  We study a two-speed model with variant drift fields, which generalizes the
Goldstein-Taylor model but the two advection fields are not necessarily in
proportion. Due to the lack of a local equilibrium structure of the steady
state, prevailing hypocoercivity methods could not be directly applied to such
a system. To prove the exponential convergence to equilibrium for this model,
we use a recent generalization of the Gearhart-Pr\"uss theorem [Sci. China
Math. 64 (2021), no. 3, 507-518] to gain semigroup bounds, where the relative
entropy estimate plays a central role in gaining desired resolvent estimates
via a compactness argument.
