Eyink’s research interests include mathematical physics, fluid mechanics, turbulence, dynamical systems, partial differential equations, non-equilibrium statistical physics, geophysics and climate, astrophysics, and plasma physics. In addition to research on turbulence, he has contributed to the renormalization group in quantum field theory, to statistical physics (thermal fluctuations and variational principles in non-equilibrium statistical mechanics), to data assimilation in geophysics, and to magnetic reconnection in astrophysics and space physics.

One of Eyink’s most influential works was a 1994 paper that provided the first published proof of a claim on fluid singularities made by Lars Onsager decades earlier and that gave the modern formulation of what is now called the “Onsager conjecture.” Eyink continues to work in this direction, publishing in 2018 a series of three papers in Physical Review X extending Onsager’s exact analysis to turbulence incompressible fluids, relativistic fluids and kinetic plasmas. In addition, a 2011 study by Eyink in the journal Physical Review E presented the first numerical evidence for the predicted phenomenon of “spontaneous stochasticity,” according to which two beads dropped into identical locations in the same turbulent flow will end up in different locations. A 2013 study that he co-authored in Nature found that the flux-freezing theorem for ideal plasmas is violated by this turbulence effect, allowing rapid changes of magnetic field-line topology during explosive events such as solar flares.

Current projects include studies of phase-space entropy cascade using spacecraft data on turbulent plasma in the terrestrial magnetosheath and stochastic Lagrangian dynamics of vorticity in wall-bounded turbulent flows.