[Paper Review] Are there really phase transitions in 1-d heat conduction models?

Lei Yang, Peter Grassberger|arXiv (Cornell University)|Jun 6, 2003

Advanced Thermodynamics and Statistical Mechanics3 citations

TL;DR

This paper challenges claims of finite-temperature phase transitions in 1D heat conduction models, specifically the Frenkel-Kontorova (FK) and coupled rotator models, where previous studies suggested infinite conductivity at certain temperatures. Using high-precision, large-scale Langevin dynamics simulations with long integration times and system sizes up to N=8192, the authors show that apparent conductivity divergences are actually slow crossovers due to harmonic limits at T→0 and T→∞, not true phase transitions.

ABSTRACT

Recently, it has been claimed (O. V. Gendelman and A. V. Savin, Phys. Rev. Lett. {\bf 84}, 2381 (2000); A.V.Savin and O.V.Gendelman, arXiv: cond-mat/0204631 (2002)) that two nonlinear classical 1-d lattice models show transitions, at finite temperatures, where the heat conduction changes from being finite to being infinite. These are the well known Frenkel-Kontorova (FK) model and a model for coupled rotators. For the FK model we give strong theoretical arguments why such a phase transition is not to be expected. For both models we show numerically that the effects observed by Gendelman {\it et al.} are not true phase transitions but are rather the expected cross-overs associated to the conductivity divergence as $T o 0$ and (for the FK model) $T o\infty$.

Motivation & Objective

• To test the claim that the Frenkel-Kontorova (FK) model exhibits a finite-temperature phase transition where heat conductivity changes from finite to infinite.
• To investigate whether the coupled rotator model shows a similar transition, as previously reported by Gendelman and Savin.
• To resolve discrepancies between earlier simulations and theoretical expectations regarding heat conduction in 1D nonlinear lattices.
• To determine whether observed conductivity divergences are genuine phase transitions or artifacts of finite-size effects and non-equilibrium dynamics.
• To clarify the role of temperature-dependent phonon scattering and localization in determining thermal conductivity in 1D systems.

Proposed method

• Performed large-scale Langevin dynamics simulations on 1D chains of up to 8192 particles with N₀ = 40 thermostatted boundary layers.
• Used a leap-frog integrator with time step 0.05 and integration times up to 5×10⁸ units to ensure steady-state convergence.
• Applied white Gaussian noise thermostats with T_high and T_low to generate a constant temperature gradient (ΔT ≈ 10–20%).
• Measured heat current J and computed finite-size conductivity κ = J·N/ΔT as a function of system size N and temperature T.
• Analyzed the size dependence of κ to distinguish between true divergence (indicative of phase transition) and slow crossover behavior.
• Compared results with earlier claims by Gendelman and Savin, focusing on the behavior of κ at low and high temperatures.

Experimental results

Research questions

• RQ1 Do the Frenkel-Kontorova and coupled rotator models exhibit true finite-temperature phase transitions where heat conductivity changes from finite to infinite?
• RQ2 Is the observed divergence in conductivity at low and high temperatures in these models a genuine phase transition or a crossover due to harmonic limits?
• RQ3 How does the size dependence of conductivity κ(N) at fixed T distinguish between finite conductivity and divergence?
• RQ4 What is the role of localized excitations and phonon scattering in blocking heat transport at low temperatures in the rotator model?
• RQ5 Why do earlier simulations report infinite conductivity in certain temperature ranges, and is this due to insufficient system size or integration time?

Key findings

• For the Frenkel-Kontorova model with ε=1.0, conductivity κ remains finite across all temperatures studied (T=0.125 to T=128), with no evidence of divergence or phase transition.
• At T=0.125 and T=128, κ increases slowly with N but appears to saturate for N>1000, indicating a crossover rather than a true divergence.
• For ε=3.0 and ε=10.0, κ is finite at all finite temperatures and shows no sign of divergence, even at low T, contradicting earlier claims.
• In the rotator model, conductivity increases with N for T=0.2, 0.3, 0.45, and 0.6, but the curves eventually flatten at large N, indicating finite conductivity at all finite T.
• The apparent divergence in earlier studies is attributed to insufficient system sizes and integration times, not to a true phase transition.
• The observed behavior at T→0 and T→∞ is consistent with the system becoming effectively harmonic, leading to slow crossover behavior, not a critical transition.