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

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.

Quantum local-equilibrium approach to dissipative hydrodynamics

“…(A16)-(A19). By combining the quantum integral fluctuation theorem (32) with the Peierls-Bogoliubov inequality (33), we deduce that Σt t ≥ 0. As a consequence of Eq.…”

“…B tr e  and the normalization condition tr e  = tr e −ς(λt)+ Σt = tr ˆ t = 1, the Peierls-Bogoliubov inequality implies that tr e −ς(λt)+ Σt e −ς(λt) e ς(λt)− Σt ≥ e − Σt t (A19) or, equivalently, the expression (33), which is the quantum version of classical Jensen's inequality e x ≥ e x with x = −Σ t (Γ). h. The entropy time derivative (35).…”

“…In crystals, where the symmetries under three-dimensional spatial translations are broken, there are three Nambu-Goldstone modes emerging from the crystalline long-range order. Such an order is described by some microscopic order field xα (r), given for instance by the displacement vector in crystals [18,19,[30][31][32][33]. Its gradient ûaα ≡ ∇ a xα obeys a local conservation equation such as Eq.…”

“…) which are comparable to the equations (2) holding for the fundamentally conserved quantities. In crystals, the order fields are given by the three Cartesian components of the displacement vector, which can be expressed in terms of the microscopic particle densities and the spatially periodic equilibrium densities of the different particle species in the crystalline lattice [18,19,[30][31][32][33].…”