Coxeter groups such as symmetric groups appear naturally in many areas of algebraic combinatorics. In particular, subsets of Coxeter groups known as Kazhdan--Lusztig (KL) cells carry interesting representation theoretic information. We study KL cells of $\mathbf{a}$-value 2, where $\mathbf{a}$ refers to an $\mathbb{N}$-valued function defined on Coxeter groups by Lusztig that is constant on each cell. Our results may be viewed as an extension of Lusztig's results on cells of $\mathbf{a}$-value 1 from 1983. Our main tool is heaps, which are certain labeled posets introduced by Viennot to help visualize Cartier and Foata's commutation monoids. Specifically, we note that all elements of $\mathbf{a}$-value 2 are fully commutative in a sense of Stembridge and hence have associated heaps, then we use heaps to classify Coxeter groups with finitely many elements of $\mathbf{a}$-value 2; for such Coxeter groups, we also use heaps to enumerate and characterize all KL cells of $\mathbf{a}$-value 2. This is joint work with Richard Green.