Andreev-Lifshitz Hydrodynamics Applied to an Ordinary Solid under Pressure

We have applied the Andreev-Lifshitz hydrodynamic theory of supersolids to an ordinary solid. This theory includes an internal pressure $P$, distinct from the applied pressure $P_a$ and the stress tensor $\lambda_{ik}$. Under uniform static $P_{a}$, we have $\lambda_{ik} = (P-P_{a})\delta_{ik}$. For $P_{a} \ne 0$, Maxwell relations imply that $P \sim P_{a}^{2}$. The theory also permits vacancy diffusion but treats vacancies as conserved. It gives three sets of propagating elastic modes; it also gives two diffusive modes, one largely of entropy density and one largely of vacancy density (or, more generally, defect density). For the vacancy diffusion mode (or, equivalently, the lattice diffusion mode) the vacancies behave like a fluid within the solid, with the deviations of internal pressure associated with density changes nearly canceling the deviations of stress associated with strain. We briefly consider pressurization experiments in solid $^4$He at low temperatures in light of this lattice diffusion mode, which for small $P_{a}$ has diffusion constant $D_{L} \sim P_{a}^{2}$. The general principles of the theory -- that both volume and strain should be included as thermodynamic variables, with the result that both $P$ and $\lambda_{ik}$ appear -- should apply to all solids under pressure, especially near the solid-liquid transition. The lattice diffusion mode provides an additional degree of freedom that may permit surfaces with different surface treatments to generate different responses in the bulk.