The Higgs mechanism and geometrical flows for two-manifolds

Using Perelman’s approach for geometrical flows in terms of an entropy functional, the Higgs mechanism is studied dynamically along flows defined in the space of parameters and in fields space. The model corresponds to two-dimensional gravity that incorporates torsion as the gradient of a Higgs field, and with the reflection symmetry to be spontaneously broken. The results show a discrete mass spectrum and the existence of a mass gap between the Unbroken Exact Symmetry and the Spontaneously Broken Symmetry scenarios. In the latter scenario, the geometries at the degenerate vacua correspond to conformally flat manifolds without torsion; twisted two-dimensional geometries are obtained by building perturbation theory around a ground state; the tunneling quantum probability between vacua is determined along the flows.

Block Factorization of the Relative Entropy via Spatial Mixing

We consider spin systems in the d-dimensional lattice $${\mathbb Z} ^d$$ Z d satisfying the so-called strong spatial mixing condition. We show that the relative entropy functional of the corresponding Gibbs measure satisfies a family of inequalities which control the entropy on a given region $$V\subset {\mathbb Z} ^d$$ V ⊂ Z d in terms of a weighted sum of the entropies on blocks $$A\subset V$$ A ⊂ V when each A is given an arbitrary nonnegative weight $$\alpha _A$$ α A . These inequalities generalize the well known logarithmic Sobolev inequality for the Glauber dynamics. Moreover, they provide a natural extension of the classical Shearer inequality satisfied by the Shannon entropy. Finally, they imply a family of modified logarithmic Sobolev inequalities which give quantitative control on the convergence to equilibrium of arbitrary weighted block dynamics of heat bath type.

On the construction of a family of anomalous-diffusion Fokker–Planck−Kolmogorov’s equations based on the Sharma–Taneja–Mittal entropy functional

A logical scheme for constructing thermodynamics of anomalous stochastic systems based on the nonextensive two-parameter (κ, ς) -entropy of Sharma–Taneja–Mittal (SHTM) is considered. Thermodynamics within the framework (2 - q) -statistics of Tsallis was constructed, which belongs to the STM family of statistics. The approach of linear nonequilibrium thermodynamics to the construction of a family of nonlinear equations of Fokker−Planck−Kolmogorov (FPK), is used, correlated with the entropy of the STM, in which the stationary solution of the diffusion equation coincides with the corresponding generalized Gibbs distribution obtained from the extremality (κ, ς) - entropy condition of a non-additive stochastic system. Taking into account the convexity property of the Bregman divergence, it was shown that the principle of maximum equilibrium entropy is valid for (κ, ς) - systems, and also was proved the H - theorem determining the direction of the time evolution of the non-equilibrium state of the system. This result is extended also to non-equilibrium systems that evolve to a stationary state in accordance with the nonlinear FPK equation. The method of the ansatz- approach for solving non-stationary FPK equations is considered, which allows us to find the time dependence of the probability density distribution function for non-equilibrium anomalous systems. Received diffusive equations FPК can be used, in particular, at the analysis of diffusion of every possible epidemics and pandemics. The obtained diffusion equations of the FPK can be used, in particular, in the analysis of the spread of various epidemics and pandemics.

Information Flows of Diverse Autoencoders

Deep learning methods have had outstanding performances in various fields. A fundamental query is why they are so effective. Information theory provides a potential answer by interpreting the learning process as the information transmission and compression of data. The information flows can be visualized on the information plane of the mutual information among the input, hidden, and output layers. In this study, we examine how the information flows are shaped by the network parameters, such as depth, sparsity, weight constraints, and hidden representations. Here, we adopt autoencoders as models of deep learning, because (i) they have clear guidelines for their information flows, and (ii) they have various species, such as vanilla, sparse, tied, variational, and label autoencoders. We measured their information flows using Rényi’s matrix-based α-order entropy functional. As learning progresses, they show a typical fitting phase where the amounts of input-to-hidden and hidden-to-output mutual information both increase. In the last stage of learning, however, some autoencoders show a simplifying phase, previously called the “compression phase”, where input-to-hidden mutual information diminishes. In particular, the sparsity regularization of hidden activities amplifies the simplifying phase. However, tied, variational, and label autoencoders do not have a simplifying phase. Nevertheless, all autoencoders have similar reconstruction errors for training and test data. Thus, the simplifying phase does not seem to be necessary for the generalization of learning.

