Operator, Boltzmann, generalized transport

In Section IV, we develop the former results and we study the structure of the transport operator and of the generalized Boltzmann operator. We also analyse the irreducibility condition which appears in Prigogine s theory by using the graphs of equilibrium statistical mechanics. 

In addition, we have calculated the three-body transport operator using Cohen s formalism and, of course, we find an expression which is different from the generalized Boltzmann operator in the same formalism. 

Finally, we attack the problem of the transport coefficients, which, by definition, are calculated in the stationary or quasi-stationary state. The variation of the distribution functions during the time rc is consequently rigorously nil, which allows us to calculate these coefficients from more simple quantities than the generalized Boltzmann operators which we call asymptotic cross-sections or transport operators. 

IV. THE STRUCTURE OF THE TRANSPORT OPERATOR AND OF THE GENERALIZED BOLTZMANN OPERATOR... 

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. 

Finally, we calculate the transport operator for three particles in the Cohen formalism. We obtain, evidently, an expression which differs from that for the generalized Boltzmann operator in the same formalism. 

We would like to prove that the initial state is forgotten, and that the collision operators reach an asymptotic form for times long compared to some microscopic time. If this were true, then the normal solution method, when applied to the generalized Boltzmann equation, would lead to expressions for the transport coefficients for a dense gas that would (a) be independent of the precise initial state of the gas, (b) be independent of the time elapsed since the initial state of the gas, and (c) have a density expansion of the form... 

When we construct normal solutions of the generalized Boltzmann equation using the resummed collision operator and computes the transport coefficients for a moderately dense gas (in three dimensions), we find that the viscosity, say, has the expansion ... 

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