Miserez, Ganguly, Haussmann and Fuchs (Phys. Rev. E 106, 054125, 2022) derived, from the classical many-body Hamiltonian via the Zwanzig–Mori projection-operator formalism, the complete linear continuum mechanics of a one-component crystal with a finite concentration of point defects: isothermal and adiabatic elastic constants from equilibrium correlations and all transport coefficients (viscosities, thermal conductivity, defect diffusivity) as Green–Kubo formulae. They explicitly left open the extension beyond this regime: large deformations near the onset of plasticity, the connection to spatial projection operators that separate affine from nonaffine displacements, and extensions beyond point defects. Later literature has partially advanced neighboring ground: Hiura (Phys. Rev. E 108, 054101, 2023) microscopically derived nonlinear fluctuating hydrodynamics for the defect-free crystal, and Mabillard and Gaspard and coworkers computed elastic and transport coefficients, including vacancy-diffusion hydrodynamics, of hard-sphere crystals within a local-equilibrium/Helfand approach validated by molecular dynamics. The open problem is the central surviving objective of the source paper's outlook: a first-principles statistical-mechanics theory of defect-bearing crystals valid through large deformations up to the onset of plasticity, merging the reciprocal-space projection-operator description with the spatial affine/nonaffine decomposition, extending beyond point defects to dislocations, grain boundaries and multicomponent crystals, and quantitatively validated in simulation or colloidal experiment.

Background

A crystalline solid supports propagating sound modes and, when it contains point defects such as vacancies or interstitials, additional slow diffusion of the defect concentration. Continuum mechanics describes this by elasticity (stress proportional to strain) plus dissipative transport (viscosity, heat conduction, defect diffusion). The microscopic question is how these macroscopic laws and their coefficients follow from the underlying many-particle dynamics. The Zwanzig–Mori projection-operator formalism is a standard tool of nonequilibrium statistical mechanics that derives exact generalized Langevin equations for a chosen set of slow variables from the Hamiltonian; the price is memory kernels expressing the coupling to all eliminated fast degrees of freedom. In this framework, transport coefficients emerge as Green–Kubo formulae, i.e. time integrals of equilibrium correlation functions of microscopic currents, which in principle can be evaluated in atomistic simulations. The crucial ingredient for a crystal is the choice of slow variables: besides the conserved density, momentum density and energy density, spontaneous breaking of translational symmetry makes the displacement field (the coarse-grained shift of the lattice away from its reference positions) a slow variable; Bogoliubov's inequality, a rigorous relation between the order parameter (here the Bragg amplitude of the density modulation) and correlations, dictates this selection. Miserez, Ganguly, Haussmann and Fuchs (Phys. Rev. E 106, 054125, 2022) carried out this program for a one-component crystal with point defects: representing the displacement and defect-concentration fields through density fluctuations near the reciprocal-lattice (Bragg) peaks, they obtained all constants of linear elasticity (isothermal and adiabatic) from equilibrium correlations and Green–Kubo formulae for all transport coefficients, closing the equations with a Markovian approximation that replaces memory kernels by instantaneous Onsager coefficients and drops the fluctuating random-force term of the generalized Langevin equation. A complementary real-space approach, developed by Ganguly, Sengupta, Sollich and coworkers, uses spatial projection operators to decompose microscopic particle displacements into an affine part following the imposed macroscopic deformation and a nonaffine remainder, whose statistics are directly linked to defect precursors in colloidal crystals. The two descriptions have never been connected, although they address the same displacement field. The theory of Miserez et al. is restricted to linear response in strain and fluctuations, and the derivation itself invokes approximations whose quantitative impact — non-Markovian memory, finite-frequency corrections, omitted coupling terms higher order in the wave number — is unexplored. Subsequent work has advanced pieces of the neighboring terrain, but the nonlinear, near-yield regime of defect-bearing crystals remains without a first-principles microscopic theory, and the onset of plasticity — the irreversible rearrangements beyond yield — has not been derived from the projection-operator starting point.

