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Mathematically, the discrete Laplacian is equivalent to the trace of the Hessian. I've read in my literature review that the Force Constant Matrix (FCM) of a lattice system, which is a Hessian matrix, may be approximated by the discrete Laplacian due to this equivalence. However, It's unclear whether this is simply a mathematical trick or if there exists a physical interpretation of the approximation. In other words, what are the required constraints (if any) on the lattice system for the discrete Laplacian to approximately represent the phonon dynamics of a crystal lattice?

Edit: Here is a Phys.Rev.B article that uses the discrete Laplacian to propose a Discrete Laplacian Thermostat where a good amount of the physics is discussed, but is a bit dense. Another reference is Large-Scale Phonon Calculations Using the Real-Space Multigrid Method published in JCTC, which relates it to the Force Constant Matrix.

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  • $\begingroup$ Please give the exact reference: title, authors, chapter/section, page number; and a link if possible. $\endgroup$
    – Tobias Fünke
    Commented Oct 8 at 15:14
  • $\begingroup$ @6D-Hyper-Sphere please edit this into the post, rather than leaving it as a comment (which is deletable). $\endgroup$
    – Kyle Kanos
    Commented Oct 9 at 12:22
  • $\begingroup$ There is no article cited here. $\endgroup$
    – Matt Hanson
    Commented Oct 24 at 1:36
  • $\begingroup$ I've fixed the links. Please, let me know if you can provide any insight. $\endgroup$
    – 6D-Hyper-Sphere
    Commented Oct 28 at 15:05

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