Geometric mutual information at classical critical points

Geometric Structures Induced by Deformations of the Legendre Transform

The recent link discovered between generalized Legendre transforms and non-dually flat statistical manifolds suggests a fundamental reason behind the ubiquity of Rényi's divergence and entropy in a wide range of physical phenomena. However, these early findings still provide little intuition on the nature of this relationship and its implications for physical systems. Here we shed new light on the Legendre transform by revealing the consequences of its deformation via symplectic geometry and complexification. These findings reveal a novel common framework that leads to a principled and unified understanding of physical systems that are not well-described by classic information-theoretic quantities.

Wave propagation properties of rotationally symmetric lattices with curved beams

In this study, we design a type of rotationally symmetric lattice with curved beams and investigate the wave propagation properties of the structure. The analytical model of the structure is established to obtain the mass and stiffness matrices first. Because the dimensions of the mass and stiffness matrices will become very large if the structure is meshed with a number of small elements, we introduce the symplectic solution method to overcome the above difficulties of solving the eigenvalue problem. The effects of geometrical parameters and slenderness ratios on the distributions of bandgaps and variations of group velocities are investigated. We also numerically investigate the dynamic wave dispersion behavior and the transient responses of displacement and transmission coefficients in lattices subjected to excitations. Excellent agreement is obtained between the results obtained by the symplectic solution method and numerical simulations. The special wave-attenuation property of this type of structure is demonstrated and validated through experimental testing. The measured transmission coefficients in lattices with different geometrical parameters and slenderness ratios are in good agreement with the numerical simulations. The work provides a method for calculating wave behaviors in lattices and obtains lower bandgaps and directional wave propagation.

Entropy Governed by the Absorbing State of Directed Percolation

We investigate the informational aspect of (1+1)-dimensional directed percolation, a canonical model of a nonequilibrium continuous transition to a phase dominated by a single special state called the "absorbing" state. Using a tensor network scheme, we numerically calculate the time evolution of state probability distribution of directed percolation. We find a universal relaxation of Rényi entropy at the absorbing phase transition point as well as a new singularity in the active phase, slightly but distinctly away from the absorbing transition point. At the new singular point, the second-order Rényi entropy has a clear cusp. There we also detect a singular behavior of "entanglement entropy," defined by regarding the probability distribution as a wave function. The entanglement entropy vanishes below the singular point and stays finite above. We confirm that the absorbing state, though its occurrence is exponentially rare in the active phase, is responsible for these phenomena. This interpretation provides us with a unified understanding of time evolution of the Rényi entropy at the critical point as well as in the active phase.