Abstract
The method of time-ordered cumulants is used to investigate the behavior of heat pulses in a one-dimensional medium in which the thermal conductivity is random. A partial differential equation is obtained for the average temperature profile; it is the heat equation modified by the addition of a fourth-order spatial derivative. A solution is obtained by asymptotic series. The first two spatial moments of the average temperature profile are evaluated and are shown to tend to those of a Gaussian whent is large. Finally, an equation is obtained for the covariance function.
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References
R. F. Fox,J. Math. Phys. 17:1148 (1976).
R. F. Fox,J. Math. Phys. 13:1196 (1972).
J. Riordan,An Introduction to Combinatorial Analysis (Wiley, New York, 1958), Chapter 2, Section 8.
L. D. Landau and E. M. Lifschitz,Fluid Mechanics (Addison-Wesley, Reading, Mass., 1959), Chapter 5.
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Alfred P. Sloan Fellow.
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Fox, R.F., Barakat, R. Heat conduction in a random medium. J Stat Phys 18, 171–178 (1978). https://doi.org/10.1007/BF01014308
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DOI: https://doi.org/10.1007/BF01014308