[Paper Review] A cellular automaton model for thermal transport in low-dimensional systems

Alejandra León|arXiv (Cornell University)|Mar 20, 2026

Thermal properties of materials0 citations

TL;DR

This paper presents a cellular automaton model to simulate phonon-mediated thermal transport in low-dimensional nanostructures across ballistic, diffusive, and transition regimes, validated on graphene nanoribbons and scalable to large systems.

ABSTRACT

In this work, we formulate a theoretical model based on a cellular automaton (CA) to study thermal transport in low-dimensional nanostructures across ballistic, diffusive, and transition regimes. Unlike computationally intensive methods such as the Boltzmann Transport Equation (BTE), our model stands out for its geometrical robustness, allowing the seamless integration of substitutional impurities, vacancies, and irregular edges. We validated the model using graphene nanoribbons (AGNRs), successfully replicating the dependence of thermal conductivity on ribbon width and temperature. Results demonstrate that the model captures critical scattering and confinement effects with a linear scalability O(N). Given the increasing pressure to optimize computational resources and reduce the carbon footprint associated with AI infrastructure, this CA model emerges as a highly efficient tool for the parametric exploration and design of next-generation thermal devices.

Motivation & Objective

• Motivate the need for low-cost, physically-consistent models of phonon transport in nanostructures amid rising AI energy demands.
• Develop a coarse-grained CA framework that captures energy transfer among acoustic, optical, and flexural phonon channels.
• Enable modeling of complex geometries, defects, and irregular boundaries with minimal computational resources.
• Calibrate and validate the model using graphene nanoribbons to reproduce known trends in thermal conductivity with width and temperature.

Proposed method

• Define a CA where each cell represents an atomic site with a fixed species label and coordinates, and evolving variables N_a, N_o, N_f, N_T, T.
• Use an update rule N_i(t+1)=N_i(t)+β[ sum_{j in n(i)} N_j(t) − z_i N(t) ] to redistribute vibrational energy among neighboring cells.
• Introduce a temperature-dependent coupling β(T_m)=β_0 σ(T_m) with σ(T_m) = (1 + e^{(T_m−T_0)/ΔT})^{-1} to interpolate between ballistic and diffusive regimes.
• Calibrate temperature from the total occupation via T_i = α N_Ti with α ≈ 4.8 and distribute N_Ti among acoustic, optical, and flexural channels using fixed weighting factors w^(s) for each species.
• Relate the computed energy exchange to a thermal conductivity k = J / (dT/dx) where J is heat flux and dT/dx the gradient, within a 1D/2D discretized framework.
• Demonstrate the model's ability to map local vibrational excitations and obtain system temperatures in real time, while maintaining O(N) scalability.

Experimental results

Research questions

• RQ1 Can a coarse-grained cellular automaton reproduce the ballistic, diffusive, and transition regimes of phonon-mediated thermal transport in low-dimensional systems?
• RQ2 How do vacancies, impurities, and irregular edges influence thermal conductivity and local temperature distributions in graphene nanoribbons within the CA framework?
• RQ3 Does the CA model capture geometry-induced effects such as bottlenecks and topological variations (e.g., S-shaped ribbons) on heat transport?
• RQ4 What is the computational scalability of the CA model when system size increases transversely or longitudinally, under realistic defect densities?

Key findings

• The CA reproduces known trends: thermal conductivity increases with ribbon width and decreases with temperature in pristine AGNRs.
• Vacancies and irregular edges reduce thermal conductivity across studied temperatures, consistent with disorder studies.
• The model captures scattering and Umklapp-like effects via the temperature-dependent β, and supports a smooth ballistic-to-diffusive transition.
• An S-shaped geometry reveals a central-bottleneck behavior with a decreasing effective thermal conductivity as the central region length changes, and can show thermal isolation effects.
• The implementation demonstrates real-time temperature and heat-flux visualization with linear (O(N)) scalability for large systems.