  The study by Glauber of the time-dependent statistics of the Ising chain is
extended to the case where each spin is influenced unequally by its nearest
neighbours. The asymmetry of the dynamics implies the failure of the detailed
balance condition. The functional form of the rate at which an individual spin
changes its state is constrained by the global balance condition with respect
to the equilibrium measure of the Ising chain. The local magnetization, the
equal-time and two-time correlation functions and the linear response to an
external magnetic field obey linear equations which are solved explicitly. The
behaviour of these quantities and the relation between the correlation and
response functions are analyzed both in the stationary state and in the
zero-temperature scaling regime. In the stationary state, a transition between
two behaviours of the correlation function occurs when the amplitude of the
asymmetry crosses a critical value, with the consequence that the limit
fluctuation-dissipation ratio decays continuously from the value 1, for the
equilibrium state in the absence of asymmetry, to 0 for this critical value. At
zero temperature, under asymmetric dynamics, the system loses its critical
character, yet keeping many of the characteristic features of a coarsening
system.
