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  • Title: Stability analysis of phonon transport equations derived via the Chapman-Enskog method and transformation of variables.
    Author: Banach Z, Larecki W, Zajaczkowski W.
    Journal: Phys Rev E Stat Nonlin Soft Matter Phys; 2009 Oct; 80(4 Pt 1):041114. PubMed ID: 19905280.
    Abstract:
    Under the assumption of Callaway's model of the Boltzmann-Peierls equation, the Chapman-Enskog method for a phonon gas forms the basis to derive various hydrodynamic equations for the energy density and the drift velocity of interest when normal processes dominate over resistive ones. The first three levels of the expansion (i.e., the zeroth-, first-, and second-order approximations) are satisfactory in that they are entropy consistent and ensure linear stability of the rest state. However, the entropy density contains a weakly nonlocal term, the entropy production is a degenerate function of variables, and the next order in the Chapman-Enskog expansion gives the equations with linearly unstable rest solutions. In the context of Burnett and super-Burnett equations, a similar type of problem was recognized by several authors who proposed different ways to deal with it. Here we report on yet another possible device for obtaining more satisfactory equations. Namely, inspired by the fact that there exists no unique way to truncate the Chapman-Enskog expansion, we combine the Chapman-Enskog procedure with the method of variable transformation and subsequently find a class of epsilon -dependent transformations through which it is possible to derive the second-order equations possessing a local entropy density and nondegenerate expression for the entropy production. Regardless of this result, we also show that although the method cannot be used to construct linearly stable third-order equations, it can be used to make the originally stable first-order equations asymptotically stable.
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