Lyapunov exponents from geodesic spread in configuration space

The exact form of the Jacobi -- Levi-Civita (JLC) equation for geodesic spread is here explicitly worked out at arbitrary dimension for the configuration space manifold M_E = {q in R^N | V(q) < E} of a standard Hamiltonian system, equipped with the Jacobi (or kinetic energy) metric g_J. As the Hamiltonian flow corresponds to a geodesic flow on (M_E,g_J), the JLC equation can be used to study the degree of instability of the Hamiltonian flow. It is found that the solutions of the JLC equation are closely resembling the solutions of the standard tangent dynamics equation which is used to compute Lyapunov exponents. Therefore the instability exponents obtained through the JLC equation are in perfect quantitative agreement with usual Lyapunov exponents. This work completes a previous investigation that was limited only to two-degrees of freedom systems.Comment: REVTEX file, 10 pages, 2 figure

On the clustering phase transition in self-gravitating N-body systems

The thermodynamic behaviour of self-gravitating $N$-body systems has been worked out by borrowing a standard method from Molecular Dynamics: the time averages of suitable quantities are numerically computed along the dynamical trajectories to yield thermodynamic observables. The link between dynamics and thermodynamics is made in the microcanonical ensemble of statistical mechanics. The dynamics of self-gravitating $N$-body systems has been computed using two different kinds of regularization of the newtonian interaction: the usual softening and a truncation of the Fourier expansion series of the two-body potential. $N$ particles of equal masses are constrained in a finite three dimensional volume. Through the computation of basic thermodynamic observables and of the equation of state in the $P - V$ plane, new evidence is given of the existence of a second order phase transition from a homogeneous phase to a clustered phase. This corresponds to a crossover from a polytrope of index $n=3$, i.e. $p=K V^{-4/3}$, to a perfect gas law $p=K V^{-1}$, as is shown by the isoenergetic curves on the $P - V$ plane. The dynamical-microcanonical averages are compared to their corresponding canonical ensemble averages, obtained through standard Monte Carlo computations. A major disagreement is found, because the canonical ensemble seems to have completely lost any information about the phase transition. The microcanonical ensemble appears as the only reliable statistical framework to tackle self-gravitating systems. Finally, our results -- obtained in a ``microscopic'' framework -- are compared with some existing theoretical predictions -- obtained in a ``macroscopic'' (thermodynamic) framework: qualitative and quantitative agreement is found, with an interesting exception.Comment: 19 pages, 20 figure

Hamiltonian dynamics and geometry of phase transitions in classical XY models

The Hamiltonian dynamics associated to classical, planar, Heisenberg XY models is investigated for two- and three-dimensional lattices. Besides the conventional signatures of phase transitions, here obtained through time averages of thermodynamical observables in place of ensemble averages, qualitatively new information is derived from the temperature dependence of Lyapunov exponents. A Riemannian geometrization of newtonian dynamics suggests to consider other observables of geometric meaning tightly related with the largest Lyapunov exponent. The numerical computation of these observables - unusual in the study of phase transitions - sheds a new light on the microscopic dynamical counterpart of thermodynamics also pointing to the existence of some major change in the geometry of the mechanical manifolds at the thermodynamical transition. Through the microcanonical definition of the entropy, a relationship between thermodynamics and the extrinsic geometry of the constant energy surfaces $\Sigma_E$ of phase space can be naturally established. In this framework, an approximate formula is worked out, determining a highly non-trivial relationship between temperature and topology of the $\Sigma_E$. Whence it can be understood that the appearance of a phase transition must be tightly related to a suitable major topology change of the $\Sigma_E$. This contributes to the understanding of the origin of phase transitions in the microcanonical ensemble.Comment: in press on Physical Review E, 43 pages, LaTeX (uses revtex), 22 PostScript figure

The Fermi-Pasta-Ulam problem revisited: stochasticity thresholds in nonlinear Hamiltonian systems

