  A kinetic one-dimensional Ising model on a ring evolves according to a
generalization of Glauber rates, such that spins at even (odd) lattice sites
experience a temperature $T_{e}$ ($T_{o}$). Detailed balance is violated so
that the spin chain settles into a non-equilibrium stationary state,
characterized by multiple interactions of increasing range and spin order. We
derive the equations of motion for arbitrary correlation functions and solve
them to obtain an exact representation of the steady state. Two nontrivial
amplitudes reflect the sublattice symmetries; otherwise, correlations decay
exponentially, modulo the periodicity of the ring. In the long chain limit,
they factorize into products of two-point functions, in precise analogy to the
equilibrium Ising chain. The exact solution confirms the expectation, based on
simulations and renormalization group arguments, that the long-time,
long-distance behavior of this two-temperature model is Ising-like, in spite of
the apparent complexity of the stationary distribution.
