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arXiv:2608.11104

Exact Expressions of Entropy for Classical Non-interacting Many-body Systems

Yuheng Wu, Henrik J. Heelweg, Rigel Galgana

Published Aug 11, 2026• 0 citations
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Abstract

In the thermodynamic limit, the equilibrium state of a many-body system can be characterized by three pairs of conjugate thermodynamic variables: $E/T,V/P,N/μ$. In this limit, the thermodynamic properties in all ensembles are equivalent up to the leading order of $E,V,N$. However, for systems of finite size, this ensemble equivalence is no longer exact, and the thermodynamic properties may differ substantially among ensembles. To quantify these finite-size effects rigorously, it is desirable to develop a universal ensemble theory applicable to systems of arbitrary size, providing exact expressions for entropy and, thereby, giving rise to the precise value of all equilibrium thermodynamic quantities. In this work, we propose a theory that determines the exact entropy expressions for classical non-interacting many-body systems of arbitrary size across all statistical ensembles, based on only two postulates: \textbf{stationarity}, requiring that the physical laws be invariant under time translation, and \textbf{unbiasedness}, requiring that the equilibrium mixed state maximize the entropy subject to the prescribed constraints. Moreover, we show that the entropy expressions obtained in different ensembles converge to the common asymptotic form $S \asymp \ln\!\left( \left( \frac{4πm e E}{3N} \right)^{3N/2} \cdot \left(\frac{V}{N}\right)^N \right)+N$, consistent with the predictions of the large deviation theory.

