Non-Additive Entropic Forms and Evolution Equations for Continuous and Discrete Probabilities.

Increasing interest has been shown in the subject of non-additive entropic forms during recent years, which has essentially been due to their potential applications in the area of complex systems. Based on the fact that a given entropic form should depend only on a set of probabilities, its time evolution is directly related to the evolution of these probabilities. In the present work, we discuss some basic aspects related to non-additive entropies considering their time evolution in the cases of continuous and discrete probabilities, for which nonlinear forms of Fokker-Planck and master equations are considered, respectively. For continuous probabilities, we discuss an H-theorem, which is proven by connecting functionals that appear in a nonlinear Fokker-Planck equation with a general entropic form. This theorem ensures that the stationary-state solution of the Fokker-Planck equation coincides with the equilibrium solution that emerges from the extremization of the entropic form. At equilibrium, we show that a Carnot cycle holds for a general entropic form under standard thermodynamic requirements. In the case of discrete probabilities, we also prove an H-theorem considering the time evolution of probabilities described by a master equation. The stationary-state solution that comes from the master equation is shown to coincide with the equilibrium solution that emerges from the extremization of the entropic form. For this case, we also discuss how the third law of thermodynamics applies to equilibrium non-additive entropic forms in general. The physical consequences related to the fact that the equilibrium-state distributions, which are obtained from the corresponding evolution equations (for both continuous and discrete probabilities), coincide with those obtained from the extremization of the entropic form, the restrictions for the validity of a Carnot cycle, and an appropriate formulation of the third law of thermodynamics for general entropic forms are discussed.

First-Principle Validation of Fourier's Law: One-Dimensional Classical Inertial Heisenberg Model.

The thermal conductance of a one-dimensional classical inertial Heisenberg model of linear size L is computed, considering the first and last particles in thermal contact with heat baths at higher and lower temperatures, Th and Tl (Th>Tl), respectively. These particles at the extremities of the chain are subjected to standard Langevin dynamics, whereas all remaining rotators (i=2,⋯,L-1) interact by means of nearest-neighbor ferromagnetic couplings and evolve in time following their own equations of motion, being investigated numerically through molecular-dynamics numerical simulations. Fourier's law for the heat flux is verified numerically, with the thermal conductivity becoming independent of the lattice size in the limit L→∞, scaling with the temperature, as κ(T)∼T-2.25, where T=(Th+Tl)/2. Moreover, the thermal conductance, σ(L,T)≡κ(T)/L, is well-fitted by a function, which is typical of nonextensive statistical mechanics, according to σ(L,T)=Aexpq(-Bxη), where A and B are constants, x=L0.475T, q=2.28±0.04, and η=2.88±0.04.

External Stimuli on Neural Networks: Analytical and Numerical Approaches.

Based on the behavior of living beings, which react mostly to external stimuli, we introduce a neural-network model that uses external patterns as a fundamental tool for the process of recognition. In this proposal, external stimuli appear as an additional field, and basins of attraction, representing memories, arise in accordance with this new field. This is in contrast to the more-common attractor neural networks, where memories are attractors inside well-defined basins of attraction. We show that this procedure considerably increases the storage capabilities of the neural network; this property is illustrated by the standard Hopfield model, which reveals that the recognition capacity of our model may be enlarged, typically, by a factor 102. The primary challenge here consists in calibrating the influence of the external stimulus, in order to attenuate the noise generated by memories that are not correlated with the external pattern. The system is analyzed primarily through numerical simulations. However, since there is the possibility of performing analytical calculations for the Hopfield model, the agreement between these two approaches can be tested-matching results are indicated in some cases. We also show that the present proposal exhibits a crucial attribute of living beings, which concerns their ability to react promptly to changes in the external environment. Additionally, we illustrate that this new approach may significantly enlarge the recognition capacity of neural networks in various situations; with correlated and non-correlated memories, as well as diluted, symmetric, or asymmetric interactions (synapses). This demonstrates that it can be implemented easily on a wide diversity of models.

d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies.

