Equation, Boltzmann, generalized reduced

In this equation, the generalized Boltzmann operator 2(i0)T (j0) is now implicitly defined in terms of reducible... 

Before leaving this chapter, we briefly look at an important quantity known as Boltzmann s entropy, and we will examine reduced forms of the Liouville equation in generalized coordinates. 

The system defined by these assumptions is evidently one-dimensional and at steady state, and in this case the general Boltzmann relation reduces to a somewhat simpler form. The appropriate equation is obtained from (7.116) in terms of the lethargy variable this may be written... 

Section III is devoted to Prigogine s theory.14 We write down the general non-Markovian master equation. This expression is non-instantaneous because it takes account of the variation of the velocity distribution function during one collision process. Such a description does not exist in the theories of Bogolubov,8 Choh and Uhlenbeck,6 and Cohen.8 We then present two special forms of this general master equation. On the one hand, when one is far from the initial instant the Variation of the distribution functions becomes slower and slower and, in the long-time limit, the non-Markovian master equation reduces to the Markovian generalized Boltzmann equation. On the other hand, the transport coefficients are always calculated in situations which are... 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

Boltzmann s tombstone in Vienna bears the famous formula 5 = k log W (W = Wahrscheinlichkeit—probability) that was a signature of his audacious concepts. The alternative formula (13.69) (which reduces to Boltzmann s in the limit of equal a priori probabilities pa) was ultimately developed by Gibbs, Shannon, and others in a more general and productive way (see Sidebar 13.4), but the key step of employing probability to trump Newtonian determinism was his. Boltzmann was long identified with efforts to establish the //-theorem and Boltzmann equation within the context of classical mechanics, but each such effort to justify the second law (or existence of atoms) in the strict framework of Newtonian dynamics proved futile. Boltzmann s deep intuition to elevate probability to a primary physical principle therefore played a key role in efforts to find improved foundation for atomic and molecular concepts in the pre-quantum era. 

Specialized to thermal equilibrium, the velocity distributions for the molecules are the Maxwell-Boltzmann distribution (a special case of the general Boltzmann distribution law). The expression for the rate constant at temperature T, k(T), can be reduced to an integral over the relative speed of the reactants. Also, as a consequence of the time-reversal symmetry of the Schrodinger equation, the ratio of the rate constants for the forward and the reverse reaction is equal to the equilibrium constant (detailed balance). 

In a typical macroscopic assumption of proportionality between polarization and applied electric field, P = e0(c — 1 )E, where e is the dielectric constant, and eq3 reduces to the traditional Poisson—Boltzmann equation (the concentrations cH and c0h being in general much smaller than ce). However, if the correlations between neighboring dipoles are taken into account, the following constitutive equation relating the polarization to the macroscopic electric field is obtained7... 

The function ffjl is derived analytically from the hard-sphere-collision integral, and readers interested in the exact forms are referred to Tables 6.1-6.3 of Chapter 6. One crucial issue is the description of the equilibrium distribution with QBMM. In fact, since the nonlinear collision source terms that drive the NDF and its moments to the Maxwellian equilibrium are approximated, the equilibrium is generally not perfectly described. The error involved is generally very small, and is reduced when the number of nodes is increased, but can be easily overcome by using some simple corrections. Details on these corrections for the isotropic Boltzmann equation test case are reported in Icardi et al. (2012). 

The method discussed in Section 3 for the derivation of the generalized Boltzmann equation can be carried over, with some simple modifications, to derive an analogous kinetic equation for d, t). However, due to the special role of the tagged particle and the fact that the corresponding initial N-particle distribution is proportional to Ui, one obtains a linear equation for 4>d (vi, t) that for low densities reduces to the Lorentz-Boltzmann equation " ... 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

Only when a complete description including all seven variables is required is it necessary to solve the Boltzmann equation in all its generalities, Frequently, simplifying assumptions and limiting conditions can be imposed which reduce the integrodifTerential equation to more tractable form. Thus much of the subject of reactor analysis is devoted to the development and the application of simplified analytical models which define, within the limits of engineering needs, the nuclear characteristics of the reactor complex. 

The general field of problems described above, except in some special areas, may be treated by the well-known methods and analytical models of mathematical physics. It has already been noted that the most general description of the neutron population usually starts with a neutron-balance relation of the Boltzmann type. The Boltzmann equation was developed in connection with the study of nonuniform gas mixtures, and the application to the neutron problem represents a considerable simplification of the general gas problem. (Whereas in gas problems all the particles are in motion, in reactor problems only the neutrons are in motion. ) The fundamental equation of reactor physics, then, is already a familiar one from the kinetic theory. Further, many of the most useful neutron models obtained from approximations to the Boltzmann equation reduce to familiar forms, such as the heat-conduction, Helmholtz, and telegraphist s equations. These simplifications result from the elimination of various independent variables in the... 

This equation relates the reaction rate (cross section) of a material at the temperature to its cross section at a lower temperature Tn- As in all the preceding analyses, it is assumed that the nuclei are distributed according to the Maxwell-Boltzmann relation (4.198). The above equation may be used then to compute the cross-section curves for a reactor operating at any temperature from the known cross-section data of the reactor material which have been determined at some temperature Tn. Note that U = Tn then Eq. (4.233) reduces to an identity [see also (4.225)]. In general, the indicated integration must be carried out in detail. There is one special case, however, which leads to an especially simple result. If the measured cross-section curve varies as l/v, which is the case for many absorbers in the low-energy range, then it is easily shown that the reaction rate is independent of the moderator temperature. For example, if we take aa v) = Co/y, where Co is some constant, then from (4.233)... 

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