DOI
10.48550/arxiv.2608.11104

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In the thermodynamic limit, the equilibrium state of a many-body system can be characterized by three pairs of conjugate thermodynamic variables: E/T,V/P,N/μE/T,V/P,N/μE/T,V/P,N/μ. In this limit, the thermodynamic properties in all ensembles are equivalent up to the leading order of E,V,NE,V,NE,V,N. However, for systems of finite size, this ensemble equivalence is no longer exact, and the thermodynamic properties may differ substantially among ensembles. To quantify these finite-size effects rigorously, it is desirable to develop a universal ensemble theory applicable to systems of arbitrary size, providing exact expressions for entropy and, thereby, giving rise to the precise value of all equilibrium thermodynamic quantities. In this work, we propose a theory that determines the exact entropy expressions for classical non-interacting many-body systems of arbitrary size across all statistical ensembles, based on only two postulates: \textbf{stationarity}, requiring that the physical laws be invariant under time translation, and \textbf{unbiasedness}, requiring that the equilibrium mixed state maximize the entropy subject to the prescribed constraints. Moreover, we show that the entropy expressions obtained in different ensembles converge to the common asymptotic form S≍ln⁡ ⁣((4πmeE3N)3N/2⋅(VN)N)+NS \asymp \ln\!\left( \left( \frac{4πm e E}{3N} \right)^{3N/2} \cdot \left(\frac{V}{N}\right)^N \right)+NS≍ln((3N4πmeE​)3N/2⋅(NV​)N)+N, consistent with the predictions of the large deviation theory.
Thermodynamics is commonly presented as a theory of macroscopic systems in stable equilibrium, built upon assumptions of extensivity and scaling with system size. In this paper, we present a universal formulation of the elementary foundations of thermodynamics, in which entropy and energy are defined and employed beyond equilibrium and without assuming extensivity. The formulation applies to all systems -- large and small, with many or few particles -- and to all states, whether equilibrium or nonequilibrium, by relying on carefully stated operational definitions and existence principles rather than macroscopic idealizations. Key thermodynamic concepts, including adiabatic availability and available energy, are developed and illustrated using the energy-entropy diagram representation of nonequilibrium states, which provides geometric insight into irreversibility and the limits of work extraction for systems of any size. A substantial part of the paper is devoted to the analysis of entropy transfer in non-work interactions, leading to precise definitions of heat interactions and heat-and-diffusion interactions of central importance in mesoscopic continuum theories of nonequilibrium behavior in simple and complex solids and fluids. As a direct consequence of this analysis, Clausius inequalities and the Clausius statement of the second law are derived in forms explicitly extended to nonequilibrium processes. The resulting framework presents thermodynamics as a universal theory whose concepts apply uniformly to all systems, large and small, and provides a coherent foundation for both teaching and modern applications.
A theory of non-equilibrium thermodynamics is developed. The general method to derive the closed set of kinetic equations for expectation values of physical quantities is developed, by making use of the functional Taylor expansion with respect to expectation values of physical quantities around the reference state. For this purpose, a definition of the thermodynamic state in the system whose state changes with time is presented in order to describe the general state of the many-body system from the unified point of view. The statistical mechanical probability distribution functions for the quantum-mechanical expectation values at the stationary state are developed, without making use of the concept of the constant of the motion. The relation between the expectation value and the most probable value for the probability distribution function is discussed, and it is found that both values are equivalent to each other, if the system is extended infinitely in space and time. Thus the condition for the thermodynamic limit may be defined in a generalized way. Our method is looked as a perturbational one in the equilibrium statistical mechanics, though the choice of the reference state is arbitrary.
For a small system at a low temperature, thermal fluctuation and quantum effect play important roles in quantum thermodynamics. Starting from micro-canonical ensemble, we generalize the Boltzmann–Gibbs statistical factor from infinite to finite systems, no matter the interactions between particles are considered or not. This generalized factor, similar to Tsallis’s q-form as a power-law distribution, has the restriction of finite energy spectrum and includes the nonextensivities of the small systems. We derive the exact expression for distribution of average particle numbers in the interacting classical and quantum nonextensive systems within a generalized canonical ensemble. This expression in the almost independent or elementary excitation quantum finite systems is similar to the corresponding ones obtained from the conventional grand-canonical ensemble. In the reconstruction for the statistical theory of the small systems, we present the entropy of the equilibrium systems and equation of total thermal energy. When we investigate the thermodynamics for the interacting nonextensive systems, we obtain the system–bath heat exchange and “uncompensated heat” which are in the thermodynamical level and independent on the detail of the system–bath coupling. For ideal finite systems, with different traps and boundary conditions, we calculate some thermodynamic quantities, such as the specific heat, entropy, and equation of state, etc. Particularly at low temperatures for the small systems, we predict some novel behaviors in the quantum thermodynamics, including internal entropy production, heat exchanges between the system and its surroundings and finite-size effects on the free energy.
Thermalization conventionally describes local properties of isolated many-body systems at equilibrium. Magic, or nonstabilizerness $\unicode{x2013}$ the resource enabling universal quantum computation $\unicode{x2013}$ is by contrast encoded in the global structure of the many-body wavefunction. We show that, despite its global nature, the magic of equilibrium pure states of chaotic many-body systems, including late-time evolved states and energy eigenstates, is universal: it is captured by the thermal Scrooge ensemble, the minimally informative ensemble of pure states consistent with the Gibbs state at the same effective temperature. Therefore, for systems with no conserved quantities other than the total energy, equilibrium magic is a function of temperature alone, independent of the initial state and other microscopic features of the equilibrium state. This yields concrete universal predictions for the stabilizer Rényi entropies (SREs). At infinite temperature, the SRE is set by Haar-like fluctuations of the Pauli spectrum, while at finite temperature energy conservation induces a volume-law thermodynamic correction controlled by the thermal Pauli spectrum. We support these predictions with analytical arguments and extensive numerical simulations. We further show that chaotic many-body systems at high temperatures possess long-range magic and entanglement that cannot be removed by finite-depth local quantum circuits. Our results establish magic as a thermodynamic property of chaotic many-body systems and suggest that Scrooge ensembles may provide a unified framework for quantum many-body resources.
The infinite-volume limit of thermodynamic functions calculated in the quantum microcanonical ensemble is shown to exist for a fairly wide class of spin systems and quantum gases. The entropy is set equal to the logarithm of the number of eigenstates in an energy interval which increases linearly with the size of the system, but is otherwise arbitrary. The limiting entropy per unit volume agrees with that calculated in the canonical formalism, and possesses certain convexity properties required for thermodynamic stability. A precise criterion, in terms of the energy spectra of large systems, is given for determining the limit of the thermodynamic entropy as the temperature approaches zero. This is not determined by the degeneracy of the ground state, contrary to the discussion of the ``third law of thermodynamics'' found in some textbooks.
Quantum many-body states that frequently appear in physics often obey an entropy scaling law, meaning that an entanglement entropy of a subsystem can be expressed as a sum of terms that scale linearly with its volume and area, plus a correction term that is independent of its size.We conjecture that these states have an efficient dual description in terms of a set of marginal density matrices on bounded regions, obeying the same entropy scaling law locally.We prove a restricted version of this conjecture for translationally invariant systems in two spatial dimensions.Specifically, we prove that a translationally invariant marginal obeying three nonlinear constraints-all of which follow from the entropy scaling law straightforwardlymust be consistent with some global state on an infinite lattice.Moreover, we derive a closed-form expression for the maximum-entropy density compatible with those marginals, deriving a variational upper bound on the thermodynamic free energy.The main assumptions of our construction are satisfied exactly by solvable models of topological order and approximately by finite-temperature Gibbs states of certain quantum spin Hamiltonians.