We numerically study the first-principle dynamics and thermostatistics of a d-dimensional classical inertial Heisenberg ferromagnetic model ( d = 1 , 2 , 3 ) with interactions decaying with the distance r i j as 1 / r i j α ( α ≥ 0 ), where the limit α = 0 ( α → ∞ ) corresponds to infinite-range (nearest-neighbour) interactions, and the ratio α / d > 1 ( 0 ≤ α / d ≤ 1 ) characterizes the short-ranged (long-ranged) regime. By means of first-principle molecular dynamics we study: (i) The scaling with the system size N of the maximum Lyapunov exponent λ in the form λ ∼ N - κ , where κ ( α / d ) depends only on the ratio α / d ; (ii) The time-averaged single-particle angular momenta probability distributions for a typical case in the long-range regime 0 ≤ α / d ≤ 1 (which turns out to be well fitted by q-Gaussians), and (iii) The time-averaged single-particle energies probability distributions for a typical case in the long-range regime 0 ≤ α / d ≤ 1 (which turns out to be well fitted by q-exponentials). Through the Lyapunov exponents we observe an intriguing, and possibly size-dependent, persistence of the non-Boltzmannian behavior even in the α / d > 1 regime. The universality that we observe for the probability distributions with regard to the ratio α / d makes this model similar to the α -XY and α -Fermi-Pasta-Ulam Hamiltonian models as well as to asymptotically scale-invariant growing networks.

Equilibrium States in Two-Temperature Systems.

Systems characterized by more than one temperature usually appear in nonequilibrium statistical mechanics. In some cases, e.g., glasses, there is a temperature at which fast variables become thermalized, and another case associated with modes that evolve towards an equilibrium state in a very slow way. Recently, it was shown that a system of vortices interacting repulsively, considered as an appropriate model for type-II superconductors, presents an equilibrium state characterized by two temperatures. The main novelty concerns the fact that apart from the usual temperature T, related to fluctuations in particle velocities, an additional temperature θ was introduced, associated with fluctuations in particle positions. Since they present physically distinct characteristics, the system may reach an equilibrium state, characterized by finite and different values of these temperatures. In the application of type-II superconductors, it was shown that θ ≫ T , so that thermal effects could be neglected, leading to a consistent thermodynamic framework based solely on the temperature θ . In the present work, a more general situation, concerning a system characterized by two distinct temperatures θ 1 and θ 2 , which may be of the same order of magnitude, is discussed. These temperatures appear as coefficients of different diffusion contributions of a nonlinear Fokker-Planck equation. An H-theorem is proven, relating such a Fokker-Planck equation to a sum of two entropic forms, each of them associated with a given diffusion term; as a consequence, the corresponding stationary state may be considered as an equilibrium state, characterized by two temperatures. One of the conditions for such a state to occur is that the different temperature parameters, θ 1 and θ 2 , should be thermodynamically conjugated to distinct entropic forms, S 1 and S 2 , respectively. A functional Λ [ P ] ≡ Λ ( S 1 [ P ] , S 2 [ P ] ) is introduced, which presents properties characteristic of an entropic form; moreover, a thermodynamically conjugated temperature parameter γ ( θ 1 , θ 2 ) can be consistently defined, so that an alternative physical description is proposed in terms of these pairs of variables. The physical consequences, and particularly, the fact that the equilibrium-state distribution, obtained from the Fokker-Planck equation, should coincide with the one from entropy extremization, are discussed.

Associating an Entropy with Power-Law Frequency of Events.

Events occurring with a frequency described by power laws, within a certain range of validity, are very common in natural systems. In many of them, it is possible to associate an energy spectrum and one can show that these types of phenomena are intimately related to Tsallis entropy S q . The relevant parameters become: (i) The entropic index q, which is directly related to the power of the corresponding distribution; (ii) The ground-state energy ε 0 , in terms of which all energies are rescaled. One verifies that the corresponding processes take place at a temperature T q with k T q ∝ ε 0 (i.e., isothermal processes, for a given q), in analogy with those in the class of self-organized criticality, which are known to occur at fixed temperatures. Typical examples are analyzed, like earthquakes, avalanches, and forest fires, and in some of them, the entropic index q and value of T q are estimated. The knowledge of the associated entropic form opens the possibility for a deeper understanding of such phenomena, particularly by using information theory and optimization procedures.

Entropic form emergent from superstatistics.

