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Friction is a phenomenon that manifests across all spatial and temporal scales, from the molecular to the macroscopic scale. It describes the dissipation of energy from the motion of particles or abstract reaction coordinates and arises in the transition from a detailed molecular-level description to a simplified, coarse-grained model. It has long been understood that time-dependent (non-Markovian) friction effects are critical for describing the dynamics of many systems, but that they are notoriously difficult to evaluate for complex physical, chemical, and biological systems. In recent years, the development of advanced numerical friction extraction techniques and methods to simulate the generalized Langevin equation has enabled exploration of the role of time-dependent friction across all scales. We discuss recent applications of these friction extraction techniques and the growing understanding of the role of friction in complex equilibrium and nonequilibrium dynamic many-body systems.

The relation between the frequency-dependent friction of a molecule in a liquid and the hydrodynamic properties of the liquid is fundamental for molecular dynamics. We investigate this connection for a water molecule moving in liquid water using all-atomistic molecular dynamics (MD) simulations and linear hydrodynamic theory. We analytically calculate the frequency-dependent friction of a sphere with finite surface slip moving in a viscoelastic compressible fluid by solving the linear transient Stokes equation, including frequency-dependent shear and volume viscosities, both determined from MD simulations of bulk liquid water. From MD simulation trajectories, we also determine the frequency-dependent friction of a single water molecule moving in liquid water, as defined by the generalized Langevin equation. The frequency dependence of the shear viscosity of liquid water requires careful consideration of hydrodynamic finite-size effects to observe the asymptotic hydrodynamic power-law tail. By fitting the effective sphere radius and the slip length, the frequency-dependent friction and velocity autocorrelation function from the transient Stokes equation and simulations quantitatively agree. This shows that the transient Stokes equation accurately describes the important features of the frequency-dependent friction of a single water molecule in liquid water and thus applies down to molecular length and time scales, provided accurate frequency-dependent viscosities are used. In contrast, for a methane molecule moving in water, the frequency-dependent friction cannot be predicted based on a homogeneous model, which, supported by the extraction of the frequency-dependent surface slip, suggests that a methane molecule is surrounded by a finite-thickness hydration layer with viscoelastic properties that differ significantly from those of bulk water.