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A numerical scheme for the quantum Boltzmann equation with stiff collision terms. (English) Zbl 1277.82046

The authors propose a new scheme for the quantum Boltzmann equation \[ \frac{\partial f}{\partial t} +v\cdot\nabla_xf=\frac{1}{\epsilon}Q(f),\;\;x\in\Omega\subset\mathbb{R}^{d_x},\;v\in \mathbb{R}^{d_v} . \] Here \(\epsilon\) is the Knudsen number which measures the degree of rarefication of the gas. The quantum collision operator \(Q_q\) is defined as \[ Q_q(f)(v)=\int\limits_{\mathbb{R}^{d_v}}\int\limits_{\scriptstyle{S^{d_v-1}}} B(v-v_*,\omega) [f'f'_*(1\pm\theta_0f)(1\pm\theta_0f_*)-ff_*(1\pm\theta_0f')(1\pm\theta_0f'_*)]d\omega dv_*, \] where \(\theta_0=\hbar^{d_v}\), \(\hbar\) is the rescaled Planck constant, \(\omega\) is the unit vector along \(v'-v'_*\); the collision kernel \(B\) is a nonnegative function that only depends on \(|v-v_*|\) and \(\cos\theta\).
The authors idea is based on the observation that the classical Maxwellian, with the temperature replaced by the (quantum) internal energy, has the same first five moments as the quantum Maxwellian.

MSC:

82C40 Kinetic theory of gases in time-dependent statistical mechanics
35Q20 Boltzmann equations
76Y05 Quantum hydrodynamics and relativistic hydrodynamics

Citations:

Zbl 1202.82066
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