Universality breakdown in anomalous Fourier's law for the one-dimensional diatomic hard-point gas: exponent identities, hidden slow fields, and the correct mesoscopic theory of anomalous heat conduction

Hurtado and Garrido (Sci. Rep. 6, 38823, 2016) used a boundary-driven profile-collapse method to measure the conductivity anomaly exponent $\alpha$ of the one-dimensional diatomic hard-point gas, finding it to depend non-monotonically on the mass ratio $\mu$ (e.g. $\alpha(2.2)=0.308$, $\alpha(3)=0.297(6)$, $\alpha(10)=0.260(14)$, tending to $\approx 1/4$ at large $\mu$), in apparent conflict with the nonlinear-fluctuating-hydrodynamics universality prediction $\gamma=\delta=1/3$ independent of $\mu$. The problem cluster the source posed remains open and actively contested: whether the profile-based, linear-response, and equilibrium exponents $\alpha$, $\gamma$, $\delta$ coincide; what the microscopic origin of the apparent breakdown is (hidden non-hydrodynamic slow fields versus crossover artifacts); what mesoscopic framework reconciles the local anomalous Fourier law with the nonlocal $L^{\alpha}$ scaling; and whether the effect generalizes to other one-dimensional models and is removed by conservative noise. A 2026 Physical Review Research paper claims to resolve the parameter-dependent exponents as crossover effects with asymptotic $\kappa\sim N^{1/3}$, but it does not confront the size-robust collapse measurements, and the dispute stands unresolved.

Background

Fourier's law of heat conduction states that the heat current $J$ is locally proportional to the temperature gradient, $J=-\kappa\,dT/dx$, with a size-independent conductivity $\kappa$. In one-dimensional (1d) systems whose dynamics conserves momentum, this law is anomalous: the effective conductivity grows with system size $L$. Two exponents conventionally characterize the anomaly. The linear-response exponent $\gamma$ is defined by the scaling of the steady current under a small boundary temperature difference, $J\sim L^{-1+\gamma}$ (equivalently $\kappa_L\sim L^{\gamma}$). The equilibrium exponent $\delta$ is defined by the long-time tail of the total heat-current autocorrelation function, $\langle J(0)J(t)\rangle\sim t^{-1+\delta}$, whose nonintegrability makes the Green-Kubo conductivity (the time integral of this correlation) diverge. Renormalization-group arguments (Narayan and Ramaswamy, 2002) and exact mode-coupling results (van Beijeren, 2012) led to the conjecture that the anomaly is universal in 1d momentum-conserving systems. Nonlinear fluctuating hydrodynamics (NFH) — a mode-coupling treatment of the fluctuating Euler equations for the three locally conserved fields (stretch, i.e. the local interparticle spacing — or, for gases such as the hard-point gas, the particle density — together with momentum and energy), decomposed into two sound modes and one heat mode — sharpened this: for generic 1d systems the sound peaks belong to the Kardar-Parisi-Zhang (KPZ) universality class, the universality class of generic one-dimensional stochastic growth, with dynamical exponent $z=3/2$ (peak widths growing as $t^{2/3}$), and the heat peak is a symmetric Lévy $5/3$ stable distribution (a probability distribution whose width grows as $t^{3/5}$), implying $\gamma=\delta=1/3$ with only non-universal amplitudes depending on the model (Spohn, 2014). Numerical tests confirm these predictions for anharmonic chains and for hard-point systems (Das et al., 2014; Mendl and Spohn, 2014).

The paradigmatic fluid model is the diatomic hard-point gas (DHP): point particles on a line with alternating masses $m$ and $M=\mu m>1$ interacting only through elastic collisions, so that the mass ratio $\mu$ is its single intrinsic parameter; $\mu=1$ is integrable (ballistic transport). Hurtado and Garrido (2016) introduced a collapse method to measure the anomaly exponent away from the linear-response regime: the gas is driven by stochastic thermal walls (boundaries that re-inject particles with Maxwellian velocities at temperatures $T_0\neq T_L$), and, under the hypotheses of an anomalous Fourier law $J=-\kappa_L(\rho,T)\,dT/dx$ with $\kappa_L\propto L^{\alpha}$ and macroscopic local equilibrium (local density and temperature tied by the equilibrium equation of state $P=\rho T$), all measured density and temperature profiles must collapse onto a universal master curve when position is rescaled as $L^{-\alpha}\psi x$ with the reduced current $\psi=J/P^{3/2}$. The observed collapse is sharp across three decades in $N$ (up to $N=10^5{+}1$) and yields $\alpha(\mu)$ with small error bars; since the linear-response limit of the anomalous Fourier law gives $J\propto L^{-1+\alpha}$, the authors conjecture $\alpha=\gamma(=\delta)$, so a $\mu$-dependent $\alpha$ would break the predicted universality. The same analysis delivers a closed-form master curve, a power-law density dependence of the conductivity $k(\rho)=a\rho^{\alpha}$, a bound on the heat current in terms of pressure, and an explanation of why standard linear-response measurements drift with system size: the effective boundary temperatures felt by the bulk converge to the wall temperatures extremely slowly, the deviation decaying as $(\log N)^{-1/2}$, an effect of boundary layers (thin regions near the walls where finite-size corrections accumulate, decoupled from the bulk by bulk-boundary decoupling).

