Mechanism of normal energy and momentum diffusion in one-dimensional momentum-conserving lattices with bounded periodic interaction potentials

Liao and Li (Phys. Rev. E 99, 062125 (2019)) showed numerically that one-dimensional momentum-conserving nonlinear lattices whose interparticle potential is bounded and periodic exhibit Gaussian (normal) diffusion of both energy and momentum at moderate temperatures — whether the potential is symmetric, like the classic coupled rotator, or fully asymmetric — with a crossover to Lévy-like energy spreading and ballistic momentum transport at low temperatures near the integrable Toda limit. This establishes periodicity, not left-right symmetry, as the structural feature that suppresses the anomalous hydrodynamic correlations otherwise generic in one dimension, but it is an empirical result without a derivation. The open problem is a predictive microscopic theory of this phenomenon: a derivation of why bounded periodicity of the interaction potential restores normal diffusive energy and momentum transport at not-very-low temperature, and a quantitative account of the crossover to the Lévy-like energy spreading and ballistic momentum transport observed at low energy density near the Toda integrable limit.

Background

A one-dimensional (1D) anharmonic lattice is a chain of classical oscillators with nearest-neighbor Hamiltonian $H=\sum_i[p_i^2/2+V(q_{i+1}-q_i)]$, where $q_i$ and $p_i$ are the displacement and momentum of the $i$-th particle and $V$ is the interaction potential. When there is no on-site (pinning) potential, total momentum $P=\sum_i p_i$ is conserved. Simulations and theory over four decades established that such momentum-conserving 1D lattices generically exhibit anomalous heat conduction: the thermal conductivity $\kappa$, measured via the Green-Kubo formula (the time integral of the equilibrium heat-current autocorrelation function), diverges with system size $N$ as $\kappa\propto N^\alpha$ with $0<\alpha<1$, rather than approaching the finite value Fourier's law requires. The microscopic signature is superdiffusive spreading of an energy perturbation: the energy mean-square displacement grows as $\langle\Delta x^2(t)\rangle_E\propto t^\beta$ with $\beta>1$, and the two exponents are linked by the relation $\alpha=\beta-1$ (Liu, Hänggi, Li, Ren and Li, Phys. Rev. Lett. 112, 040601 (2014)). The currently accepted framework, nonlinear fluctuating hydrodynamics (NLFH) — a mode-coupling treatment of fluctuating hydrodynamics for chains with three conserved fields (stretch $r_i=q_{i+1}-q_i$, momentum, and energy) — predicts for generic anharmonic chains a Lévy-distributed central heat mode and sound peaks in the Kardar-Parisi-Zhang (KPZ) universality class (the scaling class of the 1D stochastic Burgers equation), i.e., anomalous conduction.

The classic exception is the coupled rotator lattice, $V(x)=1-\cos x$, whose interaction potential is bounded and periodic; Gendelman and Savin (Phys. Rev. Lett. 84, 2381 (2000)) found that it conducts normally. Its angle variables live on a circle (they are compact), so the 'stretch' field of the NLFH framework is not globally well defined. Das and Dhar (arXiv:1411.5247) showed that its equilibrium energy and momentum correlations both spread diffusively and argued, within stochastic hydrodynamics, that normal diffusion follows because the angle variables are not a conserved field, which eliminates long-wavelength currents. This argument was disputed by Gao, Li and Li (Phys. Rev. E 93, 032130 (2016)), whose systematic study of stretch diffusion across standard 1D lattices found no clear connection between conserved quantities and heat-conduction type and noted that the rotator's stretch conservation is ambiguous due to the $2\pi$ degeneracy of the angles; Olla (Phys. Rev. E 107, L062104 (2023)) later proposed a different mechanism altogether — that the crossover scale beyond which anomalous transport appears can diverge — illustrated with a composite chain. So even for the symmetric rotator the mechanism of normal transport remains contested, and the rigorous result closest to it (Komorowski and Olla, arXiv:1601.02123, proving diffusive energy propagation for a momentum-conserving chain with added conservative momentum-exchange noise) relies on stochasticity and does not cover deterministic periodic lattices.

A parallel episode concerns asymmetric interactions. Zhong, Zhang, Wang and Zhao (Phys. Rev. E 85, 060102(R) (2012)) reported that momentum-conserving lattices with asymmetric (non-mirror-symmetric) interparticle potentials conduct normally, but Wang, Hu and Li (Phys. Rev. E 88, 052112 (2013)) showed that the apparent Fourier behavior of such asymmetric nonperiodic lattices is a finite-size effect: beyond much longer chains the power-law divergence of $\kappa$ returns, so asymmetry alone does not enforce normal conduction in the thermodynamic limit. This precedent illustrates how long pre-asymptotic finite-size windows can mimic Fourier behavior in momentum-conserving lattices.

