Alamoudi S M, Boyanovsky D, de Vega H J
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1999 Jul;60(1):94-113. doi: 10.1103/physreve.60.94.
We study the dynamics of relaxation and thermalization in an exactly solvable model of a particle interacting with a harmonic oscillator bath. Our goal is to understand the effects of non-Markovian processes on the relaxational dynamics and to compare the exact evolution of the distribution function with approximate Markovian and non-Markovian quantum kinetics. There are two different cases that are studied in detail: (i) a quasiparticle (resonance) when the renormalized frequency of the particle is above the frequency threshold of the bath and (ii) a stable renormalized "particle" state below this threshold. The time evolution of the occupation number for the particle is evaluated exactly using different approaches that yield to complementary insights. The exact solution allows us to investigate the concept of the formation time of a quasiparticle and to study the difference between the relaxation of the distribution of bare particles and that of quasiparticles. For the case of quasiparticles, the exact occupation number asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes whereas in the stable particle case, the distribution of particles does not thermalize with the bath. We derive a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation (energy conserving) are obtained from the quantum kinetic equation in the limit of wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian, and Boltzmann approximations are compared to the exact result. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important.
我们研究了一个粒子与简谐振子浴相互作用的精确可解模型中的弛豫和热化动力学。我们的目标是了解非马尔可夫过程对弛豫动力学的影响，并将分布函数的精确演化与近似马尔可夫和非马尔可夫量子动力学进行比较。详细研究了两种不同的情况：（i）当粒子的重整化频率高于浴的频率阈值时的准粒子（共振），以及（ii）低于该阈值的稳定重整化“粒子”态。使用不同的方法精确评估了粒子占据数的时间演化，这些方法能提供互补的见解。精确解使我们能够研究准粒子形成时间的概念，并研究裸粒子分布和准粒子分布弛豫之间的差异。对于准粒子的情况，精确的占据数渐近地趋向于一个统计平衡分布，由于离壳过程，该分布不同于简单的玻色 - 爱因斯坦形式，而在稳定粒子的情况下，粒子分布不会与浴达到热平衡。我们推导了一个非马尔可夫量子动力学方程，该方程对微扰级数进行了重整，并包括离壳效应。在不同的粗粒化假设下，从量子动力学方程在时间尺度广泛分离的极限中得到了包含离壳贡献的马尔可夫近似和通常的玻尔兹曼方程（能量守恒）。将非马尔可夫、马尔可夫和玻尔兹曼近似所预测的弛豫动力学与精确结果进行了比较。在宽共振以及阈值和重整化效应重要的情况下，玻尔兹曼方法被证明是失败的。
