General Boltzmann equation

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... 

Finally, in Section V, we compare the results of Bogolubov,3 Choh and Uhlenbeck, and Cohen8 with the generalized Boltzmann equation in Prigogine s formalism. The equivalence of the two methods is well known for the two-body and Cohen s three-body results 23 the demonstration of the same indentity is extended to the three-body results of Choh and Uhlenbeck and to Cohen s four-body expression. We also present the principles of the extension of this comparison for arbitrary concentration. 

Choh and Uhlenbeck6 developed Bogolubov s ideas and extended his formal results. They established a generalized Boltzmann equation which takes account of three-particle collisions. The extension of their results to higher orders in the concentration poses no problem in principle, but it appears difficult, in this formalism, to write a priori the collision term with an arbitrary number of particles. 

We then write down the equation of evolution for the distribution function in the limit of long times. This is the generalized Boltzmann equation, which, this time, is instantaneous because, in the limit of long times, the variation of the distribution function during the time interval rc becomes slow. Also, in the long-time limit, we briefly discuss the equation which gives the correlations. 

For long times Eq. (40) assumes a Markovian form and will be called the generalized Boltzmann equation ... 

Finally, we study the structure of the generalized Boltzmann operator. It can be expressed in terms of the transport operator, which allows one to obtain the virial expansion of the generalized Boltzmann equation. The remarkable point here is that the generalized Boltzmann operator can be expressed in terms of non-connected contributions to the transport operator. This happens for the correction proportional to c3 (c = concentration) and for the following terms in the virial expansion of the generalized Boltzmann operator. 

We shall demonstrate explicitly the equivalence between the results of Bogolubov, Choh and Uhlenbeck, and Cohen (BCUC) and the generalized Boltzmann equation in Prigogine s theory. But it seems useful to us to indicate beforehand some qualitative arguments which allow a physical understanding of the grounds on which this equivalence rests (for more details see ref. 24). 

If this expression is substituted in Eqs. (25) and (92), we obtain exactly the n — 2 term of the generalized Boltzmann equation (see Eqs. 85 and 88) as it appears in the Prigogine formalism. This result is thus equivalent to the formulae (24) and (25) of Cohen s theory from which we started. 

On the other hand, the evolution equations derived by Choh and Uhlenbeck (Eqs. 11 and 12) can be written in the wave vector space (see Eq. 92 with n = 3 and where is replaced by A ). Consequently, expression (114) establishes the equivalence between the result of Choh and Uhlenbeck (Eqs. 11 and 12) and the generalized Boltzmann equation in the Prigogine formalism (see Eq. (85)). 

Let us put this result in the generalized Boltzmann equation (92) as derived by Cohen. Remembering (A.44), we get identically the expression (85) which is the generalized Boltzmann equation in Prigogine s formalism. This completes the proof of the equivalence between the two theories. 

Comparison between Two Generalized Boltzmann Equations (Brocas) 11 317... 

The Quster Expansion Method and the Generalized Boltzmann Equation... 

In the analysis of the higher-order collision terms, in the generalized Boltzmann equation, it is important to determine ... 

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