Boltzmann equation elastic

Third, a further simplification of the Boltzmann equation is the use of the two-term spherical harmonic expansion [231 ] for the EEDF (also known as the Lorentz approximation), both in the calculations and in the analysis in the literature of experimental data. This two-term approximation has also been used by Kurachi and Nakamura [212] to determine the cross section for vibrational excitation of SiHj (see Table II). Due to the magnitude of the vibrational cross section at certain electron energies relative to the elastic cross sections and the steep dependence of the vibrational cross section, the use of this two-term approximation is of variable accuracy [240]. A Monte Carlo calculation is in principle more accurate, because in such a model the spatial and temporal behavior of the EEDF can be included. However, a Monte Carlo calculation has its own problems, such as the large computational effort needed to reduce statistical fluctuations. 

A more detailed view of the dynamies of a ehromatin chain was achieved in a recent Brownian dynamics simulation by Beard and Schlick [65]. Like in previous work, the DNA is treated as a segmented elastic chain however, the nueleosomes are modeled as flat cylinders with the DNA attached to the cylinder surface at the positions known from the crystallographic structure of the nucleosome. Moreover, the electrostatic interactions are treated in a very detailed manner the charge distribution on the nucleosome core particle is obtained from a solution to the non-linear Poisson-Boltzmann equation in the surrounding solvent, and the total electrostatic energy is computed through the Debye-Hiickel approximation over all charges on the nucleosome and the linker DNA. 

In the limit of small Knudsen number, the Chapman-Enskog expansion (Chapman, 1916 Enksog, 1921) of the elastic Boltzmann equation yields a first-order term for CTp of the form... 

Figure 1 shows the edf calculated for H2 by solving the relevant Boltzmann equation with the inclusion of the following inelastic processes, besides the elastic ones2 ... 

In the presence of hydrojgen atoms the Boltzmann equation should be solved for a mixture of the two species (H2, H)3 Elastic and inelastic processes involving H-atoms should now be considered. The following have been included in the calculations of Ref.2 ... 

It can be shown that if one assumes that only elastic collisions occur one can obtain an approximate solution of the Boltzmann equation for the distribution of speeds. The result is not the Maxwell-Boltzmann dis-... 

The presence of a high density of solvent molecules leads to recollisions between the potentially reactive pair of molecules. Some examples of such recollision events are shown schematically in Fig. 7.2. In Fig. 1.1a the solute molecules A and B collide elastically, and after collision of A with a solvent molecule S, the A molecule recollides with B and reacts. An event of this type is extremely unlikely in a dilute gas. The description of collision sequences of this type is outside of the scope of a Boltzmann equation, which accounts only for uncorrelated binary collision events. Collision sequences of this kind are expected to play an increasingly important role as the solvent density increases, and as we shall see, they are often the dominant contribution to the dynamics. A similar sequence of reactive events is shown in Fig. 1.2b. 

The theory in the Gaussian limit has been refined greatly to take into account the possible fluctuations of the junction points. In these approaches, the probability of an internal state of the system is the product of the probabilities Win) for each chain. The entropy is deduced by the Boltzmann equation, and the free energy by equation (26). The three main assumptions introduced in the treatment of elasticity of rubber-like materials are that the intermolecular interactions between chains are independent of the configurations of these chains and thus of the extent of deformation (125,126) the chains are Ganssian, freely jointed, and volumeless and the total number of configurations of an isotropic network is the product of the number of configurations of the individual chains. 

The original Chapman-Enskog theory is based on the Boltzmann equation (Boltzmann 1872) that was developed for structureless particles which exhibit isotropic intermolec-ular potentials. Monatomic species are a subset of polyatomic molecules that do not possess modes of internal motion and interact via elastic collisions only. This simplifies the problem dramatically because the WCUB equation (or Waldmann-Snider equation) can be replaced by the Boltzmann equation that is solved in its linearized form. 

Consider next the mean free time, r, i.e. the ratio of the mean free path and the thermal speed t =. Clearly, r is the microscopic time scale characterizing the system at hand and 7 is the macroscopic time scale characterizing this system (the simple sheared state). The ratio r/7 = 77 is a measure of the temporal scale separation in the system. Since 77 = it is an 0(1) quantity. It follows that (unless e < 1) there is no temporal scale separation in this system, irrespective of its size or the size of the grains. Consequently, one cannot a-priori employ the assumption of fast local equilibration and/or use local equilibrium as a zeroth order distribution function (both for solving the Boltzmann equation and for the study of generalized hydrodynamics of these systems [12] unless the system is nearly elastic (in which case scale separation is restored) and (in unsteady states) the rate of change of the external parameters is sufficiently slow. The latter condition severely limits the applicability of the hydrodynamic description. 

It is required to express either pair of molecular velocities before and after the interaction in terms of the other pair, and of two independent geometrical variables b and (j>) in order to complete the specification of the encounter. The original Boltzmann equation derivation considers elastic collisions in free space between two spin-less molecules of equal mass. However, due to the major interest in multicomponent mixtures, the theory outline consider elastic collisions between two spin-less mono-atomic molecules in an ideal gas mixture. The theory may be useful even if the molecules are not mono-atomic, provided that their states of internal motion (i.e., rotation and vibration) are not affected by the collisions. The two molecules under consideration are treated as point particles with respective masses m and m2. In the laboratory frame, the incoming molecule positions are denoted by ri and r2, and the particle velocities are indicated by ci and C2. The corresponding positions and velocities after the encounter are r j, and c, c, respectively. The classical trajectories for two interacting molecules presented in the laboratory system frame are viewed in Fig. 2.1. It is supposed that the particle interaction is determined by conservative potential interaction forces only. Any external forces which might act on the molecules are considered negligible compared to the potential forces involved locally in the collision. The relative position vectors in the laboratory frame are defined by ... 

Example 3. The mean free path of electrons scattered by a crystal lattice is known to iavolve temperature 9, energy E, the elastic constant C, the Planck s constant the Boltzmann constant and the electron mass M. (see, for example, (25)). The problem is to derive a general equation among these variables. 

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