Finite-size scaling, ground-state effects, equilibration regimes, and the extension to real and three-dimensional interfaces for emergent friction in two-dimensional Frenkel-Kontorova models

In the closed, energy-conserving two-dimensional vector Frenkel-Kontorova (FK) model of Norell, Fasolino, and de Wijn (Phys. Rev. E 94, 023001 (2016)), viscous friction emerges without any heuristic damping: after a velocity kick, the center-of-mass velocity decays exponentially with an effective coefficient $\eta$, with resonance peaks when the washboard frequency matches lattice phonon modes. The authors explicitly left open how these emergent coefficients, resonance widths, and equilibration rates depend on system size and whether they survive the thermodynamic limit; whether ground-state metastability alters static-friction thresholds in 2D; the mechanism and boundary of the crossover from pathological (small $\xi$, i.e. a near-vanishing transverse elastic constant) to physical behavior, including very slow equilibration regimes; whether the diagonal-wavevector resonance found at $\xi = 0.5$ can occur in real materials; and how a fully three-dimensional extension with physically mapped elastic constants changes the emergent friction, and whether the additional dissipation channels it introduces bring the computed viscous coefficients closer to typical experimental magnitudes. No study known to date computes the size scaling or thermodynamic limit of the emergent viscous coefficient for this model, and no three-dimensional FK friction study exists.

Background

The Frenkel-Kontorova (FK) model describes a set of interacting particles subject to a spatially periodic substrate potential; because the sliding object is not treated as rigid, it is the simplest model in which dynamic friction is emergent, generated by phonon excitations rather than by heuristic damping. Its one-dimensional (1D) version underlies the theory of structural superlubricity, the near-vanishing static friction of incommensurate (structurally mismatched) contacts observed, for example, for graphite flakes. In the dynamic version studied by van den Ende, de Wijn, and Fasolino (J. Phys.: Condens. Matter 24, 445009 (2012)), a closed, energy-conserving 1D FK chain is given a velocity kick and the center-of-mass (CM) velocity decays as if subject to viscous damping; fitting an exponential to the ensemble-averaged decay yields an effective viscous friction coefficient $\eta$, which depends non-monotonically on the initial velocity, with resonance peaks when the washboard frequency $2\pi v/a_s$ ($a_s$ the substrate period) matches a phonon frequency of the chain.

Norell, Fasolino, and de Wijn (Phys. Rev. E 94, 023001 (2016)) extended this program to two dimensions (2D), comparing two square-lattice extensions on a separable cosine substrate potential $V_{\mathrm{ext}} = (\lambda h/2\pi)[\cos(2\pi x) + \cos(2\pi y)]$ (here $\lambda = V_0/(Ka_s^2)$ is the dimensionless ratio of substrate amplitude $V_0$ to spring stiffness $K$ at substrate period $a_s$, $h = Ka_s^2$ is the energy unit, and the source uses reduced units with time unit $\tau_0 = \sqrt{m/K}$): a scalar model, in which the $x$ and $y$ coordinates are coupled only through shear springs of strength $K_{\mathrm{shear}}$ and which in practice behaves as two decoupled 1D systems that cannot thermalize energy between the directions, and a vector model with true two-dimensional springs between nearest neighbors (strength $(1-\xi)K$) and diagonal next-nearest neighbors (strength $\xi K$). The parameter $\xi$ controls the ratio of the longitudinal to transverse elastic constants, $c_{11}/c_{44} = K/(\xi K)$ in 2D; matching the elastic constants of 3D rare-gas solids (Xe and Kr, with $c_{11}/c_{44} \approx 2$) while accounting for the higher number of next-nearest neighbors in 3D, which changes the comparison to $(1+\xi)K/(2\xi K)$, yields the physical value $\xi \approx 0.33$. The incommensurate ratio of lattice spacing to substrate period (the golden mean) is approximated by ratios of consecutive Fibonacci numbers, $r = F_{n+1}/F_n$, so that periodic boundary conditions can be applied; the production size was $N_x = N_y = 89$ particles (spacing $a_0 = 144/89$). Their findings: static friction is robust and essentially 1D-like in 2D; the scalar model fails to equilibrate between directions; the vector model equilibrates and shows $\eta$ qualitatively similar to 1D for physical $\xi$, but with genuinely 2D effects - extra dissipation channels from quasi-transverse phonon polarizations and from modes propagating in non-sliding directions, a qualitatively different temperature dependence, anisotropic friction in which the friction force is not parallel to the velocity so the sliding direction depends on the initial speed, and, at $\xi = 0.5$, an apparent third resonance peak arising from modes with wavevector along the lattice diagonal rather than along the sliding direction. For small $\xi$ (roughly below 0.01) the near-vanishing $c_{44}$ produces low-frequency transverse modes that absorb energy pathologically easily, giving orders-of-magnitude larger, velocity-decreasing friction; equilibration can be extremely slow; and at $\xi = 0$ with zero temperature the system does not equilibrate at all by symmetry. The computed $\eta$ values are substantially smaller than typical experimental friction coefficients, which the authors attribute in part to missing three-dimensional dissipation channels.

