  We investigate the statistical mechanics of the periodic one-dimensional
Ising chain when the number of positive spins is constrained to be either an
even or an odd number. We calculate the partition function using a
generalization of the transfer matrix method. On this basis, we derive the
exact magnetization, susceptibility, internal energy, heat capacity and
correlation function. We show that in general the constraints substantially
slow down convergence to the thermodynamic limit. By taking the thermodynamic
limit together with the limit of zero temperature and zero magnetic field, the
constraints lead to new scaling functions and different probability
distributions for the magnetization. We demonstrate how these results solve a
stochastic version of the one-dimensional voter model.
