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Anomalous transport in the Fermi-Pasta-Ulam-Tsingou model: a review and open problems

Stefano Lepri, Roberto Livi, Antonio Politi

Abstract

This review provides an up-to-date account of energy transport in Fermi-Pasta-Ulam-Tsingou (FPUT) chains, a key testbed for nonequilibrium statistical physics. We discuss the transition from the historical puzzle of thermalization to the discovery of anomalous heat transport, where the effective thermal conductivity $κ$ diverges with system size $L$ as $κ\propto L^δ$. The article clarifies the distinction between two universality classes: the FPUT-$αβ$ model, characterized by $δ= 1/3$ and linked to Kardar-Parisi-Zhang (KPZ) physics, and the symmetric FPUT-$β$ model, where numerical and theoretical evidence support $δ= 2/5$. We investigate how finite-size effects - unavoidably induced by the thermostatting protocols - can disguise the asymptotic scaling. Additionally, we analyze the role of conservative noise in preserving hydrodynamic properties and examine how proximity to integrable limits leads to long-lived quasi-particles and, thereby, to diffusive regimes over intermediate spatial scales.

Anomalous transport in the Fermi-Pasta-Ulam-Tsingou model: a review and open problems

Abstract

This review provides an up-to-date account of energy transport in Fermi-Pasta-Ulam-Tsingou (FPUT) chains, a key testbed for nonequilibrium statistical physics. We discuss the transition from the historical puzzle of thermalization to the discovery of anomalous heat transport, where the effective thermal conductivity diverges with system size as . The article clarifies the distinction between two universality classes: the FPUT- model, characterized by and linked to Kardar-Parisi-Zhang (KPZ) physics, and the symmetric FPUT- model, where numerical and theoretical evidence support . We investigate how finite-size effects - unavoidably induced by the thermostatting protocols - can disguise the asymptotic scaling. Additionally, we analyze the role of conservative noise in preserving hydrodynamic properties and examine how proximity to integrable limits leads to long-lived quasi-particles and, thereby, to diffusive regimes over intermediate spatial scales.
Paper Structure (14 sections, 23 equations, 12 figures)

This paper contains 14 sections, 23 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Generalities about energy transport in the FPUT-model
  3. A historical perspective
  4. Notations and observables
  5. Anomalous transport in the FPUT-model
  6. Finite-size effects
  7. Length of the thermostatted region
  8. Coupling strength
  9. Equilibrium simulations
  10. Correlation functions
  11. FPUT as an interface problem
  12. Adding conservative noise
  13. Transport close to the integrable limit
  14. Conclusions

Figures (12)

  • Figure 1: A one-dimensional chain of coupled oscillators interacting with two thermal reservoirs ad different temperatures $T_+$ and $T_-$. Actual implementation of the reservoirs can be either stochastic (Langevin) or deterministic (via e.g. isokinetic thermostat), see LLP03 for details.
  • Figure 2: Kinetic temperature profile $\langle p_n^2\rangle$ for an FPUT-$\beta$ chain with $L=8192$ particles and thermostats sets at $T_+ = 1.2$, $T_-=0.8$, respectively.
  • Figure 3: Left: Thermal conductivity of FPUT-$\beta$ versus lattice size $N$. (denoted by $L$here); Langevin heat bath at temperatures 1.2 and 0.8. Dashed line represents the power law $N^{2/5}$. Right: logarithmic derivative versus lattice size $N$. © (2024), The Physical Society of Japan, reprinted with permission from Ref.takatsu2024large.
  • Figure 4: Energy flux in the FPUT-$\beta$ model, versus the length $s$ of the thermostatted region, for different chain lengths.
  • Figure 5: Logarithmic derivative of the Energy flux in the FPUT-$\beta$ model.
  • ...and 7 more figures