The applicability conditions of a recently reported Central Limit
Theorem-based approximation method in statistical physics are investigated and
rigorously determined. The failure of this method at low and intermediate
temperature is proved as well as its inadequacy to disclose quantum
criticalities at fixed temperatures. Its high temperature predictions are in
addition shown to coincide with those stemming from straightforward appropriate
expansions up to (k_B T)^(-2). Our results are clearly illustrated by comparing
the exact and approximate temperature dependence of the free energy of some
exemplary physical systems.Comment: 12 pages, 1 figur
