arXiv subjects

Constantino Tsallis

Publications and source records attributed to Constantino Tsallis.

At least 19 recordsLinked to original sources

arXiv2026

Hyperstatistics

We propose a general approach, named by us hyperstatistics, to treat complex systems, in which Boltzmann-Gibbs statistics breaks down in domains of the system. Hyperstatistics preserves the concavity of nonadditive $q$-entropy. We obtain analytical closed-form expressions for the here proposed $(q, n)$-generalized Boltzmann factor $B^n_q$ considering uniform, $γ$, Log-normal, F, and the $q$-$γ$ probability distribution functions. Remarkably, for all investigated distribution functions, $B^n_q$ reduces to a $q$-exponential-type function. To demonstrate the applicability of hyperstatistics, we use a table top experiment of the discharge of a capacitor considering $γ$-distributed relaxation times, the pressure decay over time associated with the pumping of $^4$He lines of a closed cycle cryostat, midrapidity data for $p$-Pb collisions at the LHC, as well as data set for acceleration distribution in turbulent systems. Furthermore, we deduce the power-law-like dielectric response using the $q$-$γ$-distribution function. Our proposal is applicable to systems with inherent non-Boltzmann-Gibbsian statistics in domains of the system.

cond-mat.stat-mech↗

arXiv2026

Reminiscences about Hans Capel

As an homage to the memory of Hans Willem Capel, I present some personal reminiscences and thoughts that come to my mind in this special occasion.

cond-mat.stat-mech↗

arXiv2026

Turbulence: An Entropic Approach

We show that maximizing the generalized entropic functional $S_{q,δ}$ subject to standard kinetic energy constraints provides generalized canonical distributions that agree perfectly with measured probability densities of velocity differences at distance $r$ in highly-turbulent Taylor-Couette flow. The end point of the turbulent cascade is described by $δ=\frac{3}{2}$, a parameter value that also plays an important role in black-hole physics. At this point the Kolmogorov length scale $r=η$ is reached and all observable eddy structures of the turbulent flow disappear, in certain analogy to what is observed for black holes at the event horizon. Our approach generalizes statistical mechanics to more general nonadditive entropic functionals $S_{q,δ}$ such that it is applicable to turbulent flows. This approach asymptotically generates stretched $q$-exponentials as generalized canonical distributions relevant for turbulent flow, with a particular dependence of the stretching exponent $δ^{-1}$ on $q$ that follows from the well-known escort formalism in nonextensive statistical mechanics. Along this particular line in the parameter space, the physics can be described by $S_q$ on its own with suitable escort constraints, leading to the prediction $δ^{-1} (r) =2-q(r)$, thus allowing for a consistent thermodynamic description since $S_q$ is both trace-form and composable. We show that the above theoretically derived relation is well satisfied by measured high-precision experimental data for Taylor-Couette flow. At the Kolmogorov length scale $r=η$, the endpoint of our scenario, one has $δ=\frac{3}{2}$ and at this point the third moment of velocity differences ceases to exist and all eddies disappear. We point out various analogies with thermodynamic entropic approaches to black hole physics.

physics.flu-dyn↗

arXiv2026

A Closer Look on the Influence of Constraints Upon the Optimization of the Nonadditive Entropic Functional $S_{q}$

