弱碰撞等离子体的弛豫：连续谱、离散本征模与回声的衰减
Relaxation of weakly collisional plasma: continuous spectra, discrete eigenmodes, and the decay of echoes
AI总结：
本文对含Lenard-Bernstein碰撞算符的Boltzmann-Poisson方程进行微扰分析，精确解析了弱碰撞对连续谱、离散Landau本征模及等离子体回声衰减的影响，揭示了不同尺度下系统弛豫的物理机制。
摘要：
弱碰撞等离子体的弛豫对实验室和天体物理等离子体具有基础性意义，可用带有Lenard-Bernstein碰撞算符的Boltzmann-Poisson方程来描述。我们对这些方程进行了微扰分析，并获得了精确的解析解，解决了关于弱碰撞对连续谱、离散Landau本征模和等离子体回声衰减影响的长期争议。我们在碰撞算符中保留了阻尼和扩散项。我们发现线性响应是连续统与离散模的时间卷积，前者依赖于粒子的连续速度（对等离子体回声至关重要），后者描述整个系统的相干振荡。由于集体效应或波粒相互作用（Landau阻尼）以及碰撞耗散，离散模随时间呈指数阻尼。连续统也被碰撞阻尼，但方式略有不同。在碰撞时间（碰撞频率$ν_{\mathrm{c}}$的倒数）内，连续统衰减由粒子速度扩散驱动，在$\sim ν^{-1/3}_{\mathrm{c}}$的时间尺度上呈三次指数衰减。然而，超过碰撞时间后，连续统衰减由速度阻尼和位置扩散驱动，在$\sim ν_{\mathrm{c}}$的时间尺度上呈指数衰减。这种缓慢的指数衰减在尺度$λ$与平均自由程$λ_{\mathrm{c}}$相当时对扰动阻尼最强，但在更大尺度上非常缓慢。这建立了局部热平衡（流体极限的本质），并增强了等离子体回声的可探测性。长期衰减在$λ<λ_{\mathrm{c}}$时由离散模驱动，但在$λ>λ_{\mathrm{c}}$时由缓慢衰减的连续统和最小阻尼模驱动。
英文摘要：
The relaxation of a weakly collisional plasma, which is of fundamental interest to laboratory and astrophysical plasmas, can be described by the Boltzmann-Poisson equations with the Lenard-Bernstein collision operator. We perform a perturbative analysis of these equations, and obtain exact analytic solutions that resolve long-standing controversies regarding the impact of weak collisions on the continuous spectra, discrete Landau eigenmodes, and the decay of plasma echoes. We retain both damping and diffusion terms in the collision operator. We find that the linear response is a temporal convolution of a continuum that depends on the continuous velocities of particles (crucial for the plasma echo) and discrete modes that describe coherent oscillations of the entire system. The discrete modes are exponentially damped over time due to collective effects or wave-particle interactions (Landau damping) as well as collisional dissipation. The continuum is also damped by collisions, but somewhat differently. Up to a collision time, the inverse of the collision frequency $ν_{\mathrm{c}}$, the continuum decay is driven by the diffusion of particle velocities and occurs cubic exponentially over a timescale $\sim ν^{-1/3}_{\mathrm{c}}$. After a collision time, however, the continuum decay is driven by the damping of velocities and diffusion of positions, and occurs exponentially over a timescale $\sim ν_{\mathrm{c}}$. This slow exponential decay damps perturbations the most on scales $λ$ comparable to the mean free path $λ_{\mathrm{c}}$, but very slowly on larger scales. This establishes local thermal equilibrium, the essence of the fluid limit, and enhances the detectability of the plasma echo. The long-term decay is driven by the discrete modes for $λ<λ_{\mathrm{c}}$, but, by the slowly decaying continuum and the least damped mode for $λ>λ_{\mathrm{c}}$.
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