受限系统偶极振荡的稀有碰撞热化
Thermalization of dipole oscillations in confined systems by rare collisions
摘要：
我们研究了空间受限的相互作用费米子系统中的质心或偶极振荡的弛豫。随着约束频率$ω_{\perp}$固定的粒子被认为是沿着一个（准1D）或两个（准2D）的空间维度自由移动。我们已经集中在罕见的碰撞制度，这样的非弹性碰撞率，$1/τ_{in} \ll ω_{\perp}$。偶极振荡弛豫率，$1/τ_{\perp}$得到在三个不同的水平：直接微扰理论，求解积分Bethe-Salpeter方程和应用记忆函数形式主义。只要非谐性较弱，$1/τ_{\perp} \ll 1/ τ_{in}$这三种方法给出了相同的结果。在准二维情况下，$1/τ_{\perp} \neq 0$在零温度下。在准一维系统$1/τ_{\perp} \propto T^3$中，如果费米能$E_F$低于临界值$E_F < 3 ω_{\perp}/4$。除此之外，除非系统接近可积，否则速率$1/τ_{\perp}$具有类似于准二维的温度依赖性。在所有情况下，弛豫都是由沿无约束方向传播的粒子-空穴对的激发引起的，导致关系式$1/τ_{\perp} \propto 1/τ_{in}$，其中非弹性速率为$1/τ_{in} \neq 0$，因为相空间在有限激发能$\hbar ω_{\perp}$处打开。而$1/τ_{\perp} \propto τ_{in}$在流体动力学区域，$ω_{\perp} \ll 1/τ_{in}$，在罕见碰撞区域，$ω_{\perp} \gg 1/τ_{in}$，我们得到相反的趋势$1/τ_{\perp} \propto 1/τ_{in}$。
英文摘要：
We study the relaxation of the center-of-mass, or dipole oscillations in the system of interacting fermions confined spatially. With the confinement frequency $ω_{\perp}$ fixed the particles were considered to freely move along one (quasi-1D) or two (quasi-2D) spatial dimensions. We have focused on the regime of rare collisions, such that the inelastic collision rate, $1/τ_{in} \ll ω_{\perp}$. The dipole oscillations relaxation rate, $1/τ_{\perp}$ is obtained at three different levels: by direct perturbation theory, solving the integral Bethe-Salpeter equation and applying the memory function formalism. As long as anharmonicity is weak, $1/τ_{\perp} \ll 1/ τ_{in}$ the three methods are shown to give identical results. In quasi-2D case $1/τ_{\perp} \neq 0$ at zero temperature. In quasi-1D system $1/τ_{\perp} \propto T^3$ if the Fermi energy, $E_F$ lies below the critical value, $E_F < 3 ω_{\perp}/4$. Otherwise, unless the system is close to integrability, the rate $1/τ_{\perp}$ has the temperature dependence similar to that in quasi-2D. In all cases the relaxation results from the excitation of particle-hole pairs propagating along unconfined directions resulting in the relationship $1/τ_{\perp} \propto 1/τ_{in}$, with the inelastic rate $1/τ_{in} \neq 0$ as the phase-space opens up at finite energy of excitation, $\hbar ω_{\perp}$. While $1/τ_{\perp} \propto τ_{in}$ in the hydrodynamic regime, $ω_{\perp} \ll 1/τ_{in}$, in the regime of rare collisions, $ω_{\perp} \gg 1/τ_{in}$, we obtain the opposite trend $1/τ_{\perp} \propto 1/τ_{in}$.
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