The thermal-equilibrium canonical distribution is currently obtained by maximizing the Boltzmann-Gibbs-von Neumann-Shannon entropy $S_{BG}(p)=k\sum^{W}_{i=1}p_{i}\ln 1/p_{i}$ constrained to $\sum^{W}_{i=1}p_{i}=1$ and $\sum^{W}_{i=1}p_{i}\,e_{i}=U$, $e_{1}\leq\ldots\leq e_{W}$ being the energies of the $W$ possible states and $U\in[e_{1},e_{W}]$ their mean value. We revisit a generalized version of this optimization problem grounded in the nonadditive entropy $S_{q}(p)=k\,(\sum^{W}_{i=1}p_{i}^{q}-1)/(1-q)$ (frequently, though not necessarily, $q\in(0,1)$; $S_1=S_{BG}$), and the constraint $\sum^{W}_{i=1} p_{i}^{q^{\prime}}e_{i} / \sum^{W}_{i=1}p_{i}^{q^{\prime}}=U$, $q^{\prime}>0$. Sufficient conditions for existence, strict positivity, and uniqueness of solutions are derived, along with a theorem that enables their closed-form calculation. We apply these results to deepen the understanding of the two standard cases in the literature ($q^{\prime}=1$ and $q^{\prime}=q$), as well as of a new one ($q^{\prime}=2-q$). We prove that these standard cases are the only ones yielding optimizing probability distributions of $q$-exponential form. Furthermore, we define an effective temperature $T_{q,q^{\prime}}$ through a Clausius-like relation $1/T_{q,q^{\prime}}=\partial S_{q} / \partial U$ and derive a Helmholtz-like energy $F_{q,q^{\prime}}=U-T_{q,q^{\prime}}S_{q}$, with the former grounding the validity of the $0^{th}$ Principle of Thermodynamics within this generalized statistical mechanics. Finally, we show that the case with a linear constraint (i.e., $q^{\prime}=1$) with $q\in(0,1)$ (i) preserves the Third Law of Thermodynamics; (ii) can be used to model classical many-body Hamiltonian systems with arbitrarily-ranged interactions; and (iii) resembles features of low-dimensional nonlinear dynamical systems at the edge of chaos.

cond-mat.stat-mech↗

arXiv2026

Generalized Algebra Grounded on Nonadditive Entropies

The class of $N$-body complex systems with total number of microscopic states given by $W(N) \sim ν^{N^γ}\;(ν>1, \,γ> 0)$ can be thermostatistically handled with the nonadditive entropic functional $S_δ(\{p_{i}\}) = k\sum_{i=1}^W p_i \Bigl(\ln \frac{1}{p_i} \Bigr)^δ\;(δ>0,\,S_1=S_{BG})$, $S_{BG}=k\sum_{i=1}^W p_i \ln \frac{1}{p_i}$ being the Boltzmann-Gibbs functional. Indeed, $S_{δ=1/γ}(\{1/W(N)\})=k[\ln W(N)]^{\frac{1}γ} \propto N$, as mandated by thermodynamics. Another wide class is that with $W(N) \sim N^ρ\;(ρ>0)$ and a generalized statistical mechanics grounded on the nonadditive entropic functional $S_q(\{p_{i}\})=k\sum_{i=1}^W p_i \ln_q \frac{1}{p_i} \;(q\in \mathbb{R},\;S_1=S_{BG})$, with $\ln_q z =\frac{z^{1-q}-1}{1-q}\; (z\geq0,\;q\in\mathbb{R},\;\ln_1 z=\ln z)$, satisfactorily handles such systems with $q=1-1/ρ$. Furthermore, for this class, the size of the corresponding admissible phase space is characterized by $\ln_q (x\otimes_q y) =\ln_q x + \ln_q y,\, x,y\geq1,\,q\leq 1$, and the $q$-product $x\otimes_q y=[x^{1-q}+y^{1-q}-1]^{\frac{1}{1-q}}_{+}\;(x\otimes_1 y=xy)$ also leads to the definition of a $q$-algebra. The entropic functional $S_{q,δ}(\{p_{i}\})=k\sum_{i=1}^W p_i \Bigl(\ln_q \frac{1}{p_i} \Bigr)^δ\;(q\in\mathbb{R},δ>0)$ unifies both cases above: $S_{q,1}=S_q$, $S_{1,δ}=S_δ$ and $S_{1,1}=S_{BG}$. In this paper, we generalize the $q$-algebra associated with $S_{q}$ to a new one associated with $S_{q,δ}$, namely the $(q,δ)$-algebra.

cond-mat.stat-mech↗

arXiv2026

Brain criticality through nonadditive entropic analysis of electroencephalograms

On the grounds of nonadditive entropies -- appropriate for complex systems -- we investigate the electroencephalogram amplitudes of typical and ADHD children. The corresponding probability distributions are $q$-Gaussians, i.e., $ρ(x) \propto e_q^{-βx^2} \equiv [1+(q-1) βx^2]^{1/(1-q)}$, where $(q,β)$ are, respectively, the entropic index characterizing complexity and the inverse width. We show that $q$ tends to monotonically vary with $β$ for both typical and ADHD subjects, thus revealing critical behavior of the brain. Moreover, we verify that ADHD subjects have a higher complexity than the typical ones. Consistently, biomarkers for objective phychyatric diagnosis could emerge along this path. We show that $q$ tends to monotonically vary with $β$ for both typical and ADHD subjects, thus revealing critical behavior of the brain. Moreover, we verify that ADHD subjects have a higher complexity than the typical ones. Consistently, biomarkers for objective phychyatric diagnosis could emerge along this path.

