On the generic increase of entropy in isolated systems

TL;DR

This work identifies a universal mechanism for entropy production in isolated quantum systems driven by random-phase interactions. By developing a resolvent-based framework, it derives self-consistent equations for the overlap distribution p^{μ i}_n and demonstrates that steady-state entropy arises from two pathways: interaction-induced energy broadening and temporal coarse-graining over small energy gaps. The authors introduce a hierarchical set of ansätze—Lorentzian for the bulk, Gaussian for the tail, and a Voigt-like enhanced form—and validate them with large-scale numerical simulations of a nonintegrable Ising spin chain, showing logarithmic entropy scaling with interaction bandwidth. The framework unifies observational entropy with von Neumann entropy dynamics and provides predictive tools for entropy management in quantum technologies, while delineating the regime of validity set by the random-phase condition and potential deviations in integrable or MBL systems.

Abstract

This study establishes a universal mechanism for entropy production in isolated quantum systems governed by interactions that induce random-phase fluctuations. By developing a resolvent-based framework, we demonstrate that steady-state entropy generically arises from many-body interactions, independent of specific coupling details, provided the coherent accumulation of systematic biases does not overwhelm the random-phase fluctuations. Analytical arguments reveal that entropy generation is driven by two universal pathways: interaction-induced energy broadening and temporal coarse-graining over exponentially small energy gaps. To fully characterize the probability distribution, we introduce both Lorentzian and Gaussian ansatz, analyzing the bulk and tail behaviors respectively, and derive corresponding self-consistent equations for the distribution parameters. Numerical simulations of nonintegrable Ising spin chains confirm the predicted logarithmic entropy scaling and validate the self-consistent equations for energy shift and broadening parameters. By combining Lorentzian and Gaussian profiles into an enhanced ansatz, we further refine the description of the distribution, unifying observational entropy concepts with von Neumann entropy dynamics and providing predictive tools for thermodynamic behavior in quantum many-body systems. Our findings resolve longstanding debates about interaction-dependent entropy scaling and offer pathways for entropy control in quantum technologies.

Figures (9)

• Figure 1: (a) Probability distribution $p^{\mu i}_n$ ($i=1$) versus eigenstate index $n$ (sorted by energy) for the composite system. (b) Binned distribution $P(\mathcal{M}_{\lambda,\Delta})$ constructed by summing probabilities within energy intervals $\mathcal{M}_{\lambda,\Delta}$, where $\Delta = 0.5$, following Eq. \ref{['Lorentz']}. The binned probability is expressed as $P(\mathcal{M}_{\lambda,\Delta}) = \int_{\lambda-\Delta/2}^{\lambda+\Delta/2} d\lambda_m e^{S(\lambda_m)} p^{\mu i}(\lambda_m)$, with $\overline{\lambda}_n$ representing the average eigenenergy within each interval. Solid curves show Lorentzian fits to the binned distributions. Fitting parameters: For $\epsilon_\mu=6.324$, we obtain $a_{\mu i}+\Delta_{\mu i}=4.661$ and $\chi_{\mu i}=0.753$; for $\epsilon_\mu=0.583$, the parameters are $a_{\mu i}+\Delta_{\mu i}=-0.986$ and $\chi_{\mu i}=0.606$.
• Figure 2: Interaction strength distribution analysis. We calculate $F^2_{i,j}(\epsilon_\mu, \mathcal{M}_{\epsilon_\nu,\Delta}) := \sum_{\epsilon_\kappa \in \mathcal{M}_{\epsilon_\nu,\Delta}} |V_{\mu i,\kappa j}|^2=\int_{\epsilon_\nu+\Delta/2}^{\epsilon_\nu-\Delta/2}d\epsilon_\kappa f^2_{i,j}(\epsilon_\mu,\delta)$ with $\epsilon_\mu = 0.583$ and $\Delta = 1.0$. The horizontal axis denotes $\epsilon_\nu$. Fitting results yield: $F^2_{1,1} = F^2_{2,2} = 0.001 \times \{\tanh[1.930(x+2.985)] + \tanh[1.930(4.216-x)]\}$ and $F^2_{1,2} = F^2_{2,1} = 0.133 \times \{\tanh[1.930(x+2.985)] + \tanh[1.930(4.216-x)]\}$, demonstrating distinct coupling channels.
• Figure 3: Fitted parameters of interaction strength distribution. The interaction strength distribution $F^2_{i,j}(\epsilon_\mu, \mathcal{M}_{\epsilon_\nu,\Delta})$ for different $i,j,\epsilon_\mu$ is fitted to the function $\mathcal{F}/2 \times \{\tanh[\vartheta(x+\epsilon^{\text{L}})] + \tanh[\vartheta(\epsilon^{\text{U}}-x)]\}$. Outliers ($\approx0.4\%$ of total data) were excluded for clarity. (a) Lower and upper bounds $(\epsilon^{\text{L}},\epsilon^{\text{U}})$ of the interaction energy range, indicating the region of significant interaction strength. The fits yield an upper bound $\epsilon^{\text{U}}_{\mu}\approx0.921\epsilon_\mu+3.440$ and a lower bound $\epsilon^{\text{L}}_{\mu}\approx0.931\epsilon_\mu-3.450$. (b) The interaction strength $\mathcal{F}$ shows weak dependence on $\epsilon_\mu$, with averages $\overline{\mathcal{F}}\approx0.288$ for $i\neq j$ and $\overline{\mathcal{F}}\approx0.003$ for $i= j$.
• Figure 4: (a) Lorentzian width $\chi_{\mu i}$ versus bath energy $\epsilon_\mu$. (b) Linear regression of shifted energies yields $a_{\mu 1} + \Delta_{\mu 1} = 0.998\epsilon_\mu - 1.610$ and $a_{\mu 2} + \Delta_{\mu 2} = 0.998\epsilon_\mu + 1.608$, confirming energy-dependent shifts.
• Figure 5: Self-consistency validation for $i=2$. (a) Orange: Directly fitted $\chi_{\mu i}$; Blue: Reconstructed values via \ref{['consist', 'chslam']}. (b) Orange: Extracted $\Delta_{\mu i} - V_{\mu i}$; Blue: Calculated values using \ref{['consist2', 'chslam']}. In both (a) and (b), we used $\eta_i = (-1)^{i-1}2.63$.