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Contributions to Nonequilibrium Thermodynamics. II. Fluctuation Theory for the Boltzmann Equation
1.R. F. Fox and G. E. Uhlenbeck, Phys. Fluids 13, 1893 (1970).
2.M. Bixon and R. Zwanzig, Phys. Rev. 187, 267 (1969) have independently derived the fluctuation terms in the Boltzmann equation and have also shown that they lead to the Landau‐Lifshitz formula for the fluctuating stress tensor and heat flux vector in hydrodynamics. Since these authors use a different method and do not put the theory in the general framework of the stationary Gaussian Markov processes, it did not seem superfluous to present our derivation.
2.Compare also F. L. Hinton, Phys. Fluids 13, 857 (1970).
3.For a short review of the Boltzmann equation see G. E. Uhlenbeck and G. W. Ford, in Lectures in Statistical Mechanics (American Mathematical Society, Providence, Rhode Island, 1963), Chap. IV.
4.D. Hilbert, Theorie der Integralgleichungen (Teubner Verlag, Leipzig, 1912), Chap. 22.
5.D. Enskog, Doctoral dissertation, University of Uppsala (1917), p. 140.
5.Also compare L. Waldmann, in Handbuch der Physik, edited by S. Flügge (Springer‐Verlag, Berlin, 1958), Vol. 12, p. 366.
6.C. S. Wang Chang and G. E. Uhlenbeck, Engineering Research Institute, University of Michigan (1952).
6.This report will be reprinted in the Studies of Statistical Mechanics (North‐Holland, Amsterdam, 1970), Vol. 5. Also compare L. Waldmann, Ref. 5, Sec. 38.
7.L. Kuscer and M. M. R. Williams, Phys. Fluids 10, 1922 (1967).
8.See also S. Chapman and T. G. Cowling, The Mathematical Theory of Non‐Uniform Gases (Cambridge University Press, New York, 1961).
9.We use Greek letters for vector or tensor indices. In the numbering of the eigenfunctions we therefore write α for 3, 4.
10.No summation over k is intended. The integrand of the double integral over p and which one obtains by writing out the left‐hand side of (34), is odd under the reflection if and have opposite parity.
11.See the discussion in the book by S. Chapman and T. G. Cowling, Ref. 8, Chap. 15.
12.Note that in (42) the same matrix Q appears in and Such a symmetry relationship, which also remains in higher approximations, is reminiscent of the Onsager reciprocity relations.
13.C. S. Wang Chang, Engineering Research Institute, University of Michigan (1948). This report will be reprinted in Vol. 5 of the Studies in Statistical Mechanics (North‐Holland Amsterdam, 1970).
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