  We consider the thermodynamics of a uniformly charged polyelectrolyte with
harmonic bonds. For such a system there is at high temperatures an approximate
scaling of global properties like the end-to-end distance and the interaction
energy with the chain-length divided by the temperature. This scaling is broken
at low temperatures by the ultraviolet divergence of the Coulomb potential. By
introducing a renormalization of the strength of the nearest- neighbour
interaction the scaling is restored, making possible an efficient blocking
method for emulating very large polyelectrolytes using small systems. The high
temperature behaviour is well reproduced by the analytical high-$T$ expansions
even for fairly low temperatures and system sizes. In addition, results from
low-$T$ expansions, where the coefficients have been computed numerically, are
presented. These results approximate well the corresponding Monte Carlo results
at realistic temperatures. A corresponding analysis of screened chains is
performed. The situation here is complicated by the appearance of an additional
parameter, the screening length. A window is found in parameter space, where
scaling holds for the end-to-end distance. This window corresponds to
situations where the range of the potential interpolates between the bond
length and the size of the chain. This scaling behaviour, which is verified by
Monte Carlo results, is consistent with Flory scaling. Also for the screened
chain a blocking approach can be devised, that performs well for low
temperatures, whereas the low-$T$ expansion is inaccurate at realistic
temperatures.
