  We study the impact of out-of-equilibrium, dissipative effects on the
dynamics of inspiraling neutron stars. We find that modeling dissipative
processes (such as those from the stars internal effective fluid viscosity)
requires that one introduce a new tidal deformability parameter--the
dissipative tidal deformability--which modifies the phase of gravitational
waves emitted during the inspiral phase of a neutron star binary. We show that
the dissipative tidal deformability corrects the gravitational-wave phase at 4
post-Newtonian order for quasi-circular binaries. This correction receives a
large finite-size enhancement by the stellar compactness, analogous to the case
of the tidal deformability. Moreover, the correction is not degenerate with the
time of coalescence, which also enters at 4PN order, because it contains a
logarithmic frequency-dependent contribution. Using a simple Fisher analysis,
we show that physically allowed values for the dissipative tidal deformability
may be constrained by measurements of the phase of emitted gravitational waves
to roughly the same extent as the (electric-type, quadrupolar) tidal
deformability. Finally, we show that there are no out-of-equilibrium,
dissipative corrections to the tidal deformability itself. We conclude that
there are at least two relevant tidal deformability parameters that can be
constrained with gravitational-wave phase measurements during the late inspiral
of a neutron star binary: one which characterizes the adiabatic tidal response
of the star, and another which characterizes the leading-order
out-of-equilibrium, dissipative tidal response. These findings open a window to
probe dissipative processes in the interior of neutron stars with gravitational
waves.
