The transverse velocity time correlation function ${\tilde{C}}_T(k,\omega )$ with $k$ and $\omega$ being the wave number and the frequency, respectively, is a fundamental quantity in determining the transverse mechanical and transport properties of materials. In ordinary liquids, a nonzero value of ${\tilde{C}}_T(k,0)$ is inevitably linked to viscous material flows. Even in solids where significant material flows are precluded due to almost frozen positional degrees of freedom, our molecular dynamics simulations reveal that ${\tilde{C}}_T(k,0)$ takes a nonzero value, whereby the time integration of the velocity field shows definite diffusive behavior with diffusivity ${\tilde{C}}_T(k,0)/3$. This behavior is attributed to viscous transport accompanying a small random convection of the velocity field (the inertia effect), and the resultant viscosity is measurable in the Eulerian description: the constituent particles that substantially carry momenta fluctuate slightly around their reference positions. In the Eulerian description, the velocity field is explicitly associated with such fluctuating instantaneous particle positions, whereas in the Lagrangian description, this is not the case. The present study poses a fundamental problem for continuum mechanics: reconciling liquid and solid descriptions in the limit of the infinite structural relaxation time.

• Received 8 October 2019
• Revised 25 August 2021
• Accepted 18 October 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.245901