Renormalized holographic entanglement entropy in Lovelock gravity

We study the renormalization of Entanglement Entropy in holographic CFTs dual to Lovelock gravity. It is known that the holographic EE in Lovelock gravity is given by the Jacobson-Myers (JM) functional. As usual, due to the divergent Weyl factor in the Fefferman-Graham expansion of the boundary metric for Asymptotically AdS spaces, this entropy functional is infinite. By considering the Kounterterm renormalization procedure, which utilizes extrinsic boundary counterterms in order to renormalize the on-shell Lovelock gravity action for AAdS spacetimes, we propose a new renormalization prescription for the Jacobson-Myers functional. We then explicitly show the cancellation of divergences in the EE up to next-to-leading order in the holographic radial coordinate, for the case of spherical entangling surfaces. Using this new renormalization prescription, we directly find the C−function candidates for odd and even dimensional CFTs dual to Lovelock gravity. Our results illustrate the notable improvement that the Kounterterm method affords over other approaches, as it is non-perturbative and does not require that the Lovelock theory has limiting Einstein behavior.

Entanglement entropy in cubic gravitational theories

We derive the holographic entanglement entropy functional for a generic gravitational theory whose action contains terms up to cubic order in the Riemann tensor, and in any dimension. This is the simplest case for which the so-called splitting problem manifests itself, and we explicitly show that the two common splittings present in the literature — minimal and non-minimal — produce different functionals. We apply our results to the particular examples of a boundary disk and a boundary strip in a state dual to 4- dimensional Poincaré AdS in Einsteinian Cubic Gravity, obtaining the bulk entanglement surface for both functionals and finding that causal wedge inclusion is respected for both splittings and a wide range of values of the cubic coupling.

Holographic entanglement entropy for perturbative higher-curvature gravities

The holographic entanglement entropy functional for higher-curvature gravities involves a weighted sum whose evaluation, beyond quadratic order, requires a complicated theory-dependent splitting of the Riemann tensor components. Using the splittings of general relativity one can obtain unambiguous formulas perturbatively valid for general higher-curvature gravities. Within this setup, we perform a novel rewriting of the functional which gets rid of the weighted sum. The formula is particularly neat for general cubic and quartic theories, and we use it to explicitly evaluate the corresponding functionals. In the case of Lovelock theories, we find that the anomaly term can be written in terms of the exponential of a differential operator. We also show that order-n densities involving nR Riemann tensors (combined with n−nR Ricci’s) give rise to terms with up to 2nR− 2 extrinsic curvatures. In particular, densities built from arbitrary Ricci curvatures combined with zero or one Riemann tensors have no anomaly term in their functionals. Finally, we apply our results for cubic gravities to the evaluation of universal terms coming from various symmetric regions in general dimensions. In particular, we show that the universal function characteristic of corner regions in d = 3 gets modified in its functional dependence on the opening angle with respect to the Einstein gravity result.

Entanglement between two disjoint universes

We use the replica method to compute the entanglement entropy of a universe without gravity entangled in a thermofield-double-like state with a disjoint gravitating universe. Including wormholes between replicas of the latter gives an entropy functional which includes an “island” on the gravitating universe. We solve the back-reaction equations when the cosmological constant is negative to show that this island coincides with a causal shadow region that is created by the entanglement in the gravitating geometry. At high entanglement temperatures, the island contribution to the entropy functional leads to a bound on entanglement entropy, analogous to the Page behavior of evaporating black holes. We demonstrate that the entanglement wedge of the non-gravitating universe grows with the entanglement temperature until, eventually, the gravitating universe can be entirely reconstructed from the non-gravitating one.

Concepts and applications in functional diversity

The use of functional analyses in ecology has grown exponentially over the past two decades, broadening our understanding of biological diversity and its change across space and time. Virtually all ecological sub-disciplines recognize the critical value of looking at species and communities from a functional perspective, and this has led to a proliferation of methods for estimating contrasting dimensions of functional diversity. Differences between these methods and their development generated terminological inconsistencies and confusion about the selection of the most appropriate approach for addressing any particular ecological question, hampering the potential for comparative studies and meta-analyses. We show that two general mathematical frameworks for estimating functional diversity are prevailing: those based on dissimilarity matrices (e.g., Rao entropy, functional dendrograms) and those relying on multidimensional spaces, constructed as either binary (convex hulls) or probabilistic hypervolumes. We review these frameworks, discuss their strengths and weaknesses, and provide an overview of the main R packages allowing to perform these calculations. In parallel, we propose a ‘periodic table’ of functional diversity metrics quantifying the richness, divergence, and regularity of species or individuals under each framework. Therefore, this overview offers a roadmap for confidently approaching functional diversity analyses both theoretically and practically.