Distribution functions perturbative expansion

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

Equations (7)-(9) are well suited for numerical evaluation with arbitrary functions defining the spatial and temporal distributions of the laser pulse. In addition, the system (7)-(9) is rather convenient for analytical treatment. In particular, one can develop further perturbative expansion of (7)-(8) in terms of the fine structure constant a. In the leading order, this yields nonrelativistic formulas which agree with those formerly derived by us in [19]. For unchirped laser signals (i.e., t) = 0) these reduce further to the result of [31] by expanding all quantities in powers of the laser intensity I(r,t). Extensive numerical tests carried out by us for various forms of the chirp pulse en-... 

The Chapman-Enskog procedure to approximate the distribution functions /j, /e by a linear perturbation ansatz, the Landau form of the Coulomb collision integral together with the small mass ratio me/m expansions in the classical work of Braginskii results in the friction term R... 

By insertion of the statistical distribution function expansion (100) into the perturbed statistical average (98) and with regard to the definition (129) of a fluctuation, we obtain to within the third approximation incluavely ... 

The Enskog [24] expansion method for the solution of the Boltzmann equation provides a series approximation to the distribution function. In the zero order approximation the distribution function is locally Maxwellian giving rise to the Euler equations of change. The first order perturbation results in the Navier-Stokes equations, while the second order expansion gives the so-called Burnett equations. The higher order approximations provide corrections for the larger gradients in the physical properties like p, T and v. 

In the former case, the terms in the perturbation expansion of the Helmholtz free energy may be obtained in terms of site-site distribution functions. For... 

This distribution function is obtained straightforwardly from the known analytic expressions for the radial functions. Finally, the energy shift from first-order perturbation theory is obtained, for any finite nucleus model, in terms of a series expansion in X = 2ZR, where i is a model-specific radial nuclear size parameter, as... 

Calculation of a perturbed distribution function can be approached in various ways (1) direct solution of the Boltzmann equation for the distribution function in the perturbed system, (2) distribution-difference methods, (3) local calculations, and (4) normal-mode expansion methods. 

In eqn. 5.2.14 the coefficients of the series expansions depend on the variable x = /cr, and S v(0, surface spherical harmonics depending on the polar angles 0 and q>. The vector coefficients 1 /Jc) are independent of the position of ion j which may be considered stationary according to eqn. 5.2.6. At infinity y) and/ vanish. Besides, Pitts assumed that for r = a the perturbations may be neglected thus the ionic potentials and distribution functions on the surface of the central ion are not affected by the external field. 

In perturbation theories one relates the properties (e.g., the distribution functions or free energy) of the real system, for which the intermolecular potential energy is E, to those of a reference system where the potential is Eq, usually by an expansion in powers of the perturbation potential Ei=E - Eq. 

Uniform expansion model. The difficulty in the perturbation calculation can be improved by a simple scheme of calculation. In this scheme, it is assumed ffiat the expansion of the chain is represented by the expansion of the bond length, i.e., that the distribution function of R is well approximated by... 

Provided that they arc normalized, G and Gh produce the distribution function of h, i.e. t hey must be proportional to each other. Fhe factor of proportionality Z is chosem so that some remaining principal parts in the e expansion arc absorbed. I hcreforc, Z. Zi, and the relation v vs vq lead to micro-macro relationships, which absorb all the principal parts of the bare perturbation series, if there is a macroscopic distribution function with two macroscopically-controllablc parameters, viz., the chain length and the excluded volume of a. segment. I hus,... 

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