Problem Statement

Extend the microscopic statistical-mechanics derivation of continuum mechanics for nonideal (defect-bearing) crystals beyond the linear-response, small-strain regime established by the Zwanzig–Mori projection-operator formalism, up to and including the onset of plasticity. Concretely, the open problem comprises jointly linked objectives with a single scientific goal — a first-principles, quantitatively tested theory of the mechanics of defect-rich crystals: (1) derive the microscopic fluctuating hydrodynamic equations for a crystal with a finite concentration of point defects under large deformations, nonlinear in strain and fluctuations and reducing to the known linear theory (Phys. Rev. E 106, 054125) in the small-strain limit, including the role of memory kernels beyond the Markovian approximation and of the neglected random-force (fluctuating-elasticity) terms near yield; (2) establish how this reciprocal-space projection description relates to the real-space spatial projection operators that separate affine macroscopic deformations from nonaffine local motions, so that the two displacement decompositions are shown to be consistent and their convergence can describe the onset of plasticity (irreversible rearrangements) in defect-rich as well as nearly defect-free solids; and (3) evaluate the resulting microscopic expressions for elastic moduli and transport coefficients (viscosities, thermal conductivity, defect diffusivity) in atomistic simulations or colloidal experiments for specific defect-bearing crystalline materials — beyond the perfect and vacancy-doped hard-sphere crystals already treated in the local-equilibrium framework — and compare quantitatively with independent measurements. Extensions beyond point defects (dislocations, grain boundaries, multicomponent crystals) within the same formalism remain part of the problem's open scope and lie outside the mandatory completion criteria stated below. The problem is complete when all three objectives are established: the derived nonlinear equations, their reduction and consistency checks (1); the consistency of the reciprocal-space projection description with the real-space spatial (affine/nonaffine) projection operators, including the plasticity-onset criterion (2); and at least one quantitative validation for a defect-bearing crystal beyond the perfect and vacancy-doped hard-sphere crystals already treated in the local-equilibrium framework (3).

References

  1. Florian Miserez, Saswati Ganguly, Rudolf Haussmann, Matthias Fuchs, "Continuum mechanics of nonideal crystals: Microscopic approach based on projection-operator formalism", Phys. Rev. E 106, 054125 (2022), DOI: 10.1103/PhysRevE.106.054125, arXiv:2205.08405, https://arxiv.org/abs/2205.08405 (note: the arXiv v1 listing retains the earlier preprint title "Continuum mechanics for the elastic properties of crystals: Microscopic approach based on projection-operator formalism", while the published version carries the title quoted here)
  2. Ken Hiura, "Microscopic derivation of nonlinear fluctuating hydrodynamics for crystalline solid", Phys. Rev. E 108, 054101 (2023), DOI: 10.1103/PhysRevE.108.054101, arXiv:2303.17227, https://arxiv.org/abs/2303.17227
  3. Joël Mabillard, Pierre Gaspard, "Hydrodynamic properties of the perfect hard-sphere crystal: microscopic computations with Helfand moments", J. Stat. Mech. (2024), DOI: 10.1088/1742-5468/ad1be0, arXiv:2311.00757, https://arxiv.org/abs/2311.00757
  4. Joël Mabillard, Pierre Gaspard, "Elastic and transport coefficients of the perfect hard-sphere crystal from the poles of the hydrodynamic spectral functions", J. Stat. Mech. (2024), DOI: 10.1088/1742-5468/ad2b5e, arXiv:2312.11697, https://arxiv.org/abs/2312.11697
  5. Joël Mabillard, Pierre Gaspard, "Vacancy diffusion and the hydrodynamics of crystals", Phys. Rev. E 111, 034103 (2025), DOI: 10.1103/PhysRevE.111.034103, arXiv:2408.07399, https://arxiv.org/abs/2408.07399
  6. Arsène Yerle, Pierre Gaspard, Joël Mabillard, "Dynamic structure factor of a monatomic cubic crystal", J. Stat. Mech. (2026), DOI: 10.1088/1742-5468/ae3d2a, arXiv:2511.09308, https://arxiv.org/abs/2511.09308
  7. Saswati Ganguly, Surajit Sengupta, Peter Sollich, "Statistics of non-affine defect precursors: tailoring defect densities in colloidal crystals using external fields", Soft Matter 11, 4517 (2015), DOI: 10.1039/C5SM00301F, https://doi.org/10.1039/C5SM00301F
  8. Pankaj Popli, Sayantani Kayal, Peter Sollich, Surajit Sengupta, "Exploring the link between crystal defects and nonaffine displacement fluctuations", Phys. Rev. E 100, 033002 (2019), DOI: 10.1103/PhysRevE.100.033002, https://doi.org/10.1103/PhysRevE.100.033002