The Fermi-Pasta-Ulam $\alpha$-model of harmonic oscillators with cubic anharmonic interactions is studied from a statistical mechanical point of view. Systems of N= 32 to 128 oscillators appear to be large enough to suggest statistical mechanical behavior. A key element has been a comparison of the maximum Lyapounov coefficient $\lambda_{max}$ of the FPU $\alpha$-model and that of the Toda lattice. For generic initial conditions, $\lambda_{max}(t)$ is indistinguishable for the two models up to times that increase with decreasing energy (at fixed N). Then suddenly a bifurcation appears, which can be discussed in relation to the breakup of regular, soliton-like structures. After this bifurcation, the $\lambda_{max}$ of the FPU model appears to approach a constant, while the $\lambda_{max}$ of the Toda lattice appears to approach zero, consistent with its integrability. This suggests that for generic initial conditions the FPU $\alpha$-model is chaotic and will therefore approach equilibrium and equipartition of energy. There is, however, a threshold energy density $\epsilon_c(N)\sim 1/N^2$, below which trapping occurs; here the dynamics appears to be regular, soliton-like and the approach to equilibrium - if any - takes longer than observable on any available computer. Above this threshold the system appears to behave in accordance with statistical mechanics, exhibiting an approach to equilibrium in physically reasonable times. The initial conditions chosen by Fermi, Pasta and Ulam were not generic and below threshold and would have required possibly an infinite time to reach equilibrium.Comment: 24 pages, REVTeX, 8 PostScript figures. Published versio

Clinical features and outcomes of elderly hospitalised patients with chronic obstructive pulmonary disease, heart failure or both

Background and objective: Chronic obstructive pulmonary disease (COPD) and heart failure (HF) mutually increase the risk of being present in the same patient, especially if older. Whether or not this coexistence may be associated with a worse prognosis is debated. Therefore, employing data derived from the REPOSI register, we evaluated the clinical features and outcomes in a population of elderly patients admitted to internal medicine wards and having COPD, HF or COPD + HF. Methods: We measured socio-demographic and anthropometric characteristics, severity and prevalence of comorbidities, clinical and laboratory features during hospitalization, mood disorders, functional independence, drug prescriptions and discharge destination. The primary study outcome was the risk of death. Results: We considered 2,343 elderly hospitalized patients (median age 81 years), of whom 1,154 (49%) had COPD, 813 (35%) HF, and 376 (16%) COPD + HF. Patients with COPD + HF had different characteristics than those with COPD or HF, such as a higher prevalence of previous hospitalizations, comorbidities (especially chronic kidney disease), higher respiratory rate at admission and number of prescribed drugs. Patients with COPD + HF (hazard ratio HR 1.74, 95% confidence intervals CI 1.16-2.61) and patients with dementia (HR 1.75, 95% CI 1.06-2.90) had a higher risk of death at one year. The Kaplan-Meier curves showed a higher mortality risk in the group of patients with COPD + HF for all causes (p = 0.010), respiratory causes (p = 0.006), cardiovascular causes (p = 0.046) and respiratory plus cardiovascular causes (p = 0.009). Conclusion: In this real-life cohort of hospitalized elderly patients, the coexistence of COPD and HF significantly worsened prognosis at one year. This finding may help to better define the care needs of this population

Antidiabetic Drug Prescription Pattern in Hospitalized Older Patients with Diabetes

Objective: To describe the prescription pattern of antidiabetic and cardiovascular drugs in a cohort of hospitalized older patients with diabetes. Methods: Patients with diabetes aged 65 years or older hospitalized in internal medicine and/or geriatric wards throughout Italy and enrolled in the REPOSI (REgistro POliterapuie SIMI—Società Italiana di Medicina Interna) registry from 2010 to 2019 and discharged alive were included. Results: Among 1703 patients with diabetes, 1433 (84.2%) were on treatment with at least one antidiabetic drug at hospital admission, mainly prescribed as monotherapy with insulin (28.3%) or metformin (19.2%). The proportion of treated patients decreased at discharge (N = 1309, 76.9%), with a significant reduction over time. Among those prescribed, the proportion of those with insulin alone increased over time (p = 0.0066), while the proportion of those prescribed sulfonylureas decreased (p &lt; 0.0001). Among patients receiving antidiabetic therapy at discharge, 1063 (81.2%) were also prescribed cardiovascular drugs, mainly with an antihypertensive drug alone or in combination (N = 777, 73.1%). Conclusion: The management of older patients with diabetes in a hospital setting is often sub-optimal, as shown by the increasing trend in insulin at discharge, even if an overall improvement has been highlighted by the prevalent decrease in sulfonylureas prescription

