Thermodynamic state variables from a minimal set of quantum constituents

TL;DR

This work shows that macroscopic equilibrium variables $P$, $T$, and $S$ can be derived from the spectral and eigenvector structure of minimal quantum systems in two dimensions, via chaotic single-particle dynamics and coupled two-particle chaos. Pressure is obtained from the Hellmann-Feynman relation applied to energy-level shifts with boundary parameters, while heat, entropy, and temperature emerge from equilibration of interacting subsystems and diagonal-approximation of reduced states. The results reproduce Boyle-Mariotte-like behavior and demonstrate a microscopic underpinning of the first and second laws, with clear definitions of work $\delta W = -P dA$ and heat $\delta Q = T dS$ in this setting. The approach provides a transparent realization of the eigenstate thermalization hypothesis and highlights how thermodynamics can be inferred directly from quantum spectral data and unitary dynamics in a minimal, solvable model.

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''.

Figures (4)

• Figure 1: Test of Boyle-Mariotte's law, for a single particle moving freely in the elementary domain of a Sinai billiard, with side lengths $L_x = 1.09\,\mathcal{L}$, $L_y = 1.00\,\mathcal{L}$, quarter circle radius $R = 0.5\,\mathcal{L}$, and resulting volume $A$. To verify the isotropy of pressure $P$, the latter is extracted from individual eigenenergies of the billiard via (\ref{['HF']}), with $\lambda=L_x$ ($L_y$), to obtain $P_x$ ($P_y$). Linear fits $f_{P_{x,y}A}(E) = a_{x,y}E + b_{x,y}$ to the products $P_{x,y}A$ for the 800 lowest eigenenergies of the billiard, to verify Boyle-Mariotte's law in (a) (and the inset), give $a_x = 1.010$, $b_x\mathcal{L}^2 = -0.016\times10^3$, $a_y = 1.008$, $b_y\mathcal{L}^2 = -0.010\times10^3$, in very good agreement with the expected $a=1$, $b\mathcal{L}^2=0$. (b) Relative fluctuations $\Delta P_xA = |P_x(E)A - f_{P_xA}(E)|/|f_{P_xA}(E)|$ of $P_xA$, averaged over 25 subsequent eigenstates for each point, as a function of energy, for three grid sizes $N$ of the triangulation employed in the finite element diagonalization routine brugger2024wulf2024kantz2025weber2025holm2025. Apart from the low energy range, where a sudden decrease of $\Delta P_x A$ with $N$ indicates a lack of numerical convergence for too coarse a discretization, $\Delta P_x A$ smoothly decreases with energy, as to be expected, for decreasing de Broglie wavelengths. The results in (a) were obtained for $N=100$, $2N$ grid points along the Sinai billiard's quarter-circle, a length variation $\delta L_{x,y} = 0.01\,\mathcal{L}$, and third order polynomial interpolation between the resulting eigenvalues, to extract the $P_{x,y}$ from the energy levels' derivatives. Analogous results of comparable quality are obtained for different ratios $L_x/L_y$ of the billiard's side lengths kantz2025.
• Figure 2: Two particles of identical mass, $m\equiv 1$, placed into two adjacent 2D boxes separated by an immobile wall of finite width $b=0.001\,\mathcal{L}$, with fixed side lengths $L_x^{(\ell)} = 1.1\,\mathcal{L}$, $L_x^{(r)} = 1.3\,\mathcal{L}$, $L_y=1.4\,\mathcal{L}$ , where $L_x^{(\ell)}\ne L_x^{(r)}$ to reduce symmetries, and perfectly reflecting boundary conditions are imposed. The particles, initially prepared in a product state $\ket{\Psi_0}=\ket{\epsilon^{(\ell )}_{\rm m_0},\epsilon^{({r})}_{\rm n_0}}$ of uncoupled single-particle energy eigenstates of the left and right box, respectively, exchange energy via the Coulomb interaction $V_\text{int}$, leading to time dependent local energy expectation values $\langle E^{(\ell ,{r})}(t)\rangle$.
• Figure 3: Time evolution of the energy expectation values $\langle E^{(\ell ,{r})}(t)\rangle$, for initial states $\ket{\Psi_0}=\ket{\epsilon^{(\ell )}_{\rm m_0},\epsilon^{({r})}_{\rm n_0}}$ with initial local quantum numbers (a) $\rm m_0^{(\ell)} \equiv (2,4)$, $\mathrm{n}_0^{(r)} \equiv (1,1)$, (b) $\rm m_0^{(\ell)} \equiv (1,3)$, $\mathrm{n}_0^{(r)} \equiv (4,1)$, together with the (conserved) total energy obtained as the sum of $\langle E^{(\ell, {r})}(t)\rangle$ and $\langle V_\text{int}(t)\rangle$, for $k\mathcal{L}=-50$. (c) Distribution of the energy balance ratios $\ln(E_j^{(\ell)} / E_j^{(r)})$ for the first (energy-ordered) 1000 eigenstates, for vanishing interaction strength $k\mathcal{L}=0$, and for $k\mathcal{L}=-50$. The insets in (a,b) show the equivalent distributions which only include those (eleven (a) and five (b)) eigenstates $\ket{E_j}$ with which the respective initial state $\ket{\Psi_0}$ has an overlap $|\langle\Psi_0|E_j\rangle|^2 \geq 2\%$. The arrows in (c), in turn, indicate the energy balance ratios of those single $\ket{E_j}$ which have the largest overlap $|\langle\Psi_0|E_j\rangle|$ with the initial states evolved in (a,b). The system side lengths are, as defined before, $L_x^{(\ell)} = 1.1\,\mathcal{L}$, $L_x^{(r)} = 1.3\,\mathcal{L}$, $L_y=1.4\,\mathcal{L}$. The energy scales here covered are small compared to the ones in Fig. \ref{['fig-Boyle']}, since the numerically accessible energy range for two particles is considerably smaller than that for one particle.
• Figure 4: Absolute, $\Delta T_\text{abs}$, and relative, $\Delta T_\text{rel}$, offsets between the right and the left particle's equilibrium temperatures $T^{(\ell, r)}$ (with $k_B \equiv 1$), as inferred from the energy derivatives of $S^{(\ell,{r})}$, for initial states $\ket{\Psi_0}=\ket{\epsilon^{(\ell )}_{\rm m_0},\epsilon^{({r})}_{\rm n_0}}$ selected by two criteria: (i) Large initial energy mismatch, i.e. $\epsilon_{\text{m}_0}^{(\ell)} \gg \epsilon_{\text{n}_0}^{(r)}$ (or vice versa), and (ii) small final energy mismatch, i.e. $\overline{E^{(\ell)}} \approx \overline{E^{(r)}}$. $\Delta T_\text{abs}$ and $\Delta T_\text{rel}$ initially clearly decrease with increasing energy $E = \epsilon_{\text{m}_0}^{(\ell)} + \epsilon_{\text{n}_0}^{(r)}$, as expected as one approaches the short wavelength limit. The saturation for $E\mathcal{L}^2\geq 150$ is attributed to the finite numerical error of the eigenenergies holm2025.