Thermodynamic state variables from a minimal set of quantum constituents

Abstract: We show how the macroscopic state variables pressure, entropy and temperature of equilibrium thermodynamics can be consistently derived from the (quantum) chaotic spectral structure of one or two particles in two-dimensional domains. This provides a definition of work and heat from first principles, a microscopic underpinning of the first and second law of thermodynamics, and a transparent illustration of the ``eigenstate thermalization hypothesis''.

• The paper discovers that equilibrium thermodynamic state variables like pressure, entropy, and temperature can consistently be defined from the spectral and eigenvector structure of minimal quantum systems with one or two particles confined to two-dimensional domains.
• Single-particle chaos is sufficient to define pressure by causing delocalization of eigenstates, which is extracted via the Hellmann–Feynman theorem and confirms the isotropy of pressure and Boyle–Mariotte's law.
• The introduction of two particles interacting via an attractive Coulomb force allows for the microsopic definition of temperature and entropy when energy redistribution occurs, thereby establishing a relationship between equilibrium state variables in a pure quantum context.

The paper establishes that equilibrium thermodynamic state variables—pressure, entropy, and temperature—can be consistently defined from the spectral and eigenvector structure of a minimal quantum system comprising one or two particles confined to two-dimensional domains. The authors demonstrate that single-particle quantum chaos suffices to define pressure, while interacting two-particle chaos enables a microscopic definition of heat flux, entropy, and temperature, thereby providing first-principles derivations of work and heat and an explicit illustration of the eigenstate thermalization hypothesis (ETH) (2602.03276).

Motivation and scope

Equilibrium thermodynamics rests on the principle of equal a priori probabilities, traditionally justified by collision-induced chaotic many-body dynamics. Existing approaches—quantum thermodynamics of open systems, or studies of thermalization in unitary many-body systems (including many-body localization and scars)—either presuppose an effective open-system description or do not fully connect to standard thermodynamic state variables. The present work addresses this gap by identifying the minimal ingredients required for thermodynamics to emerge: it shows that no macroscopic ensemble is needed, only (i) classically chaotic single-particle dynamics for pressure and (ii) chaotic two-particle interaction for heat exchange. All spectral data are generated by exact finite-element diagonalization, with $\hbar = m = k_B = 1$ and energies in units of $\mathcal{L}^{-2}$.

Pressure from single-particle chaos

A rectangular billiard is integrable; its momentum components along the walls are conserved, so the generalized force on the walls is not isotropic and cannot define a consistent pressure—a point at which the authors explicitly contradict standard textbook treatments. Replacing the rectangle by a Sinai billiard breaks integrability: classical ergodicity produces delocalized, random-plane-wave-superposition eigenstates, exactly the structure assumed by ETH.

Pressure is then extracted via the Hellmann–Feynman theorem,

$P = -\frac{\partial E(\lambda)}{\partial \lambda},$

where $\lambda$ parametrizes either a straight wall length or the quarter-circle radius. The resulting $P$ is independent of the choice of $\lambda$, up to residual fluctuations set by the particle's finite de Broglie wavelength. Fitting $P_{x,y}A$ versus $E$ over the 800 lowest eigenstates yields slopes $a_x = 1.010$ and $a_y = 1.008$ with intercepts consistent with zero, confirming both isotropy of pressure and Boyle–Mariotte's law ($\mathcal{L}^{-2}$0 in two dimensions) to within about 1%. The relative fluctuations decrease smoothly with energy as the semiclassical limit is approached. This result implies that molecular chaos, usually attributed to many-body collisions, can be replaced by single-particle chaotic boundary geometry.

Heat flux, entropy, and temperature from two-particle equilibration

To introduce temperature through irreversible energy exchange rather than through statistical mechanics alone, the model is extended to two identical particles in adjacent rectangular compartments separated by an immobile wall (precluding work), coupled by an attractive Coulomb interaction $\mathcal{L}^{-2}$1. Each particle is prepared in an eigenstate of its uncoupled local Hamiltonian, and the local energy expectation values $\mathcal{L}^{-2}$2 are monitored under unitary evolution.

For initial conditions associated with unstable classical phase-space regions, energy redistributes irreversibly on short time scales, with residual fluctuations smaller than the exchanged energy. For initial conditions seeded in regular phase-space domains, no equilibration occurs and the signal's frequency content is scarce. Crucially, this dynamical dichotomy is already encoded statically in the coupled eigenstates: the distribution of logarithmic energy balance ratios $\mathcal{L}^{-2}$3 over the lowest 1000 eigenstates concentrates symmetrically around zero for $\mathcal{L}^{-2}$4, in stark contrast to the uncoupled case, and the eigenstate dominating the equilibrating dynamics sits at the center of this distribution while that dominating the non-equilibrating case sits in its wings.

Heat is identified as $\mathcal{L}^{-2}$5, expressible as $\mathcal{L}^{-2}$6 in the diagonal approximation. Local von Neumann entropies $\mathcal{L}^{-2}$7 follow from the reduced density matrices, fitted to a logarithmic dependence on the equilibrium energies (as implied by equal a priori probabilities), and inverse temperatures are obtained as energy derivatives. The central consistency check—that the emergent equilibrium corresponds to equal temperatures $\mathcal{L}^{-2}$8—is confirmed: both absolute and relative temperature offsets decrease systematically with total energy, saturating for $\mathcal{L}^{-2}$9, which the authors attribute to numerical convergence limits rather than physics. This confirms that the second law's microscopic content here follows from constraint relaxation: switching on the interaction increases the number of compatible microstates at fixed total energy.

Limitations and open questions

Several caveats bear directly on the results. First, all conclusions rest on exact diagonalization of small systems; the numerically accessible energy range for two particles is considerably smaller than for one, and the saturation of the temperature-offset decay above $P = -\frac{\partial E(\lambda)}{\partial \lambda},$0 signals finite numerical error. Second, the residual fluctuations in $P = -\frac{\partial E(\lambda)}{\partial \lambda},$1 and the finite error margins in $P = -\frac{\partial E(\lambda)}{\partial \lambda},$2 are attributed to finite de Broglie wavelength effects whose scaling the authors explicitly defer to future scrutiny. Third, the Coulomb interaction yields a classically mixed phase space, so equilibration is demonstrated only for initial conditions in the chaotic component; a complete account of how regular islands affect state-variable definitions remains open. Finally, the entropy fit assumes a logarithmic functional form inherited from the principle of equal a priori probabilities—an assumption imported rather than derived—and the extension beyond two particles and two dimensions is not addressed.

Conclusion

The paper demonstrates that pressure, entropy, and temperature emerge consistently from the spectral structure of at most two chaotic quantum constituents, yielding microscopic definitions of work ($P = -\frac{\partial E(\lambda)}{\partial \lambda},$3) and heat ($P = -\frac{\partial E(\lambda)}{\partial \lambda},$4) and securing the first and second laws without invoking ensembles or open-system coarse graining. The main open questions concern the quantitative scaling of finite-wavelength fluctuations, the role of mixed phase space, and whether the construction generalizes to larger particle numbers.