The tension is explicit: precise equilibrium simulations of the alternating hard-point gas at $\mu=3$ and $N=4096$ (Mendl and Spohn, 2014) give $\delta\approx0.33$, consistent with the universal $1/3$ but above the measured $\alpha(3)=0.297(6)$. Standard NFH leaves no room for any $\mu$-dependence of the exponents. Resolving the confrontation requires either new slow degrees of freedom beyond the three hydrodynamic fields ('hidden modes', speculated by the authors on the basis of anomalous shock-wave responses in the same model) with an extended fluctuating hydrodynamics, possibly connected to the infinite discrete (Fibonacci) family of universality classes found in lattice models with several conservation laws (Popkov et al., 2015), or a demonstration that the collapse-measured $\alpha(\mu)$ is a crossover quantity rather than the asymptotic exponent.

Problem Statement

For the one-dimensional diatomic hard-point gas (point particles with alternating masses $m$ and $M=\mu m$ colliding elastically), the paradigmatic momentum-conserving fluid, settle the anomalous-transport problem posed by Hurtado and Garrido's discovery of an apparent violation of universality in anomalous Fourier's law. The single overarching objective is to decide whether the universality breakdown is genuine and to establish the correct macroscopic (mesoscopic) theory of anomalous heat conduction for this model; its linked components are: (1) the exponent question — determine whether the profile-based anomaly exponent $\alpha$ (measured by the boundary-driven collapse method, $\kappa_L\propto L^{\alpha}$, with $\alpha(\mu)$ non-monotonic and strictly below $1/3$: $\alpha(2.2)=0.308$, $\alpha(3)=0.297(6)$, $\alpha(10)=0.260(14)$, $\alpha\to\approx1/4$ at large $\mu$), the linear-response exponent $\gamma$, and the equilibrium current-correlation exponent $\delta$ coincide as conjectured ($\alpha=\gamma=\delta$), and whether the asymptotic exponent truly depends on $\mu$ — in particular, reconcile the collapse measurements (robust over $N=10^2$–$10^5$) with the nonlinear-fluctuating-hydrodynamics universality prediction $\gamma=\delta=1/3$ independent of $\mu$ and with the existing equilibrium data ($\delta\approx0.33$ at $\mu=3$); the source also notes that standard equilibrium tests at other mass ratios, e.g. $\mu=10$, would be a desirable discriminating measurement; (2) the microscopic origin — identify additional slowly-evolving non-hydrodynamic fields ('hidden modes') in the diatomic hard-point gas, or prove their absence, and construct a fluctuating-hydrodynamics framework that incorporates them (or demonstrates their irrelevance), for example along the lines of the Fibonacci family of universality classes; (3) the mesoscopic description — derive the equations that reconcile the local anomalous Fourier law $J=-\kappa_L(\rho,T)\,dT/dx$ with the explicit nonlocal $L^{\alpha}$ scaling, whether via the anomalous non-diffusive hydrodynamic scaling $x\to x/L^{1-\alpha}$, $t\to t/L^{2-3\alpha}$, fractional diffusion equations, or Lévy-walk-based transport, and test them against the measured universal master curves, the conductivity density dependence $k(\rho)=a\rho^{\alpha}$, and the pressure bound on the current; and (4) the generality and robustness — determine whether the non-universal behavior extends to other paradigmatic one-dimensional models (Fermi-Pasta-Ulam chains, i.e. one-dimensional chains of coupled anharmonic oscillators, hard-square or shoulder potentials) or is specific to hard-point dynamics, and whether adding conservative noise (stochastic momentum-conserving perturbations) removes hidden slow fields and restores universality. The objective is met when a verified macroscopic theory of anomalous heat conduction for this model predicts (or demonstrably explains) the measured exponents and master curves, and jointly accounts for the profile-based, linear-response, and equilibrium measurements, whether or not those exponents turn out to be numerically equal (components (1)-(3) are required for completion; component (4), generality and robustness across other models, is exploratory and not required), thereby settling whether one-dimensional anomalous transport universality holds or fails for this class of systems.

References

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