Generalized Boltzmann Equation for Molecules with Internal States
1.See, e.g., S. Chapman and T. G. Cowling, The Mathematical Theory of Non‐Uniform Gases (Cambridge U.P., Cambridge, England, 1970), 3rd ed.
2.C. S. Wang Chang and G. E. Uhlenbeck, Univ. of Michigan Rept. CM‐681, 1951.
2.See also C. S. Wang Chang, G. E. Uhlenbeck, and J. deBoer, in Studies in Statistical Mechanics, edited by J. deBoer and G. E. Uhlenbeck (North‐Holland, Amsterdam, 1964), Vol. 2.
3.L. Waldmann, Z. Naturforsch. 12a, 660 (1957).
3.See also Handbuch der Physik, edited by S. Flugge (Springer, Berlin, 1958), Vol. 12.
4.R. F. Snider, J. Chem. Phys. 32, 1051 (1960).
5.J. J. M. Beenakker and F. R. McCourt, Ann. Rev. Phys. Chem. 21, 47 (1970).
6.M. W. Thomas and R. F. Snider, J. Statist. Phys. 2, 61 (1969).
7.F. M. Chen and R. F. Snider, J. Chem. Phys. 50, 4082 (1968).
8.See, for example, A. Tip and F. R. McCourt, Physica 52, 109 (1971).
9.Angular momentum conservation is accomplished in this way because no localization of the collision operator is assumed but only that the collisional and macroscopic motions are simultaneously (but independently) occurring.
10.By “superoperator” it is meant “an operator acting on operators.” This convention was introduced by J. A. Crawford, Nuovo Cimento 10, 698 (1958), and is useful in distinguishing between an operator that acts on H and an operator which changes an operator‐on‐H into a new operator‐on‐H, that is, transforms operators.
11.Boltzmann statistics are assumed throughout.
12.J. M. Jauch, B. Misra, and A. G. Gibson, Helv. Phys. Acta 41, 513 (1968).
13.Here is the superoperator analogue of the Møller wave operator, Eq. (18).
14.This way of deriving and writing the Boltzmann equation was described in a shortened version at the 1969 Cornell Symposium on Kinetic Equations, “The Boltzmann Equation and Gaseous N. M.R.” by R. F. Snider, Conference Proceedings, R. L. Liboff and N. Rostoker, editors (Gordon and Breach, London) to be published.
15.See, for example, Refs. 3, 6, and in particular, Eq. (35) of Ref. 4.
16.U. Fano, Phys. Rev. 131, 259 (1963).
17.The authors disagree with Eqs. (27a)‐(31) of the article by A. Ben‐Reuven, Phys. Rev. 141, 34 (1966).
17.This is traced to the lack of a complex conjugate in Eq. (24) of Ben‐Reuven’s paper, namely . We thank Professor Ben‐Reuven for recent correspondence pointing out that this had been corrected by J. Fiutak.
18.See, for example, R. Schatten, Ergebnisse der Mathematik and Ihrer Grenzgebiete (Springer, Berlin, 1960), Vol. 27.
19.S. Hess, Z. Naturforsch. 22a, 187 (1967).
20.A. Tip, Physica (to be published).
21.An example of an oscillatory approach to equilibrium is given by J. Philippot and D. Walgraef in Statistical Mechanics, Foundations and Applications, edited by T. A. Bak (Benjamin, New York, 1967).
22.L. Waldmann, Z. Naturforsch. 15a, 19 (1960).
23.R. F. Snider, J. Math. Phys. 5, 1580 (1964).
24.The expansion of Inρ can be calculated with the formulas of R. M. Wilcox, J. Math. Phys. 8, 962 (1966).
25.F. M. Chen and R. F. Snider, J. Chem. Phys. 46, 3937 (1967);
25.F. M. Chen and R. F. Snider, 48, 3185 (1968)., J. Chem. Phys.
26.This is the approximation of retaining only those tensorial parts of a diagonal collision superoperator matrix element which would occur if the intermolecular potential were spherically symmetric.
27.A. Ben‐Reuven, Phys. Rev. Letters 14, 349 (1965).
28.A. Tip, Phys. Letters A30, 147 (1969).
29.See, e.g., R. F. Snider and C. F. Curtiss, Phys. Fluids 1, 122 (1958),
29.where there are density corrections to the transport coefficients within the binary collision approximation but these depend sensitively on the matching of collision and free trajectories, R. F. Snider and F. R. McCourt, Phys. Fluids 6, 1020 (1963).
30.V. P. Silin, Zh. Eksp. Teor. Fiz. 38, 1771 (1959);
30.[V. P. Silin, Sov. Phys. JETP 11, 1277 (1960)].
31.There can be a tendency to interpret θ as a transformation from H to its dual (transformation from ket to bra states in Dirac terminology) and from to H. Such an interpretation merely complicates the fundamental concepts rather than simplifying them. Here very explicitly, θ is an antilinear transformation from H to H.
32.See, e.g., E. P. Wigner, Group Theory and Its Applications to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959).
33.An adjoint is defined, see also Eq. (47), by the equation and this is not possible for θ because of its antilinearity. In contrast, an “antiadjoint” can be defined to satisfy With Eq. (A3), the antiadjoint can be identified with that is see also Eqs. (A12) and (A13).
34.See, e.g., R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That (Benjamin, New York, 1964), p. 17. For Hermitian operators there is no distinction between Θ and
35.Note that so the cyclic invariance of the trace is not valid for antilinear operators.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
Article metrics loading...
Full text loading...
Most read this month
Most cited this month