Operator, Boltzmann, generalized

The time-dependent operator G00(r) describes the most general collision process between the particles in the system it generalizes to an arbitrary system the well-known Boltzmann collision operator for dilute gases. 

In Section II, we summarize the ideas and the results of Bogolubov,3 Choh and Uhlenbeck,6 and Cohen.8 Bogolubov and Choh and Uhlenbeck solved the hierarchy equations and derived two- and three-body generalized Boltzmann operators Cohen used a cluster expansion method and obtained two-, three-, and four-body explicit results which he was able to extend to arbitrary concentrations. 

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. 

Let us mention first the work of Stecki who expanded Bogolubov s results in a series in A28 and who with Taylor showed that this expansion is identical to all orders in A with the generalized Boltzmann operator (85).29 Since the method is rather different from the virial expansions which we present here, we give in Appendix A.III the major thoughts of this general work valid for any concentration. 

Choh and Uhlenbeck s result and the generalized Boltzmann operator for n = 3 in the Prigogine formalism (Section VB-2). 

We have also studied the generalized Boltzmann operator for four particles in the Choh-Uhlenbeck formalism (see, for example, this expression in the work of Cohen8). Since the paper by Stecki and Taylor,29 it seems clear that the results of Bogolubov and those of Choh and Uhlenbeck (which are derived from them) must be equivalent to the Prigogine formalism. However, it is very difficult to show this equivalence explicitly in the concentration form that we have presented here. Indeed, in the Choh and Uhlenbeck method, the streaming operators appear with their Fourier components < k S(i n) k )and< k 0 S(i -n) k >, and the products of these operators are then expressed in terms of the Y(1 > for several complex variables. This renders the mathematical operations extremely complicated (see Eq. 111). 

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

It is less forbidding than it looks. The first three lines are the linearized Boltzmann operator acting on the factor u(rin the factorial cumulant the next three lines are the same operator acting on the factor u(r2,p2) lines seven and eight represent the sources of the fluctuations and on the last line the flow terms have been added for both factors. The equation has therefore the general form (VIII.6.8), when A is identified with the linearized Boltzmann operator including the streaming term. 

In the above expression, C (pi z) is the finite frequency generalization of the Boltzmann-Lorentz collision operator. Cq1 (pi z) can be described by the finite frequency generalization of the Choh-Uhlenbeck collision operator. [57]. This operator describes the dynamical correlations created by the collisions between three particles. Using the above-mentioned description the expression of (pi z) can be shown to be written as [57]... 

In Eq. (10), E nt s(u) and Es(in) are the s=x,y,z components of the internal electric field and the field in the dielectric, respectively, and p u is the Boltzmann density matrix for the set of initial states m. The parameter tmn is a measure of the line-width. While small molecules, N<pure solid show well-defined lattice-vibrational spectra, arising from intermolecular vibrations in the crystal, overlap among the vastly larger number of normal modes for large, polymeric systems, produces broad bands, even in the crystalline state. When the polymeric molecule experiences the molecular interactions operative in aqueous solution, a second feature further broadens the vibrational bands, since the line-width parameters, xmn, Eq. (10), reflect the increased molecular collisional effects in solution, as compared to those in the solid. These general considerations are borne out by experiment. The low-frequency Raman spectrum of the amino acid cystine (94) shows a line at 8.7 cm- -, in the crystalline solid, with a half-width of several cm-- -. In contrast, a careful study of the low frequency Raman spectra of lysozyme (92) shows a broad band (half-width 10 cm- -) at 25 cm- -,... 

The form of the Boltzmann-Enskog collision operator is thus specified out task is to find its generalization. We denote the general collision operator by /l (1, 1 t ), where we have allowed for the possibility that it may be nonlocal in time as well as space. The general kinetic equation may then be written as... 

The general form of an inhomogeneous Boltzmann equation with singular operator, is... 

To make it possible to deal with systems with many degrees of freedom, the Boltzmann operator/time evolution operators are represented by a Feynman path integraland the path integral evaluated by a Monte Carlo random walk method. It is in general not feasible to do this for real values of the time t, however, because the integrand of the path integral would be oscillatory. We thus first calculate for real values of t it, i.e., pure imaginary time,... 

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