Shannon information entropy, soliton clusters and Bose-Einstein condensation in log gravity

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.