cond-mat.stat-mech↗

arXiv2026

Superstatistical Approach to Turbulent Circulation Fluctuations

Recent investigations of turbulent circulation fluctuations have uncovered substantial insights into the statistical organization of flow structures and revealed unexpected geometric features of turbulent intermittency. Of particular interest here is the observation that circulation probability distribution functions admit a superstatistical representation, namely a description based on "ensembles of Boltzmann-Gibbs ensembles". A fundamental phenomenological ingredient of this approach, which serves as a natural starting point for modeling, relies on the strong correlation between the dissipation field and the spatial distribution of elementary circulation-carrying structures, i.e., small-scale vortices. Within the language of superstatistics, this corresponds to characterizing circulation statistics through an appropriate choice of conditioned (Boltzmann-like) distributions and mixing distributions. We show that the superstatistical class of q-exponentials, known to have broad applicability in a wide range of multiscale and non-equilibrium systems, provides an accurate description of the observed circulation statistics in homogeneous and isotropic turbulence. This finding opens avenues for exploring the statistical structure of the turbulent cascade in the context of non-extensive statistical mechanics, rooted in the concept of non-additive entropies.

physics.flu-dyn↗

arXiv2026

Deep brain microelectrode signal: $q$-statistical approach

We characterize the amplitude statistics of intraoperative microelectrode recordings (MERs) obtained during deep brain stimulation (DBS) surgery in 46 patients with Parkinson's disease, using 184 recordings equally balanced between inside and outside the subthalamic nucleus (STN). The probability density of every recording is quantitatively well described by a $q$-Gaussian (grounded on a nonadditive entropic functional), $ρ(x) \propto [1 + β(q-1) x^2]^{-1/(q-1)}$, with $q > 1$ in all cases, reflecting persistent long-range temporal correlations inconsistent with Gaussian dynamics. Within the superstatistics framework, the slowly fluctuating local variance visible in the raw MER signals is a physical mechanism that directly generates the $q > 1$ form. Beyond individual fits, $q$ and $β$ collapse across all 184 recordings onto the single functional constraint $q = 3 - 1.85\,β^{-0.33}$ ($R \approx -0.91$), a reduction to one effective degree of freedom that is the quantitative hallmark of near-critical dynamics, previously identified in scale-free network growth and in acoustic precursors of material fracture. The index $q$ is statistically indistinguishable across the STN boundary ($\langle\bar{q}_\text{out}/\bar{q}_\text{in} \rangle = 1.03$), while the inverse-widthparameter shows a modest systematic difference ($\langle\barβ_\text{out}/\barβ_\text{in} \rangle = 1.18$). Since $q > 1$ is expected for any brain structure exhibiting long-range correlations, healthy or pathological, it is the tight $q(β)$ coupling, not $q > 1$ per se, that constitutes the candidate near-criticality signature of the parkinsonian cortico-basal-ganglia-thalamocortical loop.

physics.med-ph↗

arXiv2026

Universal and non-universal facets of quantum critical phenomena unveiled along the Schmidt decomposition theorem

Critical phenomena have been extensively investigated both theoretically and experimentally in many fields, such as condensed matter physics, biology, e.g., brain criticality, and cosmology. In particular, the behaviour of response functions right at critical points (CPs) is highly topical. It turns out that in the frame of Boltzmann-Gibbs-von Neumann-Shannon approach, the extensive character of entropy breaks down at CPs. The latter implies diverging susceptibilities, which is at odds with experimental observations. Here, we investigate the influence of the spin magnitude $S$ on the quantum Grüneisen parameter $Γ^{0\text{K}}_{q}$ right at CPs for the 1D Ising model under a transverse magnetic field. Our findings are fourfold: $\textit{i}$) for higher $S$, $Γ^{0\text{K}}_{q}$ is increased, but remains finite, reflecting the enhancement of the Hilbert space dimensionality; $\textit{ii}$) the Schmidt decomposition theorem recovers the extensivity of the nonadditive $q$-entropy $S_q$ only for a $\textit{special}$ value of the entropic index $q$; $\textit{iii}$) the universality class in the frame of $S_q$ depends only on the symmetry of the system; $\textit{iv}$) we propose an experimental setup to explore finite-size effects in connection with the Hilbert space occupation at CPs. Our findings unveil both universal and non-universal aspects of quantum criticality in terms of $Γ^{0\text{K}}_{q}$ and $S_q$.