The Beck-Cohen superstatistics became an important theory in the scenario of complex systems because it generates distributions representing regions of a nonequilibrium system, characterized by different temperatures T≡ß^{-1}, leading to a probability distribution f(ß). In superstatistics, some classes have been most frequently considered for f(ß), like χ^{2}, χ^{2} inverse, and log-normal ones. Herein we investigate the superstatistics resulting from a χ_{η}^{2} distribution through a modification of the usual χ^{2} by introducing a real index η (0<η≤1). In this way, one covers two common and relevant distributions as particular cases, proportional to the q-exponential (e_{q}^{-ßx}=[1-(1-q)ßx]^{1/1-q}) and the stretched exponential (e^{-(ßx)^{η}}). Furthermore, an associated generalized entropic form is found. Since these two particular-case distributions have been frequently found in the literature, we expect that the present results should be applicable to a wide range of classes of complex systems.

Finite-size scaling of quasi-stationary-state temperature.

We numerically study, from first principles, the temperature T_{QSS} and duration t_{QSS} of the longstanding initial quasi-stationary state of the isolated d-dimensional classical inertial α-XY ferromagnet with two-body interactions decaying as 1/r_{ij}^{α} (α≥0). It is shown that this temperature T_{QSS} (defined proportional to the kinetic energy per particle) depends, for the long-range regime 0≤α/d≤1, on (α,d,U,N) with numerically negligible changes for dimensions d=1,2,3, with U the energy per particle and N the number of particles. We verify the finite-size scaling T_{QSS}-T_{∞}∝1/N^{ß} where T_{∞}≡2U-1 for U≲U_{c}, and ß appears to depend sensibly only on α/d. Our numerical results indicate that neither the scaling with N of T_{QSS} nor that of t_{QSS} depend on U.

Criticality in the duration of quasistationary state.

The duration of the quasistationary states (QSSs) emerging in the d-dimensional classical inertial α-XY model, i.e., N planar rotators whose interactions decay with the distance r_{ij} as 1/r_{ij}^{α} (α≥0), is studied through first-principles molecular dynamics. These QSSs appear along the whole long-range interaction regime (0≤α/d≤1), for an average energy per rotator UU_{c}. They are characterized by a kinetic temperature T_{QSS}, before a crossover to a second plateau occurring at the Boltzmann-Gibbs temperature T_{BG}>T_{QSS}. We investigate here the behavior of their duration t_{QSS} when U approaches U_{c} from below, for large values of N. Contrary to the usual belief that the QSS merely disappears as U→U_{c}, we show that its duration goes through a critical phenomenon, namely t_{QSS}∝(U_{c}-U)^{-ξ}. Universality is found for the critical exponent ξ≃5/3 throughout the whole long-range interaction regime.

Quasi-stationary-state duration in the classical d-dimensional long-range inertial XY ferromagnet.

A classical α-XY inertial model, consisting of N two-component rotators and characterized by interactions decaying with the distance r_{ij} as 1/r_{ij}^{α} (α≥0) is studied through first-principle molecular-dynamics simulations on d-dimensional lattices of linear size L (N≡L^{d} and d=1,2,3). The limits α=0 and α→∞ correspond to infinite-range and nearest-neighbor interactions, respectively, whereas the ratio α/d>1 (0≤α/d≤1) is associated with short-range (long-range) interactions. By analyzing the time evolution of the kinetic temperature T(t) in the long-range-interaction regime, one finds a quasi-stationary state (QSS) characterized by a temperature T_{QSS}; for fixed N and after a sufficiently long time, a crossover to a second plateau occurs, corresponding to the Boltzmann-Gibbs temperature T_{BG} (as predicted within the BG theory), with T_{BG}>T_{QSS}. It is shown that the QSS duration (t_{QSS}) depends on N, α, and d, although the dependence on α appears only through the ratio α/d; in fact, t_{QSS} decreases with α/d and increases with both N and d. Considering a fixed energy value, a scaling for t_{QSS} is proposed, namely, t_{QSS}∝N^{A(α/d)}e^{-B(N)(α/d)^{2}}, analogous to a recent analysis carried out for the classical α-Heisenberg inertial model. It is shown that the exponent A(α/d) and the coefficient B(N) present universal behavior (within error bars), comparing the XY and Heisenberg cases. The present results should be useful for other long-range systems, very common in nature, like those characterized by gravitational and Coulomb forces.

