Abstract
Based on the assumption of a kinetic equation in Г space, a stochastic differential equation of the one-particle distribution is derived without the use of the linear approximation. It is just the Boltzmann equation with a Langevin-fluctuating force term. The result is the general form of the linearized Boltzmann equation with fluctuations found by Bixon and Zwanzig and by Fox and Uhlenbeck. It reduces to the general Landau-Lifshitz equations of fluid dynamics in the presence of fluctuations in a similar hydrodynamic approximation to that used by Chapman and Enskog with respect to the Boltzmann equation.
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References
L. D. Landau and E. M. Lifshitz,Fluid Mechanics (Pergamon Press, 1959), Chapter 17.
L. D. Landau and E. M. Lifshitz,Statistical Physics (Pergamon Press, 1958), Chapter 12.
L. Onsager and S. Machlup,Phys. Rev. 91:1505 (1953).
R. F. Fox and G. E. Uhlenbeck,Phys. Fluids 13:1893, 2881 (1970).
M. Bixon and R. Zwanzig,Phys. Rev. 187:267 (1969).
M. Kac, inThe Boltzmann Equation, E. G. D. Cohen and W. Thirring, eds. (Acta Physica Austriaca, Suppl. X; Springer Verlag, 1973).
G. E. Uhlenbeck, inThe Boltzmann Equation, E. G. D. Cohan and W. Thirring, eds. (Acta Physica Austriaca, Suppl. X; Springer Verlag, 1973).
R. F. Fox,J. Math. Phys. 19:1993 (1978).
N. G. van Kampen,Phys. Lett. 50A:237 (1974).
N. G. van Kampen, inFluctuation Phenomena in Solids, R. E. Burgess, ed. (Academic Press, 1965).
J. Logan and M. Kac,Phys. Rev. A 13:458 (1976).
R. Graham,Phys. Rev. A 10:1762 (1964).
J. E. Keizer,Phys. Fluids 21:198 (1978).
F. L. Hinton,Phys. Fluids 13:857 (1970).
S. Watanabe,Stochastic Differential Equations (Sangyotosho Press, 1978) (in Japanese).
L. Arnold,Stochastic Differential Equations (Wiley, 1974).
G. E. Uhlenbeck, Appendix to M. Kac,Probability and Related Topics in Physical Sciences (Interscience, 1961).
M. Kac,Probability and Related Topics in Physical Sciences (Interscience, 1959).
R. Brout,Physica 22:509 (1956).
A. Onuki,J. Stat. Phys. 19:325 (1978).
A. V. Skorokhod,Studies in the Theory of Random Processes (Addison-Wesley, 1965), p. 34.
H. Ueyama,Physica 80A:98 (1975).
A. Onuki,J. Stat. Phys. 18:475 (1978).
R. Zwanzig, inLectures in Theoretical Physics, W. E. Brittinet al., eds. (Interscience, 1961), Vol. 3.
S. Chapman and T. G. Cowling,The Mathematical Theory of Nonuniform Gases (Cambridge Press, 1970).
L. Waldmann, inHandbuch der Physik, Vol. 12 (Springer, Berlin, 1958).
J. J. Hopfield and A. J. F. Bastin,Phys. Rev. 168:193 (1968).
M. H. Ernst, J. R. Dorfman, W. R. Hoegy, and J. M. J. van Leeuwen,Physica 45:127 (1969).
P. Langevin,Compt. Rend. 1908:530.
K. Itô,Mem. Am. Math. Soc. 4:1 (1951).
J. E. Moyal,J. Roy. Stat. Soc. B 11:150 (1949).
H. van Beijeren and M. H. Ernst,Physica 68:437 (1973).
H. H. U. Konijnendijk and J. M. J. van Leeuwen,Physica 64:342 (1973).
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This work received financial support from the Alexander von Humboldt Foundation.
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Ueyama, H. The stochastic Boltzmann equation and hydrodynamic fluctuations. J Stat Phys 22, 1–26 (1980). https://doi.org/10.1007/BF01007985
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DOI: https://doi.org/10.1007/BF01007985