有限量子体统计系综理论与热力学奇异性研究

Based on the crystalization of the thermodynamics and extension of the tradi- tional quantum statistics, we construct a new quantum statistical ensemble theory and study the singularities of the thermodynamic functions in the finite systems. It is necessary to reconstruct this theory, since the traditional theory invalids for describing the singularities of the thermodynamic functions （phase transition， negative specific heat and negative entropy）in those systems. The new theory in this proposal not only has the same results with ones in the traditional theory in the thermodynamic limit, but also can explain the thermodynamic singualities, as well as can predict the novel phenomena in the small quantum systems. We take the finite volume, finite particle numbers, finite total energy and finite energy levels without the continuous spectrum approximation and thermodynamic limits. The closed system is not isolated by introducing a microcanonical equilibrium state. The system includes the boundary effects without any approximation of the interaction with the heat bath in the thermodynamic coupling. In obtaining the unique distribution functions and general entropy expressions in the finite systems, we calculate exactly the various thermodynamic quantities. It is shown naturally the singularities and novel phenomena in the finite quantum systems, e.g., negative specific heat, negative emtropy, and noncontiumn phase transition, ect. We emphasize some new effects which can be explained physically and should in principle be observable in suitably designed experiments of the small systems in the low temperatures. The productions in this proposal for the challenge fundamentional statistical problem maybe collected to the text book of the statistical physics.

在传承热力学、拓展统计学基础上，本项目构建新的有限体统计系综理论和研究热力学函数的奇异性。传统的统计系综理论不能描述有限体中热力学函数的奇异性，有必要重建新理论。新理论不但在热力学极限情况下可退回到传统理论，而且可解释有限体所发生的热力学奇异性，并预言新奇量子统计现象。本项目取有限体积、有限粒子数、有限总能量、有限能级，不作连续谱近似，不取热力学极限。引入微正则平衡态使得封闭体系不再孤立。在有限体热力学耦合下，不作体系与热槽间相互作用的近似而自动包含体系的边界效应。分析统一的分布函数和普适的熵表示，严格计算热力学量，自然显示有限体热力学函数的奇异性和新奇现象，例如负比热、负熵、负压缩性、负磁化率和非连续相变等。物理上，相变源于相互作用，负比热和负熵分别源于有限体不可逆过程的非补偿热和熵产生，并设计实验予以证实之。这些具有挑战性的基础研究所取得的原创性成果可写入统计物理教科书。

传统统计理论对于有限体不适用，依此在有限体上的各种修正或推广都不是原创理论。 我们虽然延续了Boltzmann-Gibbs理论的统计方法，但不做Boltzmann-Gibbs统计因子的近似，而是从第一性原理重建这个统计因子。 根据这个全新的系综理论，得到了有限体系的普适的配分函数表示，尤其是全新的熵的第一性原理表示。. 我们解决了巨大数值计算所面临的困难。从数论、积分表示、生成函数和严格递推关系等方面来发展精确统计配分函数的计算以及理论方法的探讨。我们发展的生成函数方法，不但将整数幂级数展开推广到任意数作为次幂的级数展开，而且将幂级数式增长的计算量压缩到多项式增长的计算量。我们的配分函数严格递推关系方法，不但将热力学变量的整数倍缩放关系推广到任意数倍缩放关系，而且将需要枚举的不可计及的海量数据压缩到幂级数式增长的计算量。. 我们对三维各项同性谐振子势下的玻色子体系，以及囚禁在周期性边界条件的立方盒子中的玻色气体的比热进行了数值计算。 从我们理论得出的数值模拟结果可以与实验数据进行对比。预言了正则系综中比热显示的“微观相变"和相互作用引起的非连续相变。力图简要回答有限量子体的统计物理本质。