The impact of changing home blood pressure monitoring cutoff from 135/85 to 130/80 mmHg on hypertension phenotypes.

This study investigated the impact of changing abnormal home blood pressure monitoring (HBPM) cutoff from 135/85 to 130/80 mmHg on the prevalence of hypertension phenotypes, considering an abnormal office blood pressure cutoff of 140/90 mmHg. We evaluated 57 768 individuals (26 876 untreated and 30 892 treated with antihypertensive medications) from 719 Brazilian centers who performed HBPM. Changing the HBPM cutoff was associated with increases in masked (from 10% to 22%) and sustained (from 27% to 35%) hypertension, and decreases in white-coat hypertension (from 16% to 7%) and normotension (from 47% to 36%) among untreated participants, and increases in masked (from 11% to 22%) and sustained (from 29% to 36%) uncontrolled hypertension, and decreases in white-coat uncontrolled hypertension (from 15% to 8%) and controlled hypertension (from 45% to 34%) among treated participants. In conclusion, adoption of an abnormal HBPM cutoff of 130/80 mmHg markedly increased the prevalence of out-of-office hypertension and uncontrolled hypertension phenotypes.

Effectiveness of chlorthalidone/amiloride versus losartan in patients with stage I hypertension and diabetes mellitus: results from the PREVER-treatment randomized controlled trial.

AIMS: To compare the blood pressure (BP)-lowering efficacy of a chlorthalidone/amiloride combination pill with losartan, during initial management of JNC 7 Stage I hypertension in patients with type 2 diabetes mellitus. METHODS: In an a priori subgroup analysis of a randomized, double-blind, controlled trial, volunteers aged 30-70 years, with stage I hypertension and diabetes mellitus, were randomized to 12.5/2.5 mg of chlorthalidone/amiloride (N = 47) or 50 mg of losartan (N = 50), and followed for 18 months in 21 clinical centers. If BP remained uncontrolled after three months, study medication dose was doubled, and if uncontrolled after six months, amlodipine (5 and 10 mg) and propranolol (40 and 80 mg BID) were added as open label drugs in a progressive fashion. RESULTS: Systolic BP decreased to a greater extent in participants allocated to diuretics compared to losartan (P < 0.001). After 18 months of follow-up, systolic BP was 128.4 ± 10.3 mmHg in the diuretic group versus 133.5 ± 8.0 in the losartan group (P < 0.01). In the diuretic group, 36 out of 43 participants (83.7%) had a JNC 7 normal BP, compared to 31/47 (66%) in the losartan group (P = 0.089). Serum cholesterol was higher in the diuretic arm at the end of the trial. Other biochemical parameters and reports of adverse events did not differ by treatment. CONCLUSIONS: Treatment of hypertension based on a combination of chlorthalidone and amiloride is more effective for BP lowering compared to losartan in patients with diabetes mellitus and hypertension. TRIAL REGISTRATION: Clinical trials registration number: NCT00971165.

Obstructive sleep apnea is frequent in patients with hypertensive intracerebral hemorrhage and is related to perihematoma edema.

BACKGROUND: Obstructive sleep apnea (OSA) is related to increased systemic inflammation and arterial hypertension. We hypothesize that OSA is frequent in patients with acute hypertensive intracerebral hemorrhage (ICH) and is related to the perihematoma edema. METHODS: Thirty-two non-comatose patients with a hypertensive ICH underwent polysomnography in the acute phase. Perihematoma edema volume was measured on CT scans at admission, after 24 h (early control) and after 4-5 days (late control). The Spearman coefficient (r(s)) was used for correlations. RESULTS: OSA occurred in 19 (59.4%) patients. The apnea-hypopnea index was correlated with relative edema at admission CT (r(s) = 0.40; p = 0.031), early CT (r(s) = 0.46; p = 0.011) and at late CT (r(s) = 0.59; p = 0.006). CONCLUSIONS: OSA is highly frequent during the acute phase of hypertensive ICH and is related to perihematoma edema.

