The work, which appeared as Volume 55, article number 86 of the journal in November 2025, situates itself within a tradition that stretches back to Ronald Fisher’s foundational 1922 paper on the mathematical foundations of theoretical statistics. Fisher information, a quantity that measures how much information a set of measurements carries about an underlying parameter, has long been known to play a dual role in physics. In thermodynamic fluctuation theory, as developed by George Ruppeiner in his influential 1995 review in Reviews of Modern Physics, the Fisher metric acts as a Riemannian metric on the space of equilibrium states: the squared distance between two nearby states is proportional to the probability of the fluctuations connecting them. What the new paper adds is the recognition that this fluctuation geometry, when applied carefully to a gas undergoing adiabatic change, contains the classical adiabatic law in its structure.

The central technical claim is a reinterpretation of the adiabatic exponent gamma, the familiar quantity that equals the ratio of the heat capacity at constant pressure to the heat capacity at constant volume, and takes the value 5/3 for a monatomic ideal gas. In the conventional textbook treatment, gamma is introduced through the thermodynamic identities governing reversible processes. Plastino and Pennini show instead that gamma admits a new interpretation as quantifying the relative stiffness of energy and volume fluctuations. Stiffness here has a precise statistical meaning. Around any equilibrium state, the energy of a system fluctuates with a variance controlled by the heat capacity, and the volume of a gas confined by a piston fluctuates with a variance controlled by its compressibility. The exponent gamma, in the Fisher-information picture, measures the ratio of these two fluctuation scales, expressing how reluctantly the system exchanges energy with its microscopic degrees of freedom compared with how reluctantly it changes its geometry.

This reading transforms the status of the adiabatic law. If gamma measures a ratio of fluctuation stiffnesses, then the condition PV^gamma equals a constant is no longer merely a rule about heat flow; it becomes a geodesic constraint in thermodynamic state space. In the language of differential geometry, a geodesic is the shortest, straightest path between two points on a curved surface, the path a freely moving particle would trace. Plastino and Pennini demonstrate that the adiabatic trajectories of an ideal gas are precisely the geodesics of the Fisher-geometric metric that describes thermodynamic fluctuations. The law that students memorize as an algebraic identity is, on this view, the signature of a straight line in the information geometry of equilibrium fluctuations. A quasi-static adiabatic process is the process that moves through thermodynamic state space most economically, without any excess fluctuation cost.

The connection between Fisher information and fluctuations is not new in itself, and the authors are careful to build on established results rather than claim wholesale novelty. Ruppeiner’s thermodynamic fluctuation theory, the geometrical treatment of statistical mechanics developed by David Brody and Nicolas Rivier, and the general framework of information geometry codified by Shun-ichi Amari and Hiroshi Nagaoka all provide the backdrop. Plastino’s group has contributed extensively to this literature over the past decade, including work on the Hellmann-Feynman connection for relative Fisher information, the Fisher thermodynamics of quasi-probabilities, and the scaling symmetries of Fisher information with S. P. Flego and A. R. Plastino. What distinguishes the new paper is the specific bridge it constructs between this fluctuation geometry and one of the oldest dynamical invariants in classical thermodynamics.

The authors also draw on a smaller but persistent line of inquiry concerning quantum origins of the adiabatic law. Work by T. Yarman and collaborators, published in the International Journal of Physical Sciences and later in Results in Physics, argued that the constancy expressed in the adiabatic gas law is ingrained within quantum mechanics, and that the second law of thermodynamics itself can be seen as a consequence of quantum mechanical structure. The new Fisher-information analysis provides a complementary and, the authors suggest, more general route to a similar conclusion. Rather than deriving the adiabatic constancy from the quantum mechanics of a specific gas model, the information-geometric approach shows how the law emerges from the statistical structure of fluctuations themselves, whatever their ultimate microscopic origin.

The practical implications of this reframing extend into several active research areas. In finite-time thermodynamics, researchers study how much extra work must be dissipated when a thermodynamic process is carried out in finite time rather than quasistatically. Seminal contributions by Peter Salamon and R. Stephen Berry in 1983, and later by T. Schmiedl and U. Seifert and by D. A. Sivak and G. E. Crooks, established that the dissipated availability along a finite-time protocol is proportional to a thermodynamic length computed from a fluctuation metric, and that optimal protocols are geodesics of that metric. The new result places the adiabatic law squarely within this framework, suggesting that the same geometry that governs optimal finite-time control also governs the idealized adiabatic paths of classical theory. This opens a pathway toward designing protocols for cold-atom platforms and other highly controllable quantum systems where the stiffness of fluctuations can be measured and manipulated.