The authors state these limitations explicitly: the system-size dependence was not investigated systematically (the initial velocity recurrence survives for extremely long times at small sizes, e.g. of order $200000\,\tau_0$ at $N_x = N_y = 13$, while equilibration was sometimes only demonstrable in smaller, longer-simulated systems); the energy-minimization procedure does not necessarily reach the true global-minimum ground state; it is not clear whether real materials exist in which the diagonal-mode resonance could be observed; a fully 3D model with more transverse phonon branches and more next-nearest neighbors would likely show stronger, larger friction; and the slow-equilibration regimes and the crossover from pathological to physical behavior as $\xi$ increases are not fully characterized. These issues jointly define the open question of whether, and under what conditions, the emergent-friction phenomenology established at one finite size and one state-preparation protocol is an asymptotic property of the model and a physically meaningful mechanism.

Problem Statement

For the closed, energy-conserving two-dimensional vector Frenkel-Kontorova model of Norell, Fasolino, and de Wijn (square lattice with nearest-neighbor springs of strength $(1-\xi)K$ and diagonal next-nearest-neighbor springs of strength $\xi K$ on a separable cosine substrate potential, incommensurability approximated by Fibonacci rationals in periodic boxes), determine the robustness and physical reach of its emergent-friction phenomenology. The common objective is to establish whether the emergent viscous friction coefficients $\eta(v, T, \xi)$, the phonon-resonance structure of the velocity dependence, and the thermal equilibration reported at the single production size $N_x = N_y = 89$ are asymptotic properties of the model and physically meaningful mechanisms, and to carry them toward real and three-dimensional interfaces. Specifically: (1) compute or derive the system-size dependence of the effective viscous friction coefficient, the resonance widths, and the equilibration times, and determine whether and how these quantities change in the thermodynamic limit (including whether finite-size recurrences, which are observed to survive for extremely long times already at small sizes, contaminate the measured coefficients at accessible sizes); (2) determine whether preparing true global-minimum ground states, rather than the possibly metastable states reached by the energy-minimization protocol, alters the static-friction thresholds in two dimensions; (3) characterize the regimes of very slow equilibration (in particular near small but nonzero $\xi$) and identify the mechanism and parameter boundary of the crossover from pathological, ultra-high, velocity-decreasing friction at $\xi$ approaching zero to physical behavior at $\xi$ of order 0.33; (4) determine whether the additional resonance at $\xi = 0.5$, arising from phonon modes with wavevector along the lattice diagonal, can occur in a real material with physical elastic constants; and (5) extend the analysis to a fully three-dimensional FK model (the direct three-dimensional analogue of the 2D vector model: a simple-cubic lattice with nearest-neighbor springs of strength $(1-\xi)K$ and face-diagonal next-nearest-neighbor springs of strength $\xi K$, with elastic constants computed from these spring strengths and matched to a real material following the 3D rare-gas comparison given in the Background), determine how the additional transverse phonon branches and dissipation channels change the emergent friction, and whether the resulting viscous coefficients approach the typical experimental magnitudes that the 2D model falls significantly short of (the experimentally reported viscous friction coefficients are of order $1/\tau_0$; the effect of the additional dissipation channels is to be reported as the residual ratio $\eta_{\mathrm{3D}}/\eta_{\mathrm{exp}}$). Components (1)-(3) jointly decide the internal validity of the 2D results (finite size, state preparation, parameter regime); components (4)-(5) decide their external validity (real materials, dimensionality). All five components are required parts of this single objective rather than independent tasks, so the overall question is answered only by the set; each component is, however, independently refutable, and a definite finding on any one of them - including a negative one, such as showing that the diagonal-mode resonance does not survive real elastic constants - resolves that component and states what remains open.

