Maria Teresa Di Bari

Geometry and chaos on Riemann and Finsler manifolds

Abstract In this paper we discuss some general aspects of the so-called geometrodynamical approach (GDA) to Chaos and present some results obtained within this framework. We firstly derive a naive but nevertheless a general geometrization procedure, and then specialize the discussion to the description of motion within the framework of two among the most representative implementations of the approach, namely the Jacobi and Finsler geometrodynamics. In order to support the claim that the GDA is not simply a mere re-transcription of the usual dynamics, but instead can give various hints on the understanding of the qualitative behaviour of dynamical systems (DSs), we then compare, from a formal point of view, the tools used within the framework of Hamiltonian dynamics to detect the presence of Chaos with the corresponding ones used within the GDA, i.e., the tangent dynamics and the geodesic deviation equations, respectively, pointing out their general inequivalence. Moreover, to advance the mathematical analysis and to highlight both the peculiarities and the analogies of the methods, we work out two concrete applications to the study of very different, yet typical in distinct contexts, dynamical systems. The first is the well-known Henon-Heiles Hamiltonian, which allows us to exploit how the Finsler GDA is well suited not only for testing the dynamical behaviour of systems with few degrees of freedom, but even to get deeper insights into the sources of instability. We show the effectiveness of the GDA, both in recovering fully satisfactory agreement with the most well-established outcomes and also in helping the understanding of the sources of Chaos. Then, in order to point out the general applicability of the method, we present the results obtained from the geometrical description of a General Relativistic DS, namely the Bianchi IX (BIX) cosmological model, whose peculiarity is well known as its very nature has been debated for a long time. Using the Finsler GDA, we obtain results with a built-in invariance, which give evidence to the non-chaotic behaviour of this system, excluding any global exponential instability in the evolution of the geodesic deviation.

Flexibility and drug release features of lipid/saccharide nanoparticles

The effect of lipophilic additives (excipients and drugs) on the behavior of lipid/saccharide nanoparticles has been investigated by incoherent elastic neutron scattering. Temperature scans from 20 K to 350 K have been performed on lecithin/chitosan particles loaded with isopropyl myristate and cetyl-stearyl alcohol, two lipophilic molecules with different melting temperatures which are commonly added to improve drug loading efficiency. In a similar way the effect of tamoxifen citrate, a lipophilic drug frequently used in breast cancer therapy, has also been studied. The different melting points of the two excipients affect mostly the low-temperature behavior of the nanoparticles. At physiological temperatures they both improve the particle flexibility. On the other hand addition of tamoxifen leads to stiffer structures and to lower amounts of released drug. The macroscopic features of the drug release appear to be correlated to the microscopic flexibility determined by neutron scattering. The data confirm also the role of chitosan as a stiffening and stabilizing agent of the lipid particles.

Dynamical instability and statistical behaviour of N-body systems

Abstract In this paper, we argue about a synthetic characterization of the qualitative properties of generic many-degrees-of-freedom (mdf) dynamical systems (DSs) by means of a geometric description of the dynamics [Geometro-Dynamical Approach (GDA)]. We exhaustively describe the mathematical framework needed to link geometry and dynamical (in)stability, discussing in particular which geometrical quantity is actually related to instability and why some others cannot give, in general, any indication of the occurrence of chaos. The relevance of the Schur theorem to select such Geometrodynamic Indicators (GDI) of instability is then emphasized, as its implications seem to have been underestimated in some of the previous works. We then compare the analytical and numerical results obtained by us and by Pettini and coworkers concerning the FPU chain, verifying a complete agreement between the outcomes of averaging the relevant GDIs over phase space (Casetti and Pettini, 1995) and our findings (Cipriani, 1993), obtained in a more conservative way, time-averaging along geodesics. Along with the check of the ergodic properties of GDIs, these results confirm that the mechanism responsible for chaos in realistic DSs largely depends on the fluctuations of curvatures rather than on their negative values, whose occurrence is very unlikely. On these grounds we emphasize the importance of the virialization process, which separates two different regimes of instability. This evolutionary path, predicted on the basis of analytical estimates, receives clear support from numerical simulations, which, at the same time, confirm also the features of the evolution of the GDIs along with their dependence on the number of degrees of freedom, N , and on the other relevant parameters of the system, pointing out the scarce relevance of negative curvature (for N ⪢ 1) as a source of instability. The general arguments outlined above, are then concretely applied to two specific N-body problems, obtaining some new insights into known outcomes and also some new results The comparative analysis of the FPU chain and the gravitational N-body system allows us to suggest a new definition of strong stochasticity, for any DS. The generalization of the concept of dynamical time-scale, tD, is at the basis of this new criterion. We derive for both the mdf systems considered the ( N , e)-dependence of tD (e being the specific energy) of the system. In light of this, the results obtained (Cerruti-Sola and Pettini, 1995), indeed turn out to be reliable, the perplexity there raised originating from the neglected N -dependence of tD, and not to an excessive degree of approximation in the averaged equations used. This points out also the peculiarities of gravitationally bound systems, which are always in a regime of strong instability; the dimensionless quantity L1 = γ1 · tD [γ1 is the maximal Lyapunov Characteristic Number (LCN)] being always positive and independent of e, as it happens for the FPU chain only above the strong stochasticity threshold (SST). The numerical checks on the analytical estimates about the ( N , e)-dependence of GDIs, allow us to single out their scaling laws, which support our claim that, for N ⪢ 1, the probability of finding a negative value of Ricci curvature is practically negligible, always for the FPU chain, whereas in the case of the Gravitational N-body system, this is certainly true when the virial equilibrium has been attained. The strong stochasticity of the latter DS is clearly due to the large amplitude of curvature fluctuations. To prove the positivity of Ricci curvature, we need to discuss the pathologies of mathematical Newtonian interaction, which have some implications also on the ergodicity of the GDIs for this DS. We discuss the Statistical Mechanical properties of gravity, arguing how they are related to its long range nature rather than to its short scale divergencies. The N -scaling behaviour of the single terms entering the Ricci curvature show that the dominant contribution comes from the Laplacian of the potential energy, whose singularity is reflected on the issue of equality between time and static averages. However, we find that the physical N-body system is actually ergodic where the GDIs are concerned, and that the Ricci curvature associated is indeed almost everywhere (and then almost always) positive, as long as N ⪢ 1 and the system is gravitationally bound and virialized. On these grounds the equality among the above mentioned averages is restored, and the GDA to instability of gravitating systems gives fully reliable and understandable results. Finally, as a by-product of the numerical simulations performed, for both the DSs considered, it emerges that the time averages of GDIs quickly approach the corresponding canonical ones, even in the quasi-integrable limit, whereas, as expected, their fluctuations relax on much longer timescales, in particular below the SST.

