of departure in Enskog s model development ... 

Kotin and Nagasawa [6] defined the counter-ion binding in analogy to the definition of Bjerrum on ion-pair formation [36]. That is, it is assumed that a polyion is placed in an infinite volume of a neutral salt solution of uni-uni valent type and the polyion is a rod of infinite length having a charge density N/L, Moreover, it is assumed that the ionic distribution around the rod is determined from the Poisson-Boltzmann equation. Then, if one plots the distribution of counter-ions PcC ) against the distance from the axis of the polyion r,... 

The quantities n, V, and (3 /m) T are thus the first five (velocity) moments of the distribution function. In the above equation, k is the Boltzmann constant the definition of temperature relates the kinetic energy associated with the random motion of the particles to kT for each degree of freedom. If an equation of state is derived using this equilibrium distribution function, by determining the pressure in the gas (see Section 1.11), then this kinetic theory definition of the temperature is seen to be the absolute temperature that appears in the ideal gas law. 

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