Sonine approximation for collisional moments of granular gases of inelastic rough spheres

We consider a dilute granular gas of hard spheres colliding inelastically with coefficients of normal and tangential restitution $\alpha$ and $\beta$, respectively. The basic quantities characterizing the distribution function $f(\mathbf{v},\bm{\omega})$ of linear ($\mathbf{v}$) and angular ($\bm{\omega}$) velocities are the second-degree moments defining the translational ($T^\text{tr}$) and rotational ($T^\text{rot}$) temperatures. The deviation of $f$ from the Maxwellian distribution parameterized by $T^\text{tr}$ and $T^\text{rot}$ can be measured by the cumulants associated with the fourth-degree velocity moments. The main objective of this paper is the evaluation of the collisional rates of change of these second- and fourth-degree moments by means of a Sonine approximation. The results are subsequently applied to the computation of the temperature ratio $T^\text{rot}/T^\text{tr}$ and the cumulants of two paradigmatic states: the homogeneous cooling state and the homogeneous steady state driven by a white-noise stochastic thermostat. It is found in both cases that the Maxwellian approximation for the temperature ratio does not deviate much from the Sonine prediction. On the other hand, non-Maxwellian properties measured by the cumulants cannot be ignored, especially in the homogeneous cooling state for medium and small roughness. In that state, moreover, the cumulant directly related to the translational velocity differs in the quasi-smooth limit $\beta\to -1$ from that of pure smooth spheres ($\beta=-1$). This singular behavior is directly related to the unsteady character of the homogeneous cooling state and thus it is absent in the stochastic thermostat case.Comment: 14 pages, 8 figures; v2: some parts rewritten, new references added; published in a special topic decicated to Carlo Cercignan

Relative Entropy of a Freely Cooling Granular Gas

The time evolution and stationary values of the entropy per particle of a homogeneous freely cooling granular gas, relative to the maximum entropy consistent with the instantaneous translational and rotational temperatures, is analyzed by means of a Sonine approximation involving fourth-degree cumulants. The results show a rich variety of dependencies of the relative entropy on time and on the coefficients of normal and tangential restitution, including a peculiar behavior in the quasi-smooth limit.Comment: 6 pages; 2 figures; contributed paper at the 28th International Symposium on Rarefied Gas Dynamics (Zaragoza, Spain, July 9-13, 2012

Bethe states for the two-site Bose-Hubbard model: a binomial approach

We calculate explicitly the Bethe vectors states by the algebraic Bethe ansatz method with the $gl(2)$-invariant $R$-matrix for the two-site Bose-Hubbard model. Using a binomial expansion of the n-th power of a sum of two operators we get and solve a recursion equation. We calculate the scalar product and the norm of the Bethe vectors states. The form factors of the imbalance current operator are also computed.Comment: 13 page