  We considered a long-range Ising model under Glauber dynamics and calculated
the difference from the mean-field approximation in a finite-size system using
perturbation theory. To deal with the BBGKY hierarchy, we assumed that certain
types of extensive properties have a Gaussian distribution, which turned out to
be equivalent to the Kirkwood superposition approximation within the range of
first-order perturbation. After several calculations, ordinary differential
equations that describe the time development of a two-body correlation were
derived. This discussion is the generalization of our previous study which
developed a similar consideration on the infinite-range Ising model. The
results of the calculation fit those of the numerical simulations for the case
in which the decay of the interaction was sufficiently slow; however, they
exhibited different behaviors when the decay became rapid.