quant-ph↗

arXiv2025

Composing $α$-Gauss and logistic maps: Gradual and sudden transitions to chaos

We introduce the $α$-Gauss-Logistic map, a new nonlinear dynamics constructed by composing the logistic and $α$-Gauss maps. Explicitly, our model is given by $x_{t+1} = f_L(x_t)x_t^{-α} - \lfloor f_L(x_t)x_t^{-α} \rfloor $ where $f_L(x_t) = r x_t (1-x_t)$ is the logistic map and $ \lfloor \ldots \rfloor $ is the integer part function. Our investigation reveals a rich phenomenology depending solely on two parameters, $r$ and $α$. For $α< 1$, the system exhibits multiple period-doubling cascades to chaos as the parameter $r$ is increased, interspersed with stability windows within the chaotic attractor. In contrast, for $1 \leq α< 2$, the onset of chaos is abrupt, occurring without any prior bifurcations, and the resulting chaotic attractors emerge without stability windows. For $α\geq 2$, the regular behavior is absent. The special case of $α= 1$ allows an analytical treatment, yielding a closed-form formula for the Lyapunov exponent and conditions for an exact uniform invariant density, using the Perron-Frobenius equation. Chaotic regimes for $α= 1$ can exhibit gaps or be gapless. Surprisingly, the golden ratio $Φ$ marks the threshold for the disappearance of the largest gap in the regime diagram. Additionally, at the edge of chaos in the abrupt transition regime, the invariant density approaches a $q$-Gaussian with $q=2$, which corresponds to a Cauchy distribution.

nlin.CD↗

arXiv2025

Central Limit Behavior at the Edge of Chaos in the z-Logistic Map

We focus on the FeigenbaumCoulletTresser point of the dissipative one-dimensional z logistic map. We show that sums of iterates converge to q Gaussian distributions, which optimize the nonadditive entropic functional Sq under simple constraints. We derive a closedform prediction for the entropic index, and validate it numerically via data collapse for typical z values. The formula captures how the limiting law depends on the nonlinearity order and implies finite variance for z larger than 2 and divergent variance for z in between 1 and 2. These results extend edge of chaos central limit behavior beyond the standard case and provide a simple predictive law for unimodal maps with varying maximum order.

cond-mat.stat-mech↗

arXiv2025

Neurophysiological correlates to the human brain complexity through $q$-statistical analysis of electroencephalogram

The prospects of assessing neural complexity (NC) by $q$-statistics of the systemic organization of different types and levels of brain activity were studied. In 70 adult subjects, NC was assessed via the parameter $q$ of $q$-statistics, applied to the ongoing and EEG and its spectral power of 20 scalp points (channels). The NC were estimated both globally for all channels (AllCh) and locally (for each single channel) in different Functional States (FSs). The values of $q$ was compared among FSs and single channels, as well they were correlated with the power of $θ$ (4-8Hz), $β_1$ (15-25Hz) and others EEG bands, in each FS. The value of $q$ across all FSs was higher for AllCh than for the single channels FSs. Consistently with previous studies, we found a negative correlation between NC and age. The FSs did not influence the $q$ of the EEG in AllCh, although locally the FS modulated $q$ in a consistent manner (e.g., reducing $q$ in posterior sites with eyes closed). The $q$ was correlated positively with the power of the $θ$ and negatively with that of the $β_1$ band in general. These findings support the idea that, as a first approach, $q$-statistics can describe the human NC. The relationship between $q$ and $θ$ power aligns with greater NC during FSs such as listening music and resting with eyes open, which is consistent with high-order representations rather than low-informative attentional tasks (OddBall).

