Most Relevant Scienfic Contributions

Most relevant scientific contributions

General Relativity. I started my research activity in 1995 working on General Relativity. I have studied the dynamics of gravitating spinless particles, in (2+1) dimensions, within the framework of the 't Hooft's presentation of the three-dimensional space-time. The causal structure of the space-time, that is the universe, seen by a set of spinless gravitating particles, is obtained by describing the time-evolution of such system in terms of a two-dimensional space-like surface. The set of these Cauchy surfaces, which are parametrised by time, foliate the space-time. Within the 't Hooft's solution each surface, at fixed time, is taken to be piecewise flat and the corresponding tessellation is constructed by means of polygons. The presence of a point-like mass is pointed out by a conical singularity. Each polygon's edges moves at constant speed but when one of the lengths of the polygon's edges vanishes or when one of the corners hits an edge, one has a transition in which the structure of the tessellation of the spatial surface undergoes a local change. These changes are embodied in the 't Hooft's formalism. By using the 't Hooft representation, in paper 1., it has been described the decays of gravitating particles, and it has been derived the associated kinematic relations by discussing the effect of gravity on the structure of the energy-momentum conservation law. Moreover, it has been shown that the time-evolution of the momenta of a set of spinless particles, can be described in terms of a generalized space-time kinematics, based on a representation of the braid group.

In paper 2. it has been illustrated how the non-trivial topology of universe can be described within the 't Hooft's formalism. In the case of a universe with spatial topology of torus it has been discussed the relation between 't Hooft's transitions and modular transformations. Moreover, it has been constructed the universal covering of space-time and it has been computed the Hubble's constant for an expanding universe.

In paper 4. it has been studied the time evolution of a universe with the spatial topology a torus by means of a one-polygon tessellation with three-legs vertices. It has been considered the action of modular transformations on this solution, and it has been shown that the corresponding orbit is densely distributed inside the entire phase-space of the model.

Thermodynamic Phase Transitions. In my PhD Thesis I started considering the dynamical properties of Hamiltonian systems that - in the statistical mechanical canonical framework - undergo second order phase transitions. In the light of a differential-geometrical treatment of Hamiltonian dynamics, where the motions are seen as geodesics of suitable Riemannian manifolds, the “mechanical manifolds”, as well as the constant energy hyper-surfaces of phase space, these systems are found to undergo peculiar geometric transformations that, in my Thesis, have been attributed to major topological changes at the transition point. These results are reported in paper 5., where certain geometric properties of sub-manifolds of configuration space have been numerically investigated for the classicalmodels in one and two dimensions, and where it has been shown that peculiar behaviours of these geometric quantities are found only in the two dimensional system, when the phase transformation does take place. To cope with the constructive study of topological properties of high dimensional manifolds is a hard task, nevertheless a substantial confirmation of the crucial role of configuration space topology in the occurrence of phase transitions has been given in my PhD Thesis through several intermediate steps culminating with the direct numerical computation of the Euler characteristic of the relevant manifolds, in presence and in the absence of phase transitions. These results are contained in paper [6.] that reports upon numerical computation of the Euler characteristic(which is a topological invariant) of the equipotential hyper-surfacesof the configuration space of the classicalmodels in one and two dimensions. The patternversus(potential energy) reveals that a major topology change in the familytakes place in presence of a phase transition. This preliminary work together that one of paper [15.] have paved the way to proving a general theorem. In fact, more recently, I have been co-author of two papers, [25.] and [26.], where it has been given a proof of a general theorem about the necessity of a deep link between configuration space topology and the occurrence of thermodynamical phase transitions. These works enlighten of new light the problem of phase transitions, indeed the thermodynamic singularities that correspond to phase transformations, are found to necessarily stem from suitable changes in the topology of certain sub-manifolds of the configuration space. This is an independent an more fundamental mechanism with respect to the standard ones and it provides new tools and methods to tackle those transition phenomena that are presently at the forefront of the research.

Bose-Einstein Condensates Dynamics. In 2000 I started investigating the dynamics of interacting Bose-Einstein condensates by exploring the classical as well the quantum regime.

