We discuss anomalous relaxation processes of a quantum Brownian particle which interacts with an acoustic phonon field as a thermal reservoir in one-dimensional chain molecule. We derive a kinetic equation for the particle using the complex spectral representation of the Liouville-von Neumann operator. Due to the one-dimensionality, the momentum space separates into infinite sets of disjoint irreducible subspaces dynamically independent of one another. Hence, momentum relaxation occurs only within each subspace toward the Maxwell distribution. We obtain a hydrodynamic mode with transport coefficients, a sound velocity, and a diffusion coefficient, defined in each subspace. Moreover, because the sound velocity has momentum dependence, phase mixing affects the broadening of the spatial distribution of the particle in addition to the diffusion process. Due to the phase mixing, the increase rate of the mean-square displacement of the particle increases linearly with time and diverges in the long-time limit.

中文翻译：

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一维分子链中量子布朗粒子的反常扩散

我们讨论了一维布朗布朗粒子的异常弛豫过程，该布朗布朗粒子与一维链状分子中的声子声场相互作用，将其作为储热库。我们使用Liouville-von Neumann算子的复杂频谱表示来得出粒子的动力学方程。由于具有一维性，动量空间可以动态地彼此独立地分成无数个不相交的不可约子空间。因此，动量弛豫仅在朝向Maxwell分布的每个子空间内发生。我们获得在每个子空间中定义的具有传输系数，声速和扩散系数的流体力学模式。此外，由于声速与动量有关，除扩散过程外，相混合还影响颗粒空间分布的加宽。由于相混合，颗粒的均方位移的增加率随时间线性增加，并且在长时间限制内发散。