q-bio.NC↗

arXiv2024

Anomalous velocity distributions in slow quantum-tunneling chemical reactions

Recent work [Wild et al., Nature 615, 425 (2023)] has provided an experimental break-through in the realization of a quantum-tunneling reaction involving a proton transfer. The reaction $D^-+H_2 \to H^-+HD$ has an extremely slow reaction rate as it can happen only via quantum tunneling, thus requiring an extremely large density of the reactants in the ion trap. At these high densities strong deviations from Maxwell-Boltzmann statistics are observed. Here we develop a consistent generalized statistical mechanics theory for the above nonequilibrium situation involving quantum effects at high densities. The trapped ions are treated in a superstatistical way and a $q$-Maxwellian velocity distribution with a universal dependence of the entropic index $q$ on the density $n$ of the buffer gas is derived. We show that the velocity distribution of the ions is non-Maxwellian, more precisely $q$-Gaussian, i.e., $p(v) \propto v^2 [1+(q-1)\tildeβ v^2]^{1/(1-q)}$, with entropic index $q>1$ depending on the density $n$ of $H_2$ molecules, in excellent agreement with the experimental observations of Wild et al. Our theory also makes predictions on the statistics of temperature fluctuations in the ion trap which can be tested in future experiments. Through the superstatistical approach, we obtain an analytical expression for $q(n)$ which is consistent with the available experimental data, and which yields $\lim_{n\to 0}q(n)=1$, i.e. recovering the Maxwell-Boltzmann distribution in the ideal gas limit, as well as $\lim_{n\to\infty}q(n)=7/5$.

cond-mat.stat-mech↗

arXiv2024

Generalization of the Gauss Map: A jump into chaos with universal features

The Gauss map (or continued fraction map) is an important dissipative one-dimensional discrete-time dynamical system that exhibits chaotic behaviour and which generates a symbolic dynamics consisting of infinitely many different symbols. Here we introduce a generalization of the Gauss map which is given by $x_{t+1}=\frac{1}{x_t^α} - \Bigl[\frac{1}{x_t^α} \Bigr]$ where $α\geq 0$ is a parameter and $x_t \in [0,1]$ ($t=0,1,2,3,\ldots$). The symbol $[\dots ]$ denotes the integer part. This map reduces to the ordinary Gauss map for $α=1$. The system exhibits a sudden `jump into chaos' at the critical parameter value $α=α_c \equiv 0.241485141808811\dots$ which we analyse in detail in this paper. Several analytical and numerical results are established for this new map as a function of the parameter $α$. In particular, we show that, at the critical point, the invariant density approaches a $q$-Gaussian with $q=2$ (i.e., the Cauchy distribution), which becomes infinitely narrow as $α\to α_c^+$. Moreover, in the chaotic region for large values of the parameter $α$ we analytically derive approximate formulas for the invariant density, by solving the corresponding Perron-Frobenius equation. For $α\to \infty$ the uniform density is approached. We provide arguments that some features of this transition scenario are universal and are relevant for other, more general systems as well.

cond-mat.stat-mech↗

arXiv2024

Universally non-diverging Grüneisen parameter at critical points

According to Boltzmann-Gibbs (BG) statistical mechanics, the thermodynamic response, such as the isothermal susceptibility, at critical points (CPs) presents a divergent-like behavior. An appropriate parameter to probe both classical and quantum CPs is the so-called Grüneisen ratio $Γ$. Motivated by the results reported in Phys. Rev. B $\textbf{108}$, L140403 (2023), we extend the quantum version of $Γ$ to the non-additive $q$-entropy $S_q$. Our findings indicate that using $S_q$ at the unique value of $q$ restoring the extensivity of the entropy, $Γ$ is universally non-diverging at CPs. We unprecedentedly introduce $Γ$ in terms of $S_q$, being BG recovered for $q \rightarrow 1$. We thus solve a long-standing problem related to the $\textit{illusory}$ diverging susceptibilities at CPs.

cond-mat.stat-mech↗

arXiv2024

Diffusion crossover from/to $q$-statistics to/from Boltzmann-Gibbs statistics in the classical inertial $α$-XY ferromagnet