In the first regime, the dynamics of Bose-Einstein condensates in periodic potentials (optical lattices) is described by the Gross-Pitaevskii equation that, in the tight binding regime, is well approximated by a discrete non-linear Schroedinger equation. On the other hand, the quantum dynamics of such systems is described by a Bose-Hubbard model. Initially, I have investigated the classical dynamics of two interacting Bose-Einstein condensates (dimer). In spite of the integrable character of this system's dynamics, due to non-linearity, its equations of motion admit pretty interesting and unexpected dynamical behaviours. In paper 8., it has been investigated the system's phase-space structure and it has been analysed the peculiar behaviours displayed by the system as the population self-trapping. Furthermore, in part of paper 8., and in paper 9., it has been considered the dimer quantum dynamics and it has been analysed the link between classical trajectories and quantum energy level spectrum. In 10. 12. and 14. it has been shown that the apparently harmless addition of a further coupled condensate to the dimer system, is sufficient to make the dynamics of three coupled Bose-Einstein condensates (trimer) non-integrable. Indeed, in 10. 12. and 14. has been investigated the trimer classical dynamics in different inequivalent, and experimentally meaningful, configurations showing that it displays instabilities in extended regions of the phase space. Besides the chaotic nature of the trimer dynamics in the latter papers they have been investigated the collective modes associated with the system's equations and the non-linear self-trapping that emerges in the super-fluid regime.

The non-linear character of the classical equations of motion of Bose-Einstein condensates in arrays of arbitrary length, is at the base of the self-localization phenomenon that, in 23., has been predicted to take place in condensates loaded in dissipative optical lattices. The dynamical configurations that in 23. has been shown to spontaneously appear, have the nature of “breathing dynamical solutions” that are the very genuine non-linear discrete dynamical states. These macroscopic effects are, and have been, of primary importance in leading toward their experimental observation also in small (few interacting condensates) systems.

Dynamics and Thermodynamics of Classical and Quantum Systems. In paper J. Stat. Phys. (2011) 143: 824–830 I have considered a generic classical many particle system described by an autonomous Hamiltonian which, in addition, has a conserved quantity , so that the Poisson bracketvanishes. I have derived in detail the microcanonical expressions for entropy and temperature. I have shown that both of these quantities depend on multidimensional integrals over submanifolds given by the intersection of the constant energy hypersurfaces with those defined by . I have shown that temperature and higher order derivatives of entropy are microcanonical observable that, under the hypothesis of ergodicity, can be calculated as time averages of suitable functions. Finally I have derived the explicit expression of the function that gives the temperature. In paper [S. Iubini, R. Franzosi, R. Livi, G.-L. Oppo and A. Politi, “Discrete breathers and negative-temperature states”[36.]] We have shown how negative temperature states can be obtained in the Nonlinear Discrete Schroedinger Equation. The definition of the microcanonical temperature given by me in paper J. Stat. Phys. (2011) 143: 824–830, associated with the corresponding Hamiltonian, allows to obtain a consistent thermodynamic description for both positive and negative temperature states. We have also described how one can pass from positive to negative temperatures by applying energy dissipation to the Nonlinear Discrete Schroedinger Equation chain boundaries. We have found that the microscopic evolution in thermalized negative temperature states is characterized by the mechanism of focusing of particle density (and energy), characterized by the formation of localized breather states. 

Microcanonical entropy for Classical Systems. The validity of the concept of negative temperature has been recently challenged by arguing that the Boltzmann entropy (that allows negative temperatures) is inconsistent from a mathematical and statistical point of view, whereas the Gibbs entropy (that does not admit negative temperatures) provides the correct definition for the microcanonical entropy. In [42.] and [43.] we have proved that the Boltzmann entropy is thermodynamically and mathematically consistent. In [45.] we have proposed a novel definition for the microcanonical entropy that resolve the debate on the correct definition of the microcanonical entropy. In particular we have show that this entropy definition fixes the problem inherent the exact extensivity of the caloric equation. Furthermore, this entropy reproduces results which are in agreement with the ones predicted with standard Boltzmann entropy when applied to macroscopic systems. On the contrary, the predictions obtained with the standard Boltzmann entropy and with the entropy we propose, are different for small system sizes. Thus, we conclude that the Boltzmann entropy provides a correct description for macroscopic systems whereas extremely small systems should be better described with the entropy that we propose [49.].

Entanglement Characterisation. In [52.] We propose a measure of entanglement that can be computed for any pure and mixed state of an M-qudit hybrid system. The entanglement measure has the form of a distance that we derive from an adapted application of the Fubini-Study metric. This measure is invariant under local unitary transformations and defined as trace of a suitable metric that we derive, the entanglement metric g. Furthermore, the analysis of the eigenvalues of g gives information about the robustness of entanglement.