[Paper Review] Why asymmetric interparticle interaction can result in convergent heat conductivity

Shunda Chen, Yong Zhang+2 more|arXiv (Cornell University)|Sep 27, 2013

Thermal properties of materials3 citations

TL;DR

This paper demonstrates that asymmetric inter-particle interactions in one-dimensional momentum-conserving lattices can induce a rapid, non-power-law decay of the heat current autocorrelation function, leading to size-independent (convergent) heat conductivity at low temperatures. The mechanism arises because the heat mode relaxes normally (with scaling exponent γ = 0.5), while the sound mode persists, causing a non-physical divergence in the Green-Kubo formula if not properly separated; thus, the observed conductivity remains practically finite and size-independent under appropriate conditions.

ABSTRACT

We show that the asymmetric inter-particle interactions may induce rapid decay of heat current autocorrelation in one-dimensional momentum conserving lattices. When the asymmetry degree and the temperature are appropriate, the decay is sufficient rapid for resulting a convergence conductivity practically. To understand the underlying mechanism, we further studied the relaxation behavior of the hydrodynamic modes. It is shown that for lattice with symmetric potential, the heat mode relaxs in the superdiffusive manner, while in the case of asymmetric potential, the heat mode may relax in the normal manner.

Motivation & Objective

• To resolve the contradiction between theoretical predictions of divergent heat conductivity and numerical observations of size-independent conductivity in asymmetric 1D lattices.
• To clarify why rapid decay of the current-current correlation function leads to convergent heat conductivity in systems with asymmetric potentials.
• To distinguish the roles of heat and sound modes in energy transport, particularly in the context of the Green-Kubo formula.
• To demonstrate that the observed size-independent conductivity is physically meaningful and not an artifact of finite-time simulations.

Proposed method

• Numerical simulations of the FPU-αβ model with asymmetric potential V(x) = ½x² − (α/3)x³ + ¼x⁴, using periodic boundary conditions and zero total momentum.
• Computation of the energy current autocorrelation function C(t) = ⟨J(t)J(0)⟩ / ⟨J(0)J(0)⟩ in log-log scale to identify decay behavior.
• Application of the Green-Kubo formula with time truncation at τ_tail = L/v to avoid unphysical divergence from long-time tails.
• Hydrodynamic mode analysis via the Prähofer-Spohn scaling function to extract relaxation exponents γ for heat and sound modes.
• Comparison of decay behavior between symmetric (FPU-β) and asymmetric (FPU-αβ) potentials at fixed energy per particle ε = 0.1.
• Separation of heat and sound contributions in energy current to isolate the physical origin of conductivity convergence.

Experimental results

Research questions

• RQ1 Can asymmetric inter-particle interactions lead to a rapid, non-power-law decay of the heat current autocorrelation function in 1D lattices?
• RQ2 Under what conditions does the Green-Kubo formula yield a size-independent heat conductivity despite long-time tail predictions?
• RQ3 Why does the heat mode relax normally (γ = 0.5) in asymmetric lattices while the sound mode persists?
• RQ4 How does the distinction between heat and sound energy transport affect the interpretation of conductivity in the Green-Kubo framework?
• RQ5 What is the physical significance of the observed size-independent conductivity in realistic temperature regimes?

Key findings

• At low temperature (T = 0.1), the current-current correlation function decays faster than power law, approximately exponentially, with τ_e ≈ 2000, leading to a finite integral ∫₀^{τ_e} C(t)dt ≈ 123.
• Even if the tail decays as t^{-0.67}, the contribution ∫_{τ_e}^{τ_tail} C(t)dt remains small (≈1) for τ_tail = 10¹², implying no significant divergence in conductivity.
• The heat mode relaxes normally with γ = 0.5, indicating normal diffusion, which is the fundamental reason for convergent conductivity.
• The sound mode continues to contribute a power-law decay to the current correlation, which would cause divergence in the Green-Kubo formula if not separated from the heat mode.
• For system sizes up to L ≈ 10¹² (≈100 meters), the conductivity remains size-independent, indicating physical relevance at macroscopic scales.
• The conclusion holds for FPU-αβ, Lennard-Jones, and other asymmetric potential models, provided the asymmetry degree and temperature are appropriately tuned.