Quantum metrology provides another natural arena of application. Fisher information is the central quantity of estimation theory: the Cramér-Rao bound, formalized by C. R. Rao, states that the variance of any unbiased estimator of a parameter is bounded below by the inverse of the Fisher information. In quantum systems, the quantum Fisher information sets the ultimate precision of measurements of temperature, phase, or field strength, and recent work on critical quantum metrology and coherence-enhanced thermometry has exploited near-critical fluctuations to sharpen estimates. The identification of the adiabatic exponent as a Fisher-geometric quantity suggests that thermodynamic processes themselves could be characterized, and optimized, by the estimation-theoretic properties of the fluctuations they carry. A quantum heat engine driven along its adiabatic geodesic would, in this picture, be an engine whose strokes minimize the information cost of moving between thermodynamic states.

The work is also part of a broader intellectual program that the authors describe as a step toward unifying epistemic and ontic perspectives on thermodynamic order. The epistemic view treats thermodynamic quantities as statements about knowledge and information, encoded in probability distributions over microstates. The ontic view treats them as objective features of the physical world, independent of any observer. Fisher information sits naturally at the boundary: it is defined epistemically, through the sensitivity of a probability distribution to parameter changes, yet the new result shows it fixing the form of an objective dynamical invariant, the adiabatic law, that governs the behavior of real gases. The suggestion that classical thermodynamic laws can be rederived as emergent signatures of Fisher-geometric structure points toward a deeper claim, namely that the laws of thermodynamics may be information-theoretic principles all the way down.

It should be emphasized that the analysis is theoretical and, as the authors state in their data availability section, no datasets were generated or analyzed in the study. The derivation concerns the equilibrium fluctuation geometry of gases, and extending the geodesic interpretation to non-ideal systems, to genuinely irreversible processes, and to strongly quantum regimes remains a task for future work. Nevertheless, the conceptual payoff is substantial. A law discovered in the nineteenth century, taught to every physics student as a piece of algebra, turns out to encode a statement about the geometry of fluctuations, a straight path through a curved space whose curvature is set by Fisher information. The research was partially supported by FONDECYT through grant 1251928, and the authors declare no competing interests. If the Fisher-geometric program continues to bear fruit, the adiabatic law may come to be seen not as an isolated rule but as the first recognizable landmark in an information-theoretic map of thermodynamic reality.

At the core of this investigation lies the Bardeen AdS black hole, a theoretical construct that diverges significantly from the standard Schwarzschild black hole by incorporating a magnetic charge, thereby presenting a more realistic and feature-rich model. This magnetic charge endows the Bardeen black hole with a unique characteristic: it possesses a finite size rather than a singularity at its center, a feature that aligns better with quantum mechanical intuitions about the fundamental discreteness of nature. The anti-de Sitter background, a spacetime with a uniform negative curvature, further complicates the picture, introducing cosmological effects that are crucial for understanding the ultimate fate and stability of such objects within a larger, expanding universe. The interplay between the magnetic charge and the AdS curvature creates a thermodynamic landscape that is far richer and more nuanced than that of simpler black hole solutions. Understanding this landscape is paramount, as it governs how these black holes form, evolve, and interact with their surroundings, and how they might eventually evaporate or merge. The research meticulously analyzes these factors to ascertain the conditions under which the Bardeen AdS black hole remains a stable entity in the grand cosmic ballet.

The true innovation of Gogoi’s research, however, resides in its application of Kaniadakis statistics. This novel statistical framework, distinct from the classical Boltzmann-Gibbs and the quantum Fermi-Dirac and Bose-Einstein statistics, offers a generalized approach to describing systems with long-range interactions and non-extensive properties. Its unique mathematical structure, rooted in a parameter known as the Kaniadakis index, allows for a more flexible description of complex phenomena where correlations between particles or thermodynamic properties are significant. In the context of black holes, which are inherently macroscopic objects influenced by gravity’s pervasive reach, Kaniadakis statistics provides a powerful tool to analyze their thermodynamic behavior. This approach allows researchers to explore regimes of thermodynamic stability and phase transitions that might be overlooked or misrepresented by traditional statistical methods, thereby unlocking deeper insights into the microphysical underpinnings of black hole thermodynamics. The choice of Kaniadakis statistics is not arbitrary; it is a deliberate move to capture the inherent non-extensivity of gravitational systems.

