%0 Book Section %1 statphys23_0668 %A Karwowski, A.J. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K boltzmann boundary conditions equation equations gas granular hamburger moment problem statphys23 topic-7 %T Boltzmann Gas of Inelastic Spheres. Moment Equations %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=668 %X We describe a hierarchy of formal expansions that represent the Fourier transform Ff of a solution f(v) of the Boltzmann equation for granular gas with constant coefficient of restitution. The approximations are based on a solution of the Hamburger moment problem where one wishes to recover f(v) knowing finite sequences of its first moments. In particiular, we describe how to construct a hierarchy of weighted power series representations for Ff that depend on the moments of f(v) alone. The constructed expansions can be Fourier inverted term by term, to recover the series representation of f. The first two representation correspond to the Maxwellian and Gaussian expansions. They have been exploited by Grad, Jenkins and Levermore in their study of the elastic and inelastic versions of the Boltzmann equation. The next representation has a weight that depends on the first 13 moments of the Boltzmann density f and it yields modified Grad's 13 moment equations for granular gas. The principal tools in deriving the moment equations are the exact Fourier transform of the inelastic, nonlinear Boltzmann equation and the finite version of the Hamburger moment expansion. We also show that it is possible to derive boundary conditions for the moments from the micriscopic boundary conditions for the Boltzmann equation itself.

@incollection{statphys23_0668, abstract = {We describe a hierarchy of formal expansions that represent the Fourier transform F{f} of a solution f(v) of the Boltzmann equation for granular gas with constant coefficient of restitution. The approximations are based on a solution of the Hamburger moment problem where one wishes to recover f(v) knowing finite sequences of its first moments. In particiular, we describe how to construct a hierarchy of weighted power series representations for F{f} that depend on the moments of f(v) alone. The constructed expansions can be Fourier inverted term by term, to recover the series representation of f. The first two representation correspond to the Maxwellian and Gaussian expansions. They have been exploited by Grad, Jenkins and Levermore in their study of the elastic and inelastic versions of the Boltzmann equation. The next representation has a weight that depends on the first 13 moments of the Boltzmann density f and it yields modified Grad's 13 moment equations for granular gas. The principal tools in deriving the moment equations are the exact Fourier transform of the inelastic, nonlinear Boltzmann equation and the finite version of the Hamburger moment expansion. We also show that it is possible to derive boundary conditions for the moments from the micriscopic boundary conditions for the Boltzmann equation itself.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Karwowski, A.J.}, biburl = {https://www.bibsonomy.org/bibtex/22c70e9bb3e86b52c2f4df059ed6edca8/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {b09b40683bd36aa21fb6e4cb1abcfeb1}, intrahash = {2c70e9bb3e86b52c2f4df059ed6edca8}, keywords = {boltzmann boundary conditions equation equations gas granular hamburger moment problem statphys23 topic-7}, month = {9-13 July}, timestamp = {2007-06-20T10:16:26.000+0200}, title = {Boltzmann Gas of Inelastic Spheres. Moment Equations}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=668}, year = 2007 }