  Intrinsic ripples with various configurations and sizes were reported to
affect the physical and chemical properties of 2D materials. By performing
molecular dynamics simulations and theoretical analysis, we use two geometric
models of the ripple shape to explore numerically the distribution of ripples
in graphene membrane. We focus on the ratio of ripple height to its diameter
(t/D) which was recently shown to be the most relevant for chemical activity of
graphene membranes. Our result demonstrates that the ripple density decreases
as the coefficient t/D increases, in a qualitative agreement with the Boltzmann
distribution derived analytically from the bending energy of the membrane. Our
theoretical study provides also specific quantitative information on the ripple
distribution in graphene and gives new insights applicable to other 2D
materials.