The “Diabetes Comorbidome”: A Different Way for Health Professionals to Approach the Comorbidity Burden of Diabetes

(1) Background: The disease burden related to diabetes is increasing greatly, particularly in older subjects. A more comprehensive approach towards the assessment and management of diabetes’ comorbidities is necessary. The aim of this study was to implement our previous data identifying and representing the prevalence of the comorbidities, their association with mortality, and the strength of their relationship in hospitalized elderly patients with diabetes, developing, at the same time, a new graphic representation model of the comorbidome called “Diabetes Comorbidome”. (2) Methods: Data were collected from the RePoSi register. Comorbidities, socio-demographic data, severity and comorbidity indexes (Cumulative Illness rating Scale CIRS-SI and CIRS-CI), and functional status (Barthel Index), were recorded. Mortality rates were assessed in hospital and 3 and 12 months after discharge. (3) Results: Of the 4714 hospitalized elderly patients, 1378 had diabetes. The comorbidities distribution showed that arterial hypertension (57.1%), ischemic heart disease (31.4%), chronic renal failure (28.8%), atrial fibrillation (25.6%), and COPD (22.7%), were the more frequent in subjects with diabetes. The graphic comorbidome showed that the strongest predictors of death at in hospital and at the 3-month follow-up were dementia and cancer. At the 1-year follow-up, cancer was the first comorbidity independently associated with mortality. (4) Conclusions: The “Diabetes Comorbidome” represents the perfect instrument for determining the prevalence of comorbidities and the strength of their relationship with risk of death, as well as the need for an effective treatment for improving clinical outcomes

Clinical features and outcomes of elderly hospitalised patients with chronic obstructive pulmonary disease, heart failure or both

Background and objective: Chronic obstructive pulmonary disease (COPD) and heart failure (HF) mutually increase the risk of being present in the same patient, especially if older. Whether or not this coexistence may be associated with a worse prognosis is debated. Therefore, employing data derived from the REPOSI register, we evaluated the clinical features and outcomes in a population of elderly patients admitted to internal medicine wards and having COPD, HF or COPD + HF. Methods: We measured socio-demographic and anthropometric characteristics, severity and prevalence of comorbidities, clinical and laboratory features during hospitalization, mood disorders, functional independence, drug prescriptions and discharge destination. The primary study outcome was the risk of death. Results: We considered 2,343 elderly hospitalized patients (median age 81&nbsp;years), of whom 1,154 (49%) had COPD, 813 (35%) HF, and 376 (16%) COPD + HF. Patients with COPD + HF had different characteristics than those with COPD or HF, such as a higher prevalence of previous hospitalizations, comorbidities (especially chronic kidney disease), higher respiratory rate at admission and number of prescribed drugs. Patients with COPD + HF (hazard ratio HR 1.74, 95% confidence intervals CI 1.16-2.61) and patients with dementia (HR 1.75, 95% CI 1.06-2.90) had a higher risk of death at one year. The Kaplan-Meier curves showed a higher mortality risk in the group of patients with COPD + HF for all causes (p = 0.010), respiratory causes (p = 0.006), cardiovascular causes (p = 0.046) and respiratory plus cardiovascular causes (p = 0.009). Conclusion: In this real-life cohort of hospitalized elderly patients, the coexistence of COPD and HF significantly worsened prognosis at one year. This finding may help to better define the care needs of this population