Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory
Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory
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具有长程相互作用的哈密顿和布朗系统：IV。
DOI：
10.1016/j.physa.2005.06.088
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发表时间：
2004
影响因子：
3.3
通讯作者：
P. Chavanis
中科院分区：
文献类型：
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作者：
P. Chavanis
我们讨论了长程相互作用系统的动力学和热力学。我们将孤立哈密顿系统的微正则描述与随机强迫布朗系统的正则描述进行了对比。我们证明了在适当的热力学极限下，平均场近似是精确的。平衡分布函数是从静态BBGKY类层次结构中得到的积分微分方程的解。它还在适当的约束下优化了一个熵势（熵或自由能）。我们讨论这些系统的动力学理论。在N→+∞极限下，哈密顿系统由Vlasov方程描述。为了达到1/N，齐次系统的碰撞项具有Lenard-Balescu算子的形式。当忽略集体效应时，它退化为朗道算符.我们还考虑了测试粒子在场粒子浴中的运动，并导出了福克-普朗克方程的一般形式。扩散系数是各向异性的，并且取决于测试粒子的速度。这可能导致异常扩散。对于布朗系统，在N→+∞极限下，动力学方程为非局部Kramers方程.在强摩擦极限− 1，它减少到一个非局部Smoluchowski方程。我们给出了明确的结果，自引力系统，二维涡和HMF模型。我们还引入了一类广义随机过程，并推导出相应的广义Fokker-Planck方程。我们讨论了如何广义热力学的概念可以出现在复杂的系统显示异常扩散。
We discuss the dynamics and thermodynamics of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system to the canonical description of a stochastically forced Brownian system. We show that the mean-field approximation is exact in a proper thermodynamic limit. The equilibrium distribution function is solution of an integrodifferential equation obtained from a static BBGKY-like hierarchy. It also optimizes a thermodynamical potential (entropy or free energy) under appropriate constraints. We discuss the kinetic theory of these systems. In the N→+∞ limit, a Hamiltonian system is described by the Vlasov equation. To order 1/N, the collision term of a homogeneous system has the form of the Lenard-Balescu operator. It reduces to the Landau operator when collective effects are neglected. We also consider the motion of a test particle in a bath of field particles and derive the general form of the Fokker-Planck equation. The diffusion coefficient is anisotropic and depends on the velocity of the test particle. This can lead to anomalous diffusion. For Brownian systems, in the N→+∞ limit, the kinetic equation is a non-local Kramers equation. In the strong friction limit ξ→+∞, or for large times t≫ ξ− 1, it reduces to a non-local Smoluchowski equation. We give explicit results for self-gravitating systems, two-dimensional vortices and for the HMF model. We also introduce a generalized class of stochastic processes and derive the corresponding generalized Fokker-Planck equations. We discuss how a notion of generalized thermodynamics can emerge in complex systems displaying anomalous diffusion.