对数引力中的香农信息熵、孤子簇与玻色-爱因斯坦凝聚
Shannon information entropy, soliton clusters and Bose-Einstein condensation in log gravity
AI总结：
本文对临界宇宙学拓扑大质量引力对数扇区的配分函数给出概率解释，证明排列服从Ewens抽样公式，结合孤子证据揭示不稳定孤子簇生成机制，引入香农信息熵度量其演化，并论证对数引力中存在玻色-爱因斯坦凝聚。
摘要：
我们对临界宇宙学拓扑大质量引力对数扇区的位形配分函数给出了概率论解释，其中我们先前工作中考虑的 Hurwitz 数在排列的轮换分布中扮演概率的角色。特别地，我们证明了排列服从 Ewens 抽样公式分布，该公式在划分结构理论及其在碎裂扩散过程和随机树中的应用中发挥重要作用。这一新的概率论结果连同先前建立的理论中存在孤子的证据，为最初在该理论中观察到的不稳定性提供了新见解。我们论证，单粒子水平下种子孤子的不稳定传播在多粒子水平下诱发了具有根树位形的缺陷孤子簇碎片的生成，从而提供了一个无序的图景。随后，引入了该概率分布的香农信息熵，作为所生成的不稳定孤子簇演化的度量。最后，基于 Feynman 关于液氦 $λ$-相变中排列对称性的路径积分形式，我们论证了对数位形配分函数中排列轮换的存在表明了对数引力中存在玻色-爱因斯坦凝聚。
英文摘要：
We give a probabilistic interpretation of the configurational partition function of the logarithmic sector of critical cosmological topologically massive gravity, in which the Hurwitz numbers considered in our previous works assume the role of probabilities in a distribution on cycles of permutations. In particular, it is shown that the permutations are distributed according to the Ewens sampling formula which plays a major role in the theory of partition structures and their applications to diffusive processes of fragmentation, and in random trees. This new probabilistic result together with the previously established evidence of solitons in the theory provide new insights on the instability originally observed in the theory. We argue that the unstable propagation of a seed soliton at single particle level induces the generation of fragments of defect soliton clusters with rooted tree configuration at multiparticle level, providing a disordered landscape. The Shannon information entropy of the probability distribution is then introduced as a measure of the evolution of the unstable soliton clusters generated. Finally, based on Feynman's path integral formalism on permutation symmetry in the $λ$-transition of liquid helium, we argue that the existence of permutation cycles in the configurational log partition function indicates the presence of Bose-Einstein condensates in log gravity.
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