The concept of entanglement is at the heart of quantum physics. It plays a central role in all quantum phenomena involving composite systems. Interestingly, there is an intriguing idea that has attracted considerable attention recently, according to which quantum entanglement may also be essential for understanding the very emergence of time and dynamical evolution. Within this point of view, sometimes referred to as the timeless picture of quantum dynamics, the Universe is regarded as consisting of a clock and a system (or “rest of the Universe”) that are jointly in a stationary quantum state, and time evolution arises as an emergent phenomenon rooted at the entanglement between the clock and the system. Here we provide a pedagogical and self-contained exposition, at the upper undergraduate level, of the role of entanglement in this timeless evolution approach to quantum mechanics. In particular, we give a detailed explanation of how the entanglement between the clock and the system is directly and quantitatively related to the average distinguishability between the states of the system at different times.

We investigate the quantum entanglement-related features of the many-body model of Plastino and Moszkowski [N. Cimento 47 (1978) 470]. This is an exactly solvable N-body, SU2 two-level model exhibiting several quantum phase transitions. We show that these transitions happen to be also entanglement-transitions associated with different many-body Dicke states. The main properties of the model considered here make it particularly well suited to study, by recourse to exact analytical computations, some connections between quantum phase transitions and quantum entanglement-theory.

We investigate entropic aspects of the quantum entanglement dynamics of two-qubits systems interacting with an environment. In particular we consider the detection, based on the violation of classical entropic inequalities involving q-entropies, of the phenomenon of entanglement disappearance and subsequent entanglement revival during the alluded two-qubits' evolution.

A general proof of the energy equipartition theorem is given. Our derivation holds for any distribution function depending on the phase space variables only through the Hamiltonian of the system. This approach generalizes the standard theorem in two main directions. On the one hand, it considers the contribution to the total mean energy of homogeneous functions having a more general type than the ones usually discussed in the literature. On the other hand, our proof does not rely on the assumption of a Boltzmann-Gibbs exponential distribution.

The role of Tsalli's non-extensive Information Measure within an à la Jaynes Information-Theory-based formulation of Statistical Mechanics is discussed in rather detailed fashion.

We revisit Tsallis Maximum Entropy Solutions to the Vlasov-Poisson Equation describing gravitational N-body systems. We review their main characteristics and discuss their relationship with other applications of Tsallis statistics to systems with long range interactions. In the following considerations we shall be dealing witha D-dimensional space so as to be in a position to investigate possible dimensional dependences of Tsallis' parameter q. The particular and important case of the Schuster solution is studied in detail, and the pertinent Tsallis parameter q is given as a function of the space dimension. In the special case of three dimensional space we recover the value q = 7/9, that has already appeared in many applications of Tsallis' formalism involving long range forces.