We present general results for one-dimensional systems of point charges (signed
point measures) on the line with a translation invariant distribution + for which
the variance of the total charge in an interval is uniformly bounded (instead of
increasing with the interval length). When the charges are restricted to multiples
of a common unit, and their average charge density does not vanish, then the
boundedness of the variance implies translation-symmetry breakingin the
sense that there exists a function of the charge configuration that is nontrivially
periodic under translationsand hence that + is not 'mixing'.