Weak and strong chaos in Fermi-Pasta-Ulam models and beyond

We briefly review some of the most relevant results that our group obtained in the past, while investigating the dynamics of the Fermi-Pasta-Ulam (FPU) models. The first result is the numerical evidence of the existence of two different kinds of transitions in the dynamics of the FPU models: (i) A stochasticity threshold (ST), characterized by a value of the energy per degree of freedom below which the overwhelming majority of the phase space trajectories are regular (vanishing Lyapunov exponents). It tends to vanish as the number N of degrees of freedom is increased. (ii) A strong stochasticity threshold (SST), characterized by a value of the energy per degree of freedom at which a crossover appears between two different power laws of the energy dependence of the largest Lyapunov exponent, which phenomenologically corresponds to the transition between weak and strong chaotic regimes. It is stable with N. The second result is the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos. Starting this theory has been motivated by the inadequacy of the approach based on homoclinic intersections to explain the origin of chaos in systems of arbitrarily large N, or arbitrarily far from quasi-integrability, or displaying a transition between weak and strong chaos. Finally, the third result stems from the search for the transition between weak and strong chaos in systems other than FPU. Actually, we found that a very sharp SST appears as the dynamical counterpart of a thermodynamic phase transition, which in turn has led, in the light of the Riemannian theory of chaos, to the development of a topological theory of phase transitions.     

Phase Transitions and Topology Changes in Configuration Space

The relation between thermodynamic phase transitions in classical systems and topological changes in their configuration space is discussed for two physical models and contains the first exact analytic computation of a topologic invariant (the Euler characteristic) of certain submanifolds in the configuration space of two physical models. The models are the mean-field XY model and the one-dimensional XY model with nearest-neighbor interactions. The former model undergoes a second-order phase transition at a finite critical temperature while the latter has no phase transitions. The computation of this topologic invariant is performed within the framework of Morse theory. In both models topology changes in configuration space are present as the potential energy is varied; however, in the mean-field model there is a particularly strong topology change, corresponding to a big jump in the Euler characteristic, connected with the phase transition, which is absent in the one-dimensional model with no phase transition. The comparison between the two models has two major consequences: (i) it lends new and strong support to a recently proposed topological approach to the study of phase transitions; (ii) it allows us to conjecture which particular topology changes could entail a phase transition in general. We also discuss a simplified illustrative model of the topology changes connected to phase transitions using of two-dimensional surfaces, and a possible direct connection between topological invariants and thermodynamic quantities.     

Vibrational entropy and the structural organization of proteins

In this paper we analyze the vibrational spectra of a large ensemble of non-homologous protein structures by means of a novel tool, that we coin Hierarchical Network Model (HNM). Our coarse-grained scheme accounts for the intrinsic heterogeneity of force constants displayed by protein arrangements and also incorporates side chain degrees of freedom. Our analysis shows that vibrational entropy per unit residue correlates with the content of secondary structure. Furthermore, we assess the individual contribution to vibrational entropy of the novel features of our scheme as compared with the predictions of state-of-the-art network models. This analysis highlights the importance of properly accounting for the intrinsic hierarchy in force strengths typical of the different atomic bonds that build up and stabilize protein scaffolds. Finally, we discuss possible implications of our findings in the context of protein aggregation phenomena.     

Steric control on the redox chemistry of (η5-C9H7)2Yb(II)(THF)2 by 6-aryl substituted iminopyridines

A steric control on the reductive capacity of ytterbocenes towards iminopyridine ligands is described. The reaction of (η(5)-C(9)H(7))(2)Yb(THF)(2) with a series of 6-organyl-2-(aldimino)pyridyl ligands (IPy) takes place with the replacement of two THF molecules by one IPy unit. In contrast to the rich reductive ytterbocene chemistry described in the presence of the unsubstituted (aldimino)pyridyl ligand, all 6-aryl substituted IPys scrutinized hereafter are involved into the metal coordination as neutral bidentate {N,N} or tridentate {N,N,S; N,N,O} ligands, with no changes of the metal oxidation state in the final complexes. A series of Yb(II) metallocene complexes of general formula (η(5)-C(9)H(7))(2)Yb(II)(η(2) or η(3))[2,6-(i)Pr(2)(C(6)H(3))N=CH(C(5)H(3)N)-6-R)] have been isolated and completely characterized. The stereo-electronic role of the aryl substituents in the IPy ligands on the ytterbocene redox chemistry has also been addressed.     

