  This work explores fundamental statistical and thermodynamic properties of
short-and long-range-interacting systems. The purpose of this study is twofold.
Firstly, we rigorously prove that the probability distribution of arbitrary
few-body observables is restricted by a Gaussian concentration bound (or
Chernoff--Hoeffding inequality) above some threshold temperature. This bound is
then derived for arbitrary Gibbs states of systems that include long-range
interactions Secondly, we establish a quantitative relationship between the
concentration bound of the Gibbs state and the equivalence of canonical and
micro-canonical ensembles. We then evaluate the difference in the averages of
thermodynamic properties between the canonical and the micro-canonical
ensembles. Under the assumption of the Gaussian concentration bound on the
canonical ensemble, the difference between the ensemble descriptions is
upper-bounded by $\left[n^{-1} \log (n^{3/2}\Delta^{-1})\right]^{1/2}$ with $n$
being the system size and $\Delta$ being the width of the energy shell of the
micro-canonical ensemble This limit gives a non-trivial upper bound
\textit{exponentially small energy width} with respect to the system size. By
combining these two results, we prove the ensemble equivalence as well as the
weak eigenstate thermalization in arbitrary long-range-interacting systems
above a threshold temperature.
