Continuum mechanics for the elastic properties of crystals: Microscopic approach based on projection-operator formalism

TL;DR

This paper derives continuum mechanics laws for nonideal ordered solids from microscopic principles using projection-operator formalism, linking fluctuations to thermodynamic and transport properties.

Contribution

It introduces a microscopic derivation of elastic and transport properties of crystals incorporating dissipation and defects via the Zwanzig-Mori formalism.

Findings

Elastic constants from equilibrium correlations

Transport coefficients as Green-Kubo formulas

Framework applicable to atomistic simulations

Abstract

We present a microscopic derivation of the laws of continuum mechanics of nonideal ordered solids including dissipation, defect diffusion, and heat transport. Starting point is the classical many-body Hamiltonian. The approach relies on the Zwanzig-Mori projection operator formalism to connect microscopic fluctuations to thermodynamic derivatives and transport coefficients. Conservation laws and spontaneous symmetry breaking, implemented via Bogoliubov's inequality, determine the selection of the slow variables. Density fluctuations in reciprocal space encode the displacement field and the defect concentration. Isothermal and adiabatic elastic constants are obtained from equilibrium correlations, while transport coefficients are given as Green-Kubo formulae, providing the basis for their measurement in atomistic simulations or colloidal experiments. The approach and results are compared to others from the literature.