Fisher Information Reveals the Quantum Roots of Adiabatic Behavior

Few equations in physics are as familiar to students as the adiabatic law of an ideal gas, the statement that pressure times volume raised to a fixed exponent remains constant during a compression or expansion in which no heat flows. It is taught as a straightforward consequence of the first law of thermodynamics, and for more than a century it has been treated as a phenomenological rule, a compact piece of bookkeeping that connects macroscopic variables without saying much about what is happening at the microscopic level. A new theoretical study published in Foundations of Physics argues that this familiar relationship deserves a far deeper interpretation. According to A. Plastino of the Institute of Physics La Plata in Argentina and F. Pennini of the Universidad Católica del Norte in Chile and the Universidad Nacional de Mar del Plata in Argentina, the adiabatic exponent that appears in the law can be understood as a measure of the relative stiffness of energy and volume fluctuations, and the adiabatic law itself emerges as a geometric constraint, a geodesic, in the space of thermodynamic states.

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.