In this setting, Liao and Li (Phys. Rev. E 99, 062125 (2019)) proposed 1D momentum-conserving lattices with interaction potentials that are simultaneously periodic and asymmetric — a Toda-form potential $V(x)=e^{-\sin x}+\sin x-1$ (period $2\pi$) and a ratchet-form potential $V(x)=\sin(x+x_0)+\frac{1}{4}\sin[2(x+x_0)]+y_0$ — and studied the equilibrium spreading of energy and momentum fluctuations through the autocorrelation functions $C_E(i,t)$ and $C_P(i,t)$. At moderate energy density both correlators collapse, after rescaling, onto Gaussians with linearly growing mean-square displacement ($\beta=1$; reported diffusivities $D_E=1.0$, $D_P=2.3$ at energy density 0.2 for the Toda-form potential and $D_E=1.85$, $D_P=1.93$ at energy density 0.8 for the ratchet potential, for chains of up to $N=1001$ sites). At low energy density (0.05) the energy spreading becomes Lévy-walk-like (a space-time-coupled random walk with finite propagation speed, producing two propagating side peaks and $\beta>1$) and momentum spreading becomes ballistic, which the authors attribute to the potential approaching, for small relative displacement, the integrable Toda chain $V(x)=e^{-x}+x-1$ (an exactly solvable exponential-interaction chain whose perturbations have very long mean free paths). Their conclusion — periodicity (boundedness) of the interaction potential, not its symmetry, is the operative structural feature suppressing anomalous transport — is empirical: the paper provides no microscopic derivation. The low-temperature regime connects to the scaling theory of Lepri, Livi and Politi (Phys. Rev. Lett. 125, 040604 (2020)), who explained apparent normal diffusion near integrable limits (e.g., weakly perturbed Toda chains) as an intermediate regime between ballistic and asymptotically anomalous transport with crossover scales growing as power laws of the perturbation strength — but their framework covers three-conserved-field chains and does not treat periodic potentials with compact variables.

Problem Statement

Consider one-dimensional Hamiltonian lattices $H=\sum_i[p_i^2/2+V(q_{i+1}-q_i)]$ whose interaction potential $V$ is smooth, bounded, and periodic (e.g., of period $2\pi$) but spatially asymmetric, meaning $V(-x)\ne V(x)$ (not even about the bond origin), so that total momentum is conserved and no globally defined conserved stretch field exists — including, as instances from Liao and Li, the Toda-form potential $V(x)=e^{-\sin x}+\sin x-1$ and the ratchet-form potential $V(x)=\sin(x+x_0)+\frac{1}{4}\sin[2(x+x_0)]+y_0$. The two named instances differ in residual spatial symmetry: the ratchet-form potential has no mirror symmetry, $V(-x+\Delta)\ne V(x+\Delta)$ for every shift $\Delta$, whereas the Toda-form potential retains the residual mirror symmetry $V(-x+\pi/2)=V(x+\pi/2)$ (mirror symmetry about $x=\pi/2$). The symmetric coupled rotator lattice $V(x)=1-\cos x$ is the classic reference case. Following the source's convention, energy density rather than temperature is the input control parameter, with the temperature defined correspondingly (the source reports $T\approx0.048$, $0.186$, $0.672$ at energy densities $e\approx0.05$, $0.2$, $0.8$).

The problem is a predictive microscopic theory of normal versus anomalous transport in these bounded-periodic momentum-conserving lattices. The simulations establish the phenomena such a theory must explain: at moderate energy density, the energy and momentum autocorrelation functions $C_E(i,t)$ and $C_P(i,t)$ collapse, after rescaling, onto Gaussians with linearly growing mean-square displacement (normal diffusion, with the diffusivities reported above), while at low energy density, where the interaction approaches the integrable Toda limit $V(x)=e^{-x}+x-1$ and the mean free path of the energy carriers is very long, energy spreading becomes Lévy-walk-like and momentum spreading ballistic. Derive from the microscopic dynamics — for example by extending nonlinear fluctuating hydrodynamics or mode-coupling theory to compact angle variables with effectively two conserved fields, or by an equivalent construction — why bounded periodicity of the interaction potential, rather than its symmetry, restores normal diffusive transport at not-very-low temperature, and account for the crossover to the near-integrable low-energy-density regime, with quantitative predictions (diffusivities, scaling forms, crossover behavior) tied to the reported simulation data. A resolution must supply all three elements jointly: a microscopic derivation of why bounded periodicity rather than symmetry restores normal diffusive energy and momentum transport at not-very-low temperature; an account of the crossover to the near-integrable low-energy-density regime; and quantitative predictions (diffusivities, scaling forms, crossover behavior) for the reported simulation data. The derivation is required for the whole class of smooth bounded periodic asymmetric potentials defined above, whereas the quantitative predictions are required only for the Toda-form and ratchet potentials at the energy densities reported above; naming a route (for example extending nonlinear fluctuating hydrodynamics or mode-coupling theory to compact angle variables) or restating the heuristic already given in the Background does not by itself constitute a resolution, and supplying one of the three elements alone leaves the task unresolved.

References

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