Spin-glass attractor on tridimensional hierarchical lattices in the presence of an external magnetic field.

A nearest-neighbor-interaction Ising spin glass, in the presence of an external magnetic field, is studied on different hierarchical lattices that approach the cubic lattice. The magnetic field is considered as uniform or random (following either a bimodal or a Gaussian probability distribution). In all cases, a spin-glass attractor is found, in the plane magnetic field versus temperature, associated with a low-temperature phase. The physical consequences of this attractor are discussed, in view of the present scenario of the spin-glass problem.

Multicritical behavior in a random-field Ising model under a continuous-field probability distribution.

A random-field Ising model that is capable of exhibiting a rich variety of multicritical phenomena, as well as a smearing of such behavior, is investigated. The model consists of an infinite-range-interaction Ising ferromagnet in the presence of a triple Gaussian random magnetic field, which is defined as a superposition of three Gaussian distributions with the same width σ, centered at H = 0 and H = ± H(0), with probabilities p and (1-p)/2, respectively. Such a distribution is very general and recovers, as limiting cases, the trimodal, bimodal and Gaussian probability distributions. In particular, the special case of the random-field Ising model in the presence of a trimodal probability distribution (limit [Formula: see text]) is able to present a rather nontrivial multicritical behavior. It is argued that the triple Gaussian probability distribution is appropriate for a physical description of some diluted antiferromagnets in the presence of a uniform external field, for which the corresponding physical realization consists of an Ising ferromagnet under random fields whose distribution appears to be well represented in terms of a superposition of two parts, namely a trimodal and a continuous contribution. The model is investigated by means of the replica method, and phase diagrams are obtained within the replica-symmetric solution, which is known to be stable for the present system. A rich variety of phase diagrams is presented, with one or two distinct ferromagnetic phases, continuous and first-order transition lines, tricritical, fourth-order, critical end points and many other interesting multicritical phenomena. Additionally, the present model carries the possibility of destroying such multicritical phenomena due to an increase in the randomness, i.e. increasing σ, which represents a very common feature in real systems.

Ising spin glass under continuous-distribution random magnetic fields: Tricritical points and instability lines.

The effects of random magnetic fields are considered in an Ising spin-glass model defined in the limit of infinite-range interactions. The probability distribution for the random magnetic fields is a double Gaussian, which consists of two Gaussian distributions centered, respectively, at +H0 and -H0, presenting the same width sigma . It is argued that such a distribution is more appropriate for a theoretical description of real systems than its simpler particular two well-known limits, namely, the single Gaussian distribution (sigma>>H0) and the bimodal one (sigma=0) . The model is investigated by means of the replica method, and phase diagrams are obtained within the replica-symmetric solution. Critical frontiers exhibiting tricritical points occur for different values of sigma , with the possibility of two tricritical points along the same critical frontier. To our knowledge, it is the first time that such a behavior is verified for a spin-glass model in the presence of a continuous-distribution random field, which represents a typical situation of a real system. The stability of the replica-symmetric solution is analyzed, and the usual Almeida-Thouless instability is verified for low temperatures. It is verified that the higher-temperature tricritical point always appears in the region of stability of the replica-symmetric solution; a condition involving the parameters H0 and sigma , for the occurrence of this tricritical point only, is obtained analytically. Some of our results are discussed in view of experimental measurements available in the literature.

Low-dose transdermal hormone therapy does not interfere with the blood pressure of hypertensive menopausal women: a pilot study.

OBJECTIVES: To determine the effects of low-dose transdermal hormone therapy (HT) on systolic (SBP) and diastolic (DBP) blood pressure (BP) evaluated by 24-h ambulatory blood pressure monitoring (ABPM) in hypertensive postmenopausal women. METHODS: The study was conducted on 24 hypertensive postmenopausal women aged, on average, 54 years and under treatment with enalapril maleate (10-20 mg/day) combined or not with hydrochlorothiazide (25 mg/day). Thirteen women used a transdermal adhesive containing estradiol and norethisterone (25 and 125 mug active substance/day, respectively) and 11 did not receive HT. ABPM, lipid profile, and climacteric symptoms were evaluated before and 3 and 6 months after treatment. RESULTS: After 3 and 6 months of follow-up, there was a statistically significant reduction of the Blatt-Kupperman menopausal index in the treated group (19.6+/-8.3 vs. 9.6+/-5.9 vs. 9.7+/-7.0; P=0.01). No significant difference in any of the ABPM variables (areas under the systolic and diastolic curves, mean SBP and DBP, SBP and DBP loads and wakefulness-sleep variation) or in the lipid profile was observed between or within groups at the three time points studied. CONCLUSION: Low-dose transdermal HT administered for 6 months was effective in improving climacteric symptoms and did not change BP values or circadian pattern in postmenopausal women with mild-to-moderate arterial hypertension taking antihypertensive medications.