Velocity filtration and temperature inversion in a system with long-range interactions

Graphene has vast promising applications on the nanoelectronics and spintronics because of its unique magnetic and electronic properties. Making use of an ab initio spin-polarized density functional theory, implemented by the Heyd-Scuseria-Ernzerhof 06 hybrid functional method, abbreviated as HSE06, the properties of semi-metal nitrogen-substitutional graphene are investigated. From our investigations, we conclude that introducing nitrogen doping would possibly perform the spin symmetry breaking, resulting energy degeneracy at some doping configurations. The spin symmetry breaking would cause spin-polarized effects, which induce magnetic response in graphene. This paper systematically analyzes the dependence of magnetic moments and band gaps in graphene on nitrogen-substitutional doping configurations.     

Metal‐to‐Ligand Alkyl Migration Inducing Carbon–Sulfur Bond Cleavage in Dialkyl Yttrium Complexes Supported by Thiazole‐Containing Amidopyridinate Ligands: Synthesis,Characterization, and Catalytic Activity in the Intramolecular Hydroamination Reaction

Neutral Y^III dialkyl complexes supported by tridentate N^?,N,N monoanionic methylthiazole– or benzothiazole–amidopyridinate ligands have been prepared and completely characterized. Studies on their stability in solution revealed progressive rearrangement of the coordination sphere in the benzothiazole‐containing system through an unprecedented metal‐to‐ligand alkyl migration and subsequent thiazole ring opening. Attempts to synthesize hydrido species from the dialkyl precursor led to the generation of a dimeric yttrium species stabilized by a trianionic N^?,N,N^?,S^? ligand as the result of metal‐to‐ligand hydride migration with chemoselective thiazole ring opening and subsequent dimerization through intermolecular addition of the residual Y?H group to the imino fragment of a second equivalent of the ring‐opened intermediate. DFT calculations were used to elucidate the thermodynamics and kinetics of the process, in support of the experimental evidence. Finally, all isolated yttrium complexes, especially their cationic forms prepared by activation with the Lewis acid Ph₃C⁺[B(C₆F₅)₄]^?, were found to be good candidate catalysts for intramolecular hydroamination/cyclization reactions. Their catalytic performance with a number of primary and secondary amino alkenes was assessed.     

"Click" on tubes: a versatile approach towards multimodal functionalization of SWCNTs

Organic functionalization of carbon nanotube sidewalls is a tool of primary importance in material science and nanotechnology, equally from a fundamental and an applicative point of view. Here, an efficient and versatile approach for the organic/organometallic functionalization of single-walled carbon nanotubes (SWCNTs) capable of imparting multimodality to these fundamental nanostructures, is described. Our strategy takes advantage of well-established Cu-mediated acetylene-azide coupling (CuAAC) reactions applied to phenylazido-functionalized SWCNTs for their convenient homo-/heterodecoration with a number of organic/organometallic frameworks, or mixtures thereof, bearing terminal acetylene pendant arms. Phenylazido-decorated SWCNTs were prepared by chemoselective arylation of the CNT sidewalls with diazonium salts under mild conditions, and subsequently used for the copper-mediated cycloaddition protocol in the presence of terminal acetylenes. The latter reaction was performed in one step by using either single acetylene derivatives or equimolar mixtures of terminal alkynes bearing either similar functional groups (masked with orthogonally cleavable protecting groups) or easily distinguishable functionalities (on the basis of complementary analytical/spectroscopic techniques). All materials and intermediates were characterized with respect to their most relevant aspects/properties by TEM microscopy, thermogravimetric analysis coupled with MS analysis of volatiles (TG-MS), elemental analysis, cyclic voltammetry (CV), Raman and UV/Vis spectroscopy. The functional loading and related chemical grafting of both primary amino- and ferrocene-decorated SWCNTs were spectroscopically (UV/Vis, Kaiser test) and electrochemically (CV) determined, respectively.     

Microcanonical relation between continuous and discrete spin models

A relation between a class of stationary points of the energy landscape of continuous spin models on a lattice and the configurations of an Ising model defined on the same lattice suggests an approximate expression for the microcanonical density of states. Based on this approximation we conjecture that if a O(n) model with ferromagnetic interactions on a lattice has a phase transition, its critical energy density is equal to that of the n=1 case, i.e., an Ising system with the same interactions. The conjecture holds true in the case of long-range interactions. For nearest-neighbor interactions, numerical results are consistent with the conjecture for n=2 and n=3 in three dimensions. For n=2 in two dimensions (XY model) the conjecture yields a prediction for the critical energy of the Bere?inskij-Kosterlitz-Thouless transition, which would be equal to that of the two-dimensional Ising model. We discuss available numerical data in this respect.