简谐外场作用下的非中性等离子体：平衡态热力学和混沌
A pure non-neutral plasma under an external harmonic field: equilibrium thermodynamics and chaos
摘要：
受Ordenes-Huanca和Velazquez [JSTAT \textbf{093303}（2016）]的先前研究的启发，我们研究了纯非中性等离子体的简单模型：一个被限制在频率为$ω$的外部谐波场下的全同非相对论带电粒子系统。我们在连续近似的框架下进行平衡热统计分析。本研究揭示了两个渐近极限的存在：已知的布里渊稳态在零温度下，和气体的谐振子在高温的极限。该模型的非广延特性由相关的热力学极限$N\rightarrow+\infty: U/N^{7/3}=const$证明，该极限与存在牛顿引力的非相对论性点粒子的自引力系统的热力学极限一致。然后，通过数值仿真分析了该模型的动力学特性。验证了相同宏观观测量的热统计估计与时间期望值的一致性。通过数值计算的李雅普诺夫指数在已知的\emph{tangent dynamics}的框架下，系统的混沌。李雅普诺夫指数$λ$的温度依赖性在该模型的两个渐近极限中趋近于零，在它们之间的过渡期间达到最大值。本模型的混沌是非常强的，因为它的速率比微观动力学$τ_{dyn}=1/ω$的特征时间尺度更快。定性分析表明，这种强混沌性不能用碰撞事件来解释，因为它们各自的特征时标是完全不同的，$τ_{ch}\propto τ_{dyn}/N^{1/4}$和$τ_{coll}\propto τ_{dyn}$ .
英文摘要：
Motivated by the precedent study of Ordenes-Huanca and Velazquez [JSTAT \textbf{093303} (2016)], we address the study of a simple model of a pure non-neutral plasma: a system of identical non-relativistic charged particles confined under an external harmonic field with frequency $ω$. We perform the equilibrium thermo-statistical analysis in the framework of continuum approximation. This study reveals the existence of two asymptotic limits: the known Brillouin steady state at zero temperature, and the gas of harmonic oscillators in the limit of high temperatures. The non-extensive character of this model is evidenced by the associated thermodynamic limit, $N\rightarrow+\infty: U/N^{7/3}=const$, which coincides with the thermodynamic limit of a self-gravitating system of non-relativistic point particles in presence of Newtonian gravitation. Afterwards, the dynamics of this model is analyzed through numerical simulations. It is verified the agreement of thermo-statistical estimations and the temporal expectation values of the same macroscopic observables. The system chaoticity is addressed via numerical computation of Lyapunov exponents in the framework of the known \emph{tangent dynamics}. The temperature dependence of Lyapunov exponent $λ$ approaches to zero in the two asymptotic limits of this model, reaching its maximum during the transit between them. The chaos of the present model is very strong, since its rate is faster than the characteristic timescale of the microscopic dynamics $τ_{dyn}=1/ω$. A qualitative analysis suggests that such a strong chaoticity cannot be explained in terms of collision events because of their respective characteristic timescales are quite different, $τ_{ch}\propto τ_{dyn}/N^{1/4}$ and $τ_{coll}\propto τ_{dyn}$.
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