Jul 2026· European journal of physics· Vol 47, pp. 045102· 0 citations· 15 references
Physics
Abstract
Phase transitions and constrained optimization are central to statistical mechanics, yet undergraduate instruction often relies on models that obscure their combinatorial origins or require advanced mathematical machinery. The random allocation model with a power-law weight w(m)=m−β offers a minimal, exactly solvable alternative. We derive the equilibrium occupation distribution using the most probable configuration method, explicitly tracking the emergence of the Lagrange multiplier as a conjugate variable to particle density. Analysis of the entropy density s(ρ) reveals a real-space condensation transition at a finite critical density ρc(β) for β>2. The transition is third-order in the Ehrenfest sense for 23, with the exponent β playing a role mathematically analogous to spatial dimensionality in the ideal Bose gas. The derivation isolates the statistical core of condensation, providing a transparent training ground for the saddle-point method without the overhead of kinetic energy or temperature. Furthermore, the model resolves a persistent pedagogical challenge by explicitly distinguishing the concavity of the extensive entropy S(M) from the convexity of the intensive entropy density s(ρ) in the condensed phase. Because the analysis requires only elementary combinatorics and Stirling’s approximation, the system serves as a tractable example of higher-order transitions that can be integrated directly into advanced undergraduate or early graduate statistical mechanics courses.
We develop a consistent pseudo-analytic framework based on $g$-calculus for constructing deformed statistical distributions. By mapping standard algebraic operations through a monotone generator function, we systematically derive the associated pseudo-logarithmic and pseudo-exponential structures. Applying this formalism, we introduce a new family of pseudo-distributions that generalizes nonextensive statistical mechanics at both the probability density and cumulative distribution levels, recovering classical and standard nonextensive statistics as limiting cases. We investigate the thermodynamic geometry of the proposed models using the Ruppeiner metric on the equilibrium manifold. A perturbative analysis around the classical limit reveals that, to leading order, the thermodynamic scalar curvature is governed solely by the generator deformation parameter, while the nonextensivity parameter remains decoupled. To evaluate the empirical robustness of the framework, we apply the model to analyze the absolute deviations of daily West Texas Intermediate crude oil prices from their hundred-day moving average. Model comparison based on information criteria demonstrates that the proposed pseudo-distributions provide a superior description of these high-frequency financial fluctuations and their heavy-tailed characteristics compared to standard benchmarks. These results suggest that $g$-calculus offers a flexible and physically grounded mathematical tool for generating deformed statistics and analyzing their geometric properties.
Negin Bayrami, H. Mohammadzadeh, H. Agahi et al.· 0 citations
In this work, we compare three formulations of thermodynamic entropy for a non-interacting bosonic gas: (i) the grand-canonical Bose–Einstein entropy, (ii) the finite-N canonical entropy obtained from the exact partition function (Ziff, Uhlenbeck, and Kac construction), and (iii) the entropy of a multinomial distribution with Boltzmann categorical probabilities and temperature determined from Clausius’ equation. It is well-known that the grand-canonical Bose–Einstein systematically overestimates entropy of canonical systems in regimes where particle-number fluctuations are significant. The exact canonical entropy correctly enforces the particle-number constraint, but recent experimental results suggest that it also overestimates particle-number fluctuations below the crossover temperature. The multinomial distribution is less common in thermodynamics. It addresses in a mathematically exact way the puzzle of the famous −log(N!) term introduced by Gibbs as a deus ex machina in discussions of thermodynamic entropy. One remarkable consequence is that the entropy of the multinomial distribution overcomes the issue of negative entropy at low temperature that affects the Gibbs and the Sackur–Tetrode entropies at low temperature. The analysis presented in the paper shows that the multinomial distribution, equipped with a categorical distribution calibrated in such a way that the resulting multinomial entropy fits Clausius’ equation, provides accurate approximations to the canonical entropy in the classical regime, while it is smaller than the canonical entropy below the crossover temperature. One feature of the multinomial distribution is that it predicts lower peak variance of the number of particles in the ground state than the canonical distribution. This is in agreement with recent experimental results; hence, this paper identifies the thermodynamically calibrated multinomial distribution as a candidate alternative to the canonical distribution for thermodynamic bosonic entropy in finite systems.