Incoherent quasi-elastic neutron scattering study of chemical hydrogels based on poly(vinyl alcohol)

In this study we report on an incoherent QENS experiment carried out at the IRIS beamline of the ISIS facility on a chemical hydrogel based on poly (vinyl alcohol). This biocompatible synthetic polymer can be used for obtaining hydrogels with potential use in the field of biomaterials. For this purpose a detailed knowledge of the state of water caged in the polymer network is requested. We characterized the dynamics of water in the hydrogel by incoherent QENS approach and to some extent we were also able to study the dynamic properties of the polymer moiety. Incoherent QENS has proved to be a valuable investigation tool in the field of the characterization of hydrogels.

Water dynamics in physical hydrogels based on partially hydrophobized hyaluronic acid.

The dynamics of hyaluronate-based hydrogels has been investigated by quasielastic neutron scattering (QENS). Hyaluronate (HYA) has been compared, in the same conditions of temperature and polymer concentration, to a chemically modified form, HYADD, in which the backbone has been grafted with a hexadecyl (C(16)) side-chain with a degree of substitution of about 2% (mol/mol). This modification increases the hydrophobicity of the polysaccharide and leads to a stable gel already at polymer concentration of 0.3% (w/v), yielding a viscosupplementation with less quantity of polysaccharide. The time-scale covered by our measurements probes both water and segmental biopolymer motions. In both systems, the local dynamics of the network in the ps time-scale is mostly due to local reorientational motions of side groups. Such motions are not significantly affected by the small amount of aliphatic chains forming the hydrophobic junctions in HYADD. The diffusivity of water in both HYA and HYADD coincides with that of pure water within the experimental uncertainty. This result confirms previous ones on the dynamics of water in HYA solutions and it is of relevance for biomedical applications of hyaluronate-based systems because it affects the diffusive processes of metabolites and their interaction with tissues.

Dynamics of Nanostructures for Drug Delivery: the Potential of QENS

Abstract In recent biomedical studies different nanocarrier systems have been proposed for ´smart´ drug delivery. The engineering of these systems requires as a prerequisite a detailed knowledge of their structure and dynamics at the molecular level. Quasielastic neutron scattering is an ideal tool for dynamic studies of these complex systems since it provides information on molecular motions in a time window which is important to relate the local dynamics to the macroscopic functional properties of the drug vectors. Some selected examples referring to different nanostructures will be analysed in the following.

The Many Faces of Dynamical Instability

Recently, [6, 7, 3, 8], we proposed a generalization to non Riemannian manifolds of the so-called Geometro-Dynamical Approach (GDA) to Chaos, [11, 2], able to widen the applicability of the method to a considerably larger class of dynamical systems (DS’s). Here, we carry on our efforts on a pathway directed towards a synthetic and a priori characterization of the qualitative properties of generic DS’s. Although being aware that this goal is very ambitious, and that, up to now, many of the trials have been discouraging, we shouldn’t forget the theoretical as well practical relevance held by a possible successful attempt. Indeed, if only it would be concevaible to single out a synthetic indicator of (in)stability, we will be able to avoid all the consuming computations needed to empirically discover the nature of a particular orbit, perhaps noticeably different with respect to another one very nearby. Besides this, going beyond the semi-phenomenological mere recognition of the occurrence of dynamical instability, this approach could give deeper hints on its sources, even in those situations where the boundary between Order and Chaos tends to become more and more nuanced, and different tools seem to give conflicting answers. Lately, a renewed interest towards a concise description of dynamical instability1 has grown, together with the feeling of the need to look deeply at the intermingled structures underlving the transition from quasi-integrable to stochastic motions.