This special issue is focused on the applications of Fractional differential equations (FDE) and Partial differential equations (PDE) in Mathematical Physics and related field. Due to the intrinsic inter-disciplinary nature of Mathematical Physics often overlap with other areas of mathematics such as partial differential equations, special functions, and asymptotic analysis.

Fractional differential equations are a generalization of differential equation through the application of fractional calculus. Fractional differential equations have gained importance and popularity, mainly due to its demonstrated applications in science and engineering. For example, these equations are increasingly used to model problems in fluid flow, rheology, diffusion, relaxation, oscillation, anomalous diffusion, reaction-diffusion, turbulence, diffusive transport akin to diffusion, electric networks, polymer physics, chemical physics, electrochemistry of corrosion, relaxation processes in complex systems, propagation of seismic waves, dynamical processes in self-similar and porous structures and many other physical processes. The most important advantage of using fractional differential equations in these and other applications is their non-local property. It is well known that the integer order differential operator is a local operator but the fractional order differential operator is nonlocal.

Quantum entanglement happens when two distinct systems interact. The correlation between their properties remains after the systems (particles, for example) are separated in space. An observer that attempts to measure one system could also predict the measurements of another observer looking at the second entangled system located at distance. Entanglement doesn’t allow communication faster than the speed of light but allows “teleporting” a quantum copy of the system, while destroying the original. As a consequence, entanglement between two distinct points in spacetime might be able to generate a wormhole in spacetime connecting them. If spacetime can be fundamentally explained using quantum fields and quantum states generating separate spacetime pieces, connected by entanglement, gravity becomes the dynamics of the entanglement, governed by thermodynamics.

Several papers in Quantum Gravity claim that entanglement entropy plays a very important and central role in the physics of spacetime. Recent work in Quantum Gravity has suggested that entanglement is the glue between different regions of the spacetime and proposed that entanglement entropy is the main ingredient of the semi-classical structure of spacetime in Quantum Gravity. The connection between gravity, quantum theory and thermodynamics is not yet fully understood. Currently, there is no unique and clear picture of the relationship between quantum fields, gravity and thermodynamics. The papers included in this collection share a common ambition to understand this relationship, under the generic assumption that entanglement entropy plays an essential role in spacetime geometry. The relevance of its role can’t be underestimated anymore.

When considering quantum effects in curved spacetime, thermal aspects seem to become unavoidable. A possible coherent thermodynamical picture of gravity could help with the understanding of the quantum aspects of gravity. Consequently, entanglement entropy could be the central ingredient for unifying the quantum, gravity and thermodynamics. The aim of this Special Issue is to provide an idea on the current status of research in the field of Gravity and quantum entanglement, gain new insights on theoretical open problems in fundamental physics and motivate future potential research activity.

A number of novel ideas have been advanced to explore the connection between entanglement and holographic principle, wormholes, gauge theories, and the curved spacetime. This issue brings together a number of papers which explore the many aspects of the relationship between gravity, entanglement and black hole thermodynamics. We explore the special role played by entanglement entropy in the context of several approaches to quantum gravity. The submitted papers that that are accepted will cover a number of ideas in this field. We have split the Issue in a number of four different sections that address several related focus problems concerning entanglement and its relevant possible connections with Quantum Gravity and Thermodynamics:

• Section 1: Entanglement in curved spacetimes

• Section 2: Entanglement entropy and gravity

• Section 3: AdS/CFT correspondence and holographic entanglement

• Section 4: Wormholes and ER=EPR Conjecture

We encourage the reader to peruse the entire Issue, as many papers will cross the frontiers between these sections. Here, we collect various papers on this topic, aiming to provide a critical inquiry of the new developments in this field. The papers will be ordered by the date of the submission.

II-VI compound that crystallizes into one of two forms: hexagonal wurtzite (WZ) and cubic zincblende (ZB) structures. The cubic or ZB phase has a more isotropic property, higher carrier mobility, lower phonon scattering, and often better doping efficiency. The wurtzite (WZ) phase has a direct band gap. Compounds with high iconicity have WZ structure, because the ideal WZ structure has a larger Coulomb interaction energy and Madelung constant. The ground state quantum potential (GSQP) of II-VI compound is not well understood which can be studied using the maximum energy principle based upon power law Sq entropies. The probability densities in both position and momentum space corresponding to the ground state of the isotropic D-dimensional quantum harmonic oscillator are Gaussians function. The maximum-entropy formalism leads to a generalization of the Gaussian probability density, which is known as q-Gaussians. These q-Gaussians constitute some of the simplest and most important examples of maximum-Sq distributions. A particle moving under the influence of a potential V (x) is a spherically symmetric system (invariant under rotations) if the potential function depends only on the distance r from the origin. Spherical symmetry implies the conservation of the angular momentum and determines the structure of the eigenvalue spectrum of the Hamiltonian. This special issue should examine the aspects of the spherically symmetric quantum potentials V(r) for II-VI compound whose ground state wave functions are associated with q-Gaussian densities.

It is amazing to see at which frequency the Heun family of equations appeared in the last ten years in several applications in black hole, quantum, plasma, molecular and solid state physics. Despite this wide range of applications our mathematical knowledge of the properties of the Heun functions is far from being complete. We can safely say that the Heun equations will play the same role in the xxI century that the hypergeometric equation played in the XX century. A special issue on the Heun functions with particular emphasis on their properties and applications would provide an excellent opportunity to review the topic for at least two good reasons: the only related book (A. Ronveaux, Heun’s differential equation, Oxford University Press) dates back to 1995 and since then there have been several publications enlarging the findings collected in Ronveaux’s book. This special issue should also examine previously unaddressed aspects such as the connection problem for the Heun equation and its confluent cases, propose and develop new approaches, exchange perspectives and encourage new lines of research.