The maximization of statistical complexity has long been associated with the emergence of probability distributions lying between perfect order and complete disorder. While previous studies have shown that complexity-maximizing distributions exhibit a two-level structure in finite discrete systems, an analogous unified treatment for both discrete and continuous probability spaces has remained unavailable. In this work, we develop a general variational framework for a generalized statistical complexity constructed from Shannon and Renyi entropies. We derive a common stationary equation governing both discrete probability masses and continuous probability densities and prove that every stationary solution necessarily possesses exactly two probability levels, establishing a universal core-halo structure. We further demonstrate that the optimization problem reduces to a single multiplicity parameter and prove that the global complexity maximum is attained by the smallest admissible core, corresponding to a single dominant state in the discrete case and an infinitesimal core in the continuous limit. These results provide a complete analytical characterization of the complexity-maximizing distributions and reveal a common mathematical structure underlying complexity optimization in both discrete and continuous settings. The framework establishes a unified foundation for generalized statistical complexity with potential applications in statistical mechanics, information theory, and the analysis of complex systems.
The partition function is statistical mechanics'answer to an exponentially large spectrum, distilling it into a single analytic object whose temperature dependence resolves the full structure of the underlying ensemble. We show that magic, the resource separating universal quantum computation from classically simulable stabilizer dynamics, admits precisely such a description. Mapping the Pauli spectrum of a quantum state onto the energy levels of a fictitious many-body system, the Pauli gas, we construct its canonical partition function, the stabilizer partition function, and from its associated free energy a magic monotone that we call the stabilizer work. Both these objects are analytic functions of an inverse-temperature-like parameter and are efficiently estimable via Bell sampling. The framework is analytically tractable. We derive exact ensemble-averaged partition functions for Haar-random, $\nu$-compressible, and pseudomagic states, together with concentration guarantees. We show that for every value of its parameter, the stabilizer work is a faithful, Subadditive magic monotone, while remaining efficiently accessible on quantum hardware and admitting an operational interpretation. Unlike measures that probe a single moment of the Pauli distribution, the stabilizer work is intrinsically moment-generating. As the temperature is tuned from high to low, it interpolates continuously between the stabilizer 2-R\'enyi entropy and the stabilizer nullity, revealing two previously disconnected monotones as limiting cases of a single object. We demonstrate the framework on low-rank stabilizer simulation, resource interconversion, molecular ground states, and quantum many-body systems. A thermodynamics of magic is therefore not merely an analogy but a working toolkit, opening a statistical-mechanical route to magic properties of quantum systems that no single measure can access.
William E. Salazar, G. Saxena, Jack S Baker et al.· 0 citations
Geometric methods are indispensable for analyzing, predicting, and mitigating the complex behaviors inherent in nonlinear systems. In this regime, the most probable transition path minimizes the Onsager-Machlup action functional, marking the likeliest route across an energy barrier. Lifting the analysis from individual sample paths to the infinite-dimensional space of probability densities, the book recasts these transitions as Schr\"odinger bridges - optimal paths between boundary distributions defined by minimizing relative entropy - and shows that the Onsager-Machlup path emerges as a special case when metastable states are idealized as Dirac masses. Generalizing further through {\alpha}-divergences, which connect to entropies and the thermodynamic cost of nonequilibrium transitions, it introduces information geodesics as the resulting optimal density paths, offering a unified geometric account of how complex systems move between metastable regimes under uncertainty.
Jinqiao Duan, Ting Gao, Qiao Huang et al.· 0 citations
We introduce a unified entropic framework for real fluids that encompasses the van der Waals, Berthelot, Redlich Kwong, and Peng Robinson equations of state within a common thermodynamic description. The corresponding microscopic interactions are then explored using Geometrothermodynamics, GTD, through the scalar curvature $mathcal{R}$ of the equilibrium manifold. We show that curvature singularities accurately reproduce macroscopic critical phenomena, while vanishing curvature $\mathcal{R}=0$ identifies specific thermodynamic states where attractive and repulsive intermolecular forces effectively balance. Furthermore, we introduce a set of dimensionless critical-amplitude ratios $Q^i_{j}$, which reveal universal geometric features of the critical regime. Although individual critical amplitudes exhibit a logarithmic dependence on the system size, these invariant ratios organize different molecular species according to the strength of criticality and encode universal scaling features, suggesting their potential as robust classification parameters. Finally, employing Bayesian inference and Markov Chain Monte Carlo, MCMC methods, we statistically reconstruct the zero-curvature curves. The posterior distributions support the consistency of the geometric scaling behavior, demonstrating that the GTD manifold encodes non-trivial information about the underlying thermodynamical models.
Carlos E. Romero-Figueroa, J. M. Ladino, S. A. Zaldivar et al.· 0 citations