Thermodynamic stability is a critical concept for black holes, dictating whether they can exist as long-lived, coherent structures or are prone to violent fluctuations and disintegration. Gogoi’s work meticulously examines the thermodynamic potential and its derivatives for the Bardeen AdS black hole under the Kaniadakis statistical framework. By analyzing these mathematical expressions, the researchers can identify specific ranges of parameters, such as the black hole’s mass and its magnetic charge, within which the system exhibits stable thermodynamic equilibrium. Unstable regions, conversely, indicate conditions where the black hole might undergo phase transitions or even evaporate. This investigation sheds light on the precise conditions required for the formation and persistence of these astronomical enigmas, offering clues about their prevalence and behavior in different cosmic epochs. The findings suggest that the Bardeen AdS black hole, when viewed through the lens of Kaniadakis statistics, exhibits a robust stability profile across a significant range of conditions, which implies their potential widespread existence throughout the universe, contributing to the overall structure and evolution of cosmic systems.

The concept of geometric thermodynamics introduces a fascinating duality, treating thermodynamic properties as intrinsic features of the spacetime geometry itself. This perspective, pioneered by researchers like Ruppeiner, views thermodynamic variables as coordinates on a manifold whose curvature is directly related to the thermodynamic stability of the system. In this study, Gogoi applies this geometric approach to the Bardeen AdS black hole in the Kaniadakis statistical setting. By constructing the relevant thermodynamic manifold and calculating its curvature invariants, the researchers can derive information about the system’s thermodynamic behavior. Positive curvature, for instance, typically signifies stability, while negative curvature can indicate instability or phase transitions. This geometric interpretation provides a powerful visual and conceptual tool for understanding the complex thermodynamic landscape of black holes, transforming abstract thermodynamic quantities into tangible geometric properties of spacetime. This elegantly bridges the gap between the microscopic statistical behavior and the macroscopic geometric manifestation of these cosmic phenomena.

The Kaniadakis index, denoted by $K$, plays a pivotal role in this study, acting as a tunable parameter that governs the nature of the Kaniadakis statistics. As this index varies, the statistical behavior shifts, interpolating between different physical regimes. The research demonstrates how altering the Kaniadakis index influences the thermodynamic stability and phase transitions of the Bardeen AdS black hole. For specific values of $K$, the black hole may exhibit behaviors analogous to those described by Maxwell-Boltzmann statistics, while for other values, it can capture features associated with systems exhibiting strong correlations or non-additivity. This parametric dependence provides an extraordinary level of control and insight into the thermodynamic properties of black holes, suggesting that their behavior might be modulated by fundamental statistical properties of the underlying constituents of spacetime itself. The universality of these findings is immense, suggesting that this approach could be applicable to a much broader class of gravitational systems, including those at the earliest moments of the universe.

The study meticulously traces the behavior of the black hole’s heat capacity, a crucial indicator of thermodynamic stability. A positive heat capacity signifies that adding energy to the system leads to an increase in its temperature, a characteristic of stable equilibrium. Conversely, a negative heat capacity suggests instability, where adding energy causes a decrease in temperature, leading to runaway processes. Gogoi’s calculations reveal that the Bardeen AdS black hole, under Kaniadakis statistics, exhibits positive heat capacity over significant intervals of its thermodynamic parameter space, reinforcing its stability. The specific range of stability, however, is shown to be intricately dependent on the Kaniadakis index, meaning that the statistical underpinnings of the universe directly influence the survivability of these cosmic giants. Furthermore, the study identifies critical points where the heat capacity diverges or changes sign, marking the boundaries of phase transitions, much like water freezing or boiling. These critical points are of particular interest for understanding the rich thermodynamic phenomenology of black holes.

Phase transitions in black hole thermodynamics are analogous to phase transitions observed in ordinary matter, such as the boiling of water or the condensation of gases. For instance, the Hawking-Page phase transition, a well-known phenomenon where a black hole can transition into a heat bath of radiation, is intricately linked to thermodynamic stability. Gogoi’s research investigates the possibility of similar phase transitions for the Bardeen AdS black hole within the Kaniadakis statistical framework. The findings suggest that the nature and occurrence of these phase transitions are significantly influenced by the Kaniadakis index and the magnetic charge parameter. This offers a novel perspective on the dynamics of black holes, implying that their ability to transition between different thermodynamic states might be a function of fundamental statistical properties, rather than solely external environmental conditions. Such insights are crucial for understanding the formation of large-scale structures in the universe and the evolution of black holes over cosmic timescales.

