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

Abstract

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.

Integrable hierarchies, Hurwitz numbers and a branch point field in critical topologically massive gravity

“…In our case, the latter was obtained in our previous work [9]. We then identify the correct urn model to be the celebrated Hoppe urn model, leading us to a concrete inference of the presence of defects in the theory as mentioned in [13], as well as to a description of the Markov process associated to the Hoppe urn model using a recurrence relation on the Hurwitz numbers appearing in the log partition function.…”

“…• The θ-ball in Hoppe's urn model was interpreted as a (θ-biased) defect, giving a perspective of the dynamics of defect diffusion in the theory that is complementary to the rooted tree expansion of Z log [12], where the roots were identified as defects [13]. The content of the urns in Hoppe's urn model description is in one-to-one correspondence with the rooted trees, and the mutator ball in the urns corresponds to the roots.…”

“…We previously argued [13] that in the log partition function expansion over rooted trees, soliton clusters are pinned by point-defects represented by the roots as the pinning sites. The content of the urns following Hoppe's scheme corresponds to the graphical representation of the log partition function in terms of rooted trees [12].…”

“…A novel interpretation of the log partition function was established through the connection between its random combinatorial structures and a classical mutation model with numerous applications in mathematical population genetics, called the infinite-alleles model [13]. Such interpretation was possible by realizing that the Hurwitz numbers in the log partition function constitute a set of variables determined by a stochastic process, whose probability distribution is governed by a closed-form sampling formula for the infinite-alleles model called the Ewens sampling formula, with parameter θ = 1 in our case.…”

“…Acknowledgements The author would like to thank the anonymous referee in the review process of [13] for drawing his attention to the investigation of mestability in the log sector of CCTMG. The author is grateful for the time spent at the 13th Joburg Workshop on String Theory, where this work approached its conclusion.…”