Simulating sticky particles: A Monte Carlo method to sample a stratification

The way κ_ij is defined here gives it units of length. To obtain a dimensionless sticky parameter, we must write the argument of the delta function as δ(r/σ_ij − 1). However, doing so would require carrying around factors of σ_ij in our later calculations, so we instead work with a dimensional κ_ij for notational convenience.

A dimensionless sticky parameter would be defined as $κ=limρ,E→∞σ−1∫0σ+ϵe−Umorse(r)/kBTdr$⁠.

Loosely, if two manifolds X, Y with dim(Y) < dim(X) are in the stratification, and $Y⊂X¯$⁠, i.e., Y is in the closure of X, then at every point y ∈ Y, the stratification near y has to look locally like a cone, and furthermore, the topology of this local picture is the Same for all y ∈