The geometric thermodynamic curvature invariants provide a deeper understanding of the correlations between different thermodynamic quantities. For example, the Ruppeiner metric, a fundamental tool in geometric thermodynamics, encodes information about the fluctuations and correlations within a system. In this study, Gogoi calculates the curvature of the thermodynamic manifold for the Bardeen AdS black hole, and the results are shown to be dependent on the Kaniadakis index. This dependency implies that the intensity of correlations within the black hole system, as perceived through its thermodynamic properties, can be tuned by the fundamental statistical parameters of the universe. A highly curved manifold would indicate strong correlations and potential instabilities, whereas a flatter manifold suggests weaker correlations and a more stable system. This correlation-induced stability or instability has profound implications for our understanding of how matter behaves under extreme gravitational conditions.

The research also sheds light on the Hawking radiation process, the phenomenon by which black holes are predicted to emit thermal radiation and evaporate over extremely long timescales. The rate and characteristics of Hawking radiation are intimately linked to the thermodynamic properties and stability of the black hole. By analyzing the thermodynamic stability of the Bardeen AdS black hole using Kaniadakis statistics, Gogoi’s work indirectly provides insights into how Hawking radiation might proceed for these complex objects. The study suggests that the evaporation rate and the temperature of the emitted radiation could be modulated by the Kaniadakis index, implying that the very process of black hole decay might be influenced by the underlying statistical laws governing the universe. This opens up new avenues for testing theoretical models of black hole evaporation and potentially even searching for observational signatures of Kaniadakis statistics in astrophysical phenomena.

The concept of regularity in astrophysical objects is a departure from the classical singularities predicted by general relativity. Regular black holes, such as the Bardeen black hole, resolve these singularities by introducing modifications to the gravitational field at short distances. Gogoi’s study confirms the thermodynamic stability of this regular Bardeen AdS black hole using Kaniadakis statistics, further solidifying the theoretical underpinnings of these non-singular cosmic structures. The ability of such regular black holes to maintain thermodynamic equilibrium under a generalized statistical framework bolster their candidacy as more accurate representations of actual black holes observed in the universe, particularly those that might have formed in the early universe where quantum gravitational effects were dominant. This research adds significant weight to the ongoing debate about the true nature of black hole interiors and the potential non-existence of true singularities.

The implications of this research extend beyond the realm of black holes themselves, potentially impacting our understanding of quantum gravity and the very fabric of spacetime. Kaniadakis statistics, with its inherent flexibility and ability to describe non-extensive systems, might offer a vital bridge between the macroscopic world governed by general relativity and the microscopic quantum realm. Black holes, being objects of immense gravitational force and quantum significance, serve as perfect laboratories for testing such unified theories. The consistency of the Bardeen AdS black hole’s thermodynamic properties within this framework suggests that Kaniadakis statistics could be a fundamental aspect of quantum gravity, influencing how spacetime behaves at its most extreme. This could lead to a paradigm shift in theoretical physics, offering new avenues for reconciling the seemingly disparate theories of quantum mechanics and general relativity.

In essence, Gogoi’s investigation is a testament to the power of exploring exotic statistical frameworks to unravel the deepest mysteries of the cosmos. By applying Kaniadakis statistics to the Bardeen AdS black hole, the research unveils a universe where thermodynamic stability and geometric properties are intricately linked to fundamental statistical indices. This study not only deepens our understanding of black holes but also hints at a more sophisticated and interconnected universe than we currently perceive, where the rules of thermodynamics themselves might be more flexible and profound than previously imagined. The future of cosmology and theoretical physics is brimming with possibilities, and this research stands as a beacon, illuminating a path toward a more comprehensive understanding of the universe’s most enigmatic inhabitants and the fundamental laws that govern them. The ongoing quest for a unified theory of everything may well find crucial clues within the statistical nuances of cosmic phenomena like these.

Subject of Research: Thermodynamic stability and geometric thermodynamics of regular Bardeen AdS black holes using Kaniadakis statistics.

Article Title: Thermodynamic stability and geometric thermodynamics of regular Bardeen AdS black hole using Kaniadakis statistics.

Article References:
Gogoi, B.J. Thermodynamic stability and geometric thermodynamics of regular Bardeen AdS black hole using Kaniadakis statistics.
Eur. Phys. J. C 86, 95 (2026). https://doi.org/10.1140/epjc/s10052-026-15348-1

Image Credits: AI Generated

DOI: https://doi.org/10.1140/epjc/s10052-026-15348-1

Keywords**: Black Holes, Thermodynamics, Kaniadakis Statistics, Bardeen Black Hole, Anti-de Sitter Space, Geometric Thermodynamics, Stability, Phase Transitions, Quantum Gravity.