Kinetic solutions of the Boltzmann-Peierls equation and its moment systems
The authors offer a lengthy abstract, asserting that maximum entropy principle produces acceptable results if one assumes the kinetic equation of Boltzmann-Peierls for phonons. We note that phonons are treated as Bose particles. Authors argue that two-dimensional presentation suffices for required accuracy. Thus the motion takes place in \(\mathbb{R}^2\). It was not clear to reviewer what version of the Brownian motion was used, but that is perhaps the reviewer's fault, by not studying latest developments in low temperature thermodynamics. But the formulas for entropy density and entropy flux seem to indicate the direction taken by the authors. The entropy density is given by formula: \(h(f)=\eta\int\{(1+f/\eta)\ln (1+f/\eta)-f/\eta)\}\,dk\), where \(k\) is described as the ``wave vector''. The integration is over the entire space \(\mathbb{R}^2\). Here \(\eta\) is a constant equal to \(3/8\pi\). A similar formula defines the entropy flux. For any phase density \(f\) the authors seek phase density \(f_M\) which maximizes the entropy: \(h(f_M)= \max(h (f))\), while preserving the densities. In their computations of extremal values, where they generally use the gradient method, one of the main difficulties appears to be the computation of the corresponding Lagrangian multipliers. In their study they have introduced a reduced Boltzmann-Peierls equation, which they claim produces results of computations close to those of the original Boltzmann-Peierls equation, while making computations much easier. Two numerical examples complete their presentation. The authors' abstract offers adequate explanation of the main objectives of their study.
scientific article; zbMATH DE number 5077045