非广延热力学的严格结果
Rigorous results in non-extensive thermodynamics
摘要：
本文在非广延热力学的背景下研究有限自由度的量子系统。证明了用启发式方法得到的试探密度矩阵是平衡密度矩阵。如果熵参数q大于1，则存在试平衡密度矩阵要求q小于某个临界值q_c，该临界值取决于哈密顿量的本征值发散的速率。当q<2时，证明了存在唯一的平衡密度矩阵.对于0和1之间的q，使得2<q+q_c，自由能在试验密度矩阵的集合中具有至少一个最小值。如果存在唯一的平衡密度矩阵，那么它必然是试验密度矩阵之一。注意，这是一个有限秩算子，这意味着在平衡状态下，高能级的占有概率为零。
英文摘要：
This paper studies quantum systems with a finite number of degrees of freedom in the context of non-extensive thermodynamics. A trial density matrix, obtained by heuristic methods, is proved to be the equilibrium density matrix. If the entropic parameter q is larger than 1 then existence of the trial equilibrium density matrix requires that q is less than some critical value q_c which depends on the rate by which the eigenvalues of the hamiltonian diverge. Existence of a unique equilibrium density matrix is proved if in addition q<2 holds. For q between 0 and 1, such that 2<q+q_c, the free energy has at least one minimum in the set of trial density matrices. If a unique equilibrium density matrix exists then it is necessarily one of the trial density matrices. Note that this is a finite rank operator, which means that in equilibrium high energy levels have zero probability of occupancy.
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