Spironolactone Versus Clonidine as a Fourth-Drug Therapy for Resistant Hypertension: The ReHOT Randomized Study (Resistant Hypertension Optimal Treatment).

The aim of this study is to compare spironolactone versus clonidine as the fourth drug in patients with resistant hypertension in a multicenter, randomized trial. Medical therapy adherence was checked by pill counting. Patients with resistant hypertension (no office and ambulatory blood pressure [BP] monitoring control, despite treatment with 3 drugs, including a diuretic, for 12 weeks) were randomized to an additional 12-week treatment with spironolactone (12.5-50 mg QD) or clonidine (0.1-0.3 mg BID). The primary end point was BP control during office (<140/90 mm Hg) and 24-h ambulatory (<130/80 mm Hg) BP monitoring. Secondary end points included BP control from each method and absolute BP reduction. From 1597 patients recruited, 11.7% (187 patients) fulfilled the resistant hypertension criteria. Compared with the spironolactone group (n=95), the clonidine group (n=92) presented similar rates of achieving the primary end point (20.5% versus 20.8%, respectively; relative risk, 1.01 [0.55-1.88]; P=1.00). Secondary end point analysis showed similar office BP (33.3% versus 29.3%) and ambulatory BP monitoring (44% versus 46.2%) control for spironolactone and clonidine, respectively. However, spironolactone promoted greater decrease in 24-h systolic and diastolic BP and diastolic daytime ambulatory BP than clonidine. Per-protocol analysis (limited to patients with ≥80% adherence to spironolactone/clonidine treatment) showed similar results regarding the primary end point. In conclusion, clonidine was not superior to spironolactone in true resistant hypertensive patients, but the overall BP control was low (≈21%). Considering easier posology and greater decrease in secondary end points, spironolactone is preferable for the fourth-drug therapy. CLINICAL TRIAL REGISTRATION: URL: http://www.clinicaltrials.gov. Unique identifier: NCT01643434.

Effectiveness of low-dose diuretics for blood pressure reduction to optimal values in prehypertension: a randomized clinical trial.

BACKGROUND: To determine the effectiveness of low-dose diuretic therapy to achieve an optimal level of blood pressure (BP) in adults with prehypertension. METHODS: The PREVER-prevention trial was a randomized, parallel, double-blinded, placebo-controlled trial, with 18 months of follow-up, conducted at 21 academic medical centers in Brazil. Of 1772 individuals evaluated for eligibility, 730 volunteers with prehypertension who were aged 30-70 years, and who did not reach optimal blood pressure after 3 months of lifestyle intervention, were randomized to a fixed association of chlorthalidone 12.5 mg and amiloride 2.5 mg or placebo once a day. The main outcomes were the percentage of participants who achieved an optimal level of BP. RESULTS: A total of 372 participants were randomly allocated to diuretics and 358 to placebo. After 18 months of treatment, optimal BP was noted in 25.6% of the diuretic group and 19.3% in the placebo group (P < 0.05). The mean net reduction in SBP and DBP for the diuretic group compared with placebo was 2.8 mmHg (95% CI 1.1 to 4.5) and 1.1 mmHg (95% CI -0.09 to 2.4), respectively. Most participants in the active treatment group (74.5%) and in the placebo group (80.7%) continued to have BP in the prehypertension range or progressed to hypertension. CONCLUSION: Low-dose diuretic therapy increased the probability of individuals with prehypertension to achieve optimal BP but most of those treated continued to have a BP in the prehypertension range or progressed to having overt hypertension.