We study the angular diffusion in a classical $d-$dimensional inertial XY model with interactions decaying with the distance between spins as $r^{-α}$, wiht $α\geqslant 0$. After a very short-time ballistic regime, with $σ_θ^2\sim t^2$, a super-diffusive regime, for which $σ_θ^2\sim t^{α_D}$, with $α_D \simeq 1\text{.}45$ is observed, whose duration covers an initial quasistationary state and its transition to a second plateau characterized by the Boltzmann-Gibbs temperature $T_\text{BG}$. Long after $T_\text{BG}$ is reached, a crossover to normal diffusion, $σ_θ^2\sim t$, is observed. We relate, for the first time, via the expression $α_D = 2/(3 - q)$, the anomalous diffusion exponent $α_D$ with the entropic index $q$ characterizing the time-averaged angles and momenta probability distribution functions (pdfs), which are given by the so called $q-$Gaussian distributions, $f_q(x)\propto e_q(-βx^2)$, where $e_q (u) \equiv [1 + (1 - q)u]^{\frac{1}{1 - q}}$ ($e_1(u) = \exp(u)$). For fixed size $N$ and large enough times, the index $q_θ$ characterizing the angles pdf approaches unity, thus indicating a final relaxation to Boltzmann-Gibbs equilibrium. For fixed time and large enough $N$, the crossover occurs in the opposite sense.

cond-mat.stat-mech↗

arXiv2024

Identifying Attention-Deficit/Hyperactivity Disorder through the electroencephalogram complexity

There are reasons to suggest that a number of mental disorders may be related to alteration in the neural complexity (NC). Thus, quantitative analysis of NC could be helpful in classifying mental and understanding conditions. Here, focusing on a methodological procedure, we have worked with young individuals, typical and with attention-deficit/hyperactivity disorder (ADHD) whose NC was assessed using q-statistics applied to the electroencephalogram (EEG). The EEG was recorded while subjects performed the visual Attention Network Test (ANT) and during a short pretask period of resting state. Time intervals of the EEG amplitudes that passed a threshold were collected from task and pretask signals from each subject. The data were satisfactorily fitted with a stretched $q$-exponential including a power-law prefactor(characterized by the exponent c), thus determining the best $(c, q)$ for each subject, indicative of their individual complexity. We found larger values of $q$ and $c$ in ADHD subjects as compared with the typical subjects both at task and pretask periods, the task values for both groups being larger than at rest. The $c$ parameter was highly specific in relation to DSM diagnosis for inattention, where well-defined clusters were observed. The parameter values were organized in four well-defined clusters in $(c, q)$-space. As expected, the tasks apparently induced greater complexity in neural functional states with likely greater amount of internal information processing. The results suggest that complexity is higher in ADHD subjects than in typical pairs. The distribution of values in the $(c, q)$-space derived from $q$-statistics seems to be a promising biomarker for ADHD diagnosis.

q-bio.NC↗

arXiv2024

Extensive Composable Entropy for the Analysis of Cosmological Data

Along recent decades, an intensive worldwide research activity is focusing both black holes and cosmos (e.g. the dark-energy phenomenon) on the basis of entropic approaches. The Boltzmann-Gibbs-based Bekenstein-Hawking entropy $S_{BH}\propto A/l_P^2$ ($A \equiv$ area; $l_P \equiv$ Planck length) systematically plays a crucial theoretical role although it has a serious drawback, namely that it violates the thermodynamic extensivity of spatially-three-dimensional systems. Still, its intriguing area dependence points out the relevance of considering the form $W(N)\sim μ^{N^γ}\;\;(μ>1;γ>0)$, $W$ and $N$ respectively being the total number of microscopic possibilities and the number of components; $γ=1$ corresponds to standard Boltzmann-Gibbs (BG) statistical mechanics. For this $W(N)$ asymptotic behavior, we introduce here, on a group-theory basis, the entropic functional $S_{α,γ}=k \Bigl[ \frac{\ln Σ_{i=1}^W p_i^α}{1-α} \Bigr]^{\frac{1}γ} \;(α\in \mathbb{R};\,S_{1,1}=S_{BG}\equiv-k\sum_{i=1}^W p_i \ln p_i)$. This functional simultaneously is {\it extensive} (as required by thermodynamics) and {\it composable} (as required for logic consistency), $\forall (α,γ)$. We further show that $(α,γ)=(1,2/3)$ satisfactorily agrees with cosmological data measuring neutrinos, Big Bang nucleosynthesis and the relic abundance of cold dark matter particles, as well as dynamical and geometrical cosmological data sets.

cond-mat.stat-mech↗