费米子和玻色子的HMF模型
The HMF model for fermions and bosons
摘要：
研究了T=0时具有长程相互作用的量子粒子的热力学。具体来说，我们推广的哈密顿平均场（HMF）模型的费米子和玻色子的情况下。在费米子的情况下，我们认为，在适当的热力学极限成为精确的T-Fermi近似。由费米分布（或水袋分布）描述的平衡组态等价于指数n=1/2的多面体。在玻色子的情况下，我们认为哈特里近似，成为准确的适当的热力学极限。平衡组态是具有余弦相互作用的平均场薛定谔方程的解。我们表明，均匀相，这是不稳定的经典制度，成为稳定的量子制度。这通过费米子的一阶相变和玻色子的二阶相变发生，其中控制参数是归一化的普朗克常量。在费米子的情况下，均匀相是稳定的泡利不相容原理，而玻色子的稳定是由于海森堡不确定原理。因此，费米子和玻色子的热力学极限是不同的。我们指出了量子HMF模型与天体物理学中费米子和玻色子星概念之间的类比。最后，作为我们的分析的副产品，我们得到了新的结果有关的Vlasov动态稳定性的水袋分布。
英文摘要：
We study the thermodynamics of quantum particles with long-range interactions at T=0. Specifically, we generalize the Hamiltonian Mean Field (HMF) model to the case of fermions and bosons. In the case of fermions, we consider the Thomas-Fermi approximation that becomes exact in a proper thermodynamic limit. The equilibrium configurations, described by the Fermi (or waterbag) distribution, are equivalent to polytropes with index n=1/2. In the case of bosons, we consider the Hartree approximation that becomes exact in a proper thermodynamic limit. The equilibrium configurations are solutions of the mean field Schrödinger equation with a cosine interaction. We show that the homogeneous phase, that is unstable in the classical regime, becomes stable in the quantum regime. This takes place through a first order phase transition for fermions and through a second order phase transition for bosons where the control parameter is the normalized Planck constant. In the case of fermions, the homogeneous phase is stabilized by the Pauli exclusion principle while for bosons the stabilization is due to the Heisenberg uncertainty principle. As a result, the thermodynamic limit is different for fermions and bosons. We point out analogies between the quantum HMF model and the concepts of fermion and boson stars in astrophysics. Finally, as a by-product of our analysis, we obtain new results concerning the Vlasov dynamical stability of the waterbag distribution.
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