  We obtain rates of convergence in limit theorems of partial sums $S_n$ for
certain sequences of dependent, identically distributed random variables, which
arise naturally in statistical mechanics, in particular, in the context of the
Curie-Weiss models. Under appropriate assumptions there exists a real number
$\alpha$, a positive real number $\mu$, and a positive integer $k$ such that
$(S_n- n \alpha)/n^{1 - 1/2k}$ converges weakly to a random variable with
density proportional to $\exp(-\mu |x|^{2k} /(2k)!)$. We develop Stein's method
for exchangeable pairs for a rich class of distributional approximations
including the Gaussian distributions as well as the non-Gaussian limit
distributions with density proportional to $\exp(-\mu |x|^{2k} /(2k)!)$. Our
results include the optimal Berry-Esseen rate in the Central Limit Theorem for
the total magnetization in the classical Curie-Weiss model, for high
temperatures as well as at the critical temperature $\beta_c=1$, where the
Central Limit Theorem fails. Moreover, we analyze Berry-Esseen bounds as the
temperature $1/ \beta_n$ converges to one and obtain a threshold for the speed
of this convergence. Single spin distributions satisfying the
Griffiths-Hurst-Sherman (GHS) inequality like models of liquid helium or
continuous Curie-Weiss models are considered.