References

1. Jesper Norell, Annalisa Fasolino, Astrid S. de Wijn, "Emergent friction in two-dimensional Frenkel-Kontorova models", Phys. Rev. E 94, 023001 (2016), DOI 10.1103/PhysRevE.94.023001, arXiv:1604.01565, https://doi.org/10.1103/PhysRevE.94.023001
2. J. A. van den Ende, A. S. de Wijn, A. Fasolino, "The effect of temperature and velocity on superlubricity", J. Phys.: Condens. Matter 24, 445009 (2012), DOI 10.1088/0953-8984/24/44/445009, https://doi.org/10.1088/0953-8984/24/44/445009
3. Yang Yang, Wen-Shan Duan, Jian-Min Chen, Lei Yang, Jasmina Tekić, Zhi-Gang Shao, Cang-Long Wang, "Friction phenomena and phase transition in the underdamped two-dimensional Frenkel-Kontorova model", Phys. Rev. E 82, 051119 (2010), DOI 10.1103/PhysRevE.82.051119, https://doi.org/10.1103/PhysRevE.82.051119
4. Wang Cang-Long, Duan Wen-Shan, Yang Yang, Chen Jian-Min, "Application of Two-Dimensional Frenkel-Kontorova Model to Nanotribology", Commun. Theor. Phys. 54, 112 (2010), DOI 10.1088/0253-6102/54/1/22, https://doi.org/10.1088/0253-6102/54/1/22
5. Ru-Tao Li, Wen-Shan Duan, Yang Yang, Cang-Long Wang, Jian-Min Chen, "Hysteresis in the pinning-depinning transition in underdamped two-dimensional Frenkel-Kontorova model", EPL 94, 56003 (2011), DOI 10.1209/0295-5075/94/56003, https://doi.org/10.1209/0295-5075/94/56003
6. Kazumasa Shinjo, Motohisa Hirano, "Dynamics of friction: superlubric state", Surf. Sci. 283, 473 (1993), DOI 10.1016/0039-6028(93)91022-H, https://doi.org/10.1016/0039-6028(93)91022-H
7. M. Dienwiebel, G. S. Verhoeven, N. Pradeep, J. W. M. Frenken, J. A. Heimberg, H. W. Zandbergen, "Superlubricity of graphite", Phys. Rev. Lett. 92, 126101 (2004), DOI 10.1103/PhysRevLett.92.126101, https://doi.org/10.1103/PhysRevLett.92.126101
8. S. Amiri, C. A. Volkert, R. L. C. Vink, "Friction on incommensurate substrates: Role of anharmonicity and defects", Phys. Rev. E 104, 014802 (2021), DOI 10.1103/PhysRevE.104.014802, arXiv:2009.14562, https://doi.org/10.1103/PhysRevE.104.014802
9. Shubham Agarwal, Martin H. Müser, "Finite-temperature properties of the Frenkel-Kontorova model: Relation to tribological systems and fluid rheology", Phys. Rev. E 112, 045502 (2025), DOI 10.1103/ll2g-k2gq, arXiv:2507.14948, https://doi.org/10.1103/ll2g-k2gq
10. Shubham Agarwal, "Breakloose suppression in minimal friction models", Tribol. Lett. 74, 93 (2026), DOI 10.1007/s11249-026-02185-2, arXiv:2603.17076, https://doi.org/10.1007/s11249-026-02185-2
11. Serena Dipierro, Giorgio Poggesi, Enrico Valdinoci, "A quantitative rigidity result for a two-dimensional Frenkel-Kontorova model", Physica D 423, 132871 (2021), DOI 10.1016/j.physd.2021.132871, arXiv:2102.04630, https://doi.org/10.1016/j.physd.2021.132871
12. Rui Dong, Alessandro Lunghi, Stefano Sanvito, "Stiffness and atomic-scale friction in superlubricant MoS2 bilayers", J. Phys. Chem. Lett. 14, 6086 (2023), DOI 10.1021/acs.jpclett.3c01066, arXiv:2107.00486, https://doi.org/10.1021/acs.jpclett.3c01066
13. Gianluca Costagliola, Federico Bosia, Nicola M. Pugno, "A 2-D model for friction of complex anisotropic surfaces", J. Mech. Phys. Solids 112, 50 (2018), DOI 10.1016/j.jmps.2017.11.015, arXiv:1706.08055, https://doi.org/10.1016/j.jmps.2017.11.015
14. Zhenjun Zhang, Chunmei Tang, Peiqing Tong, "Dynamical thermalization of Frenkel-Kontorova model in the thermodynamic limit", Phys. Rev. E 93, 022216 (2016), DOI 10.1103/PhysRevE.93.022216, https://doi.org/10.1103/PhysRevE.93.022216
15. A. Vanossi, D. Dietzel, A. Schirmeisen, E. Meyer, R. Pawlak, T. Glatzel, M. Kisiel, S. Kawai, N. Manini, "Recent highlights in nanoscale and mesoscale friction", Beilstein J. Nanotechnol. 9, 190 (2018), DOI 10.3762/bjnano.9.190, https://doi.org/10.3762/bjnano.9.190
16. Antonio Cammarata, Elliot Perviz, Tomas Polcar, "Current perspective towards a general framework to describe and harness friction at the nanoscale", Prog. Surf. Sci. 99, 100753 (2024), DOI 10.1016/j.progsurf.2024.100753, https://doi.org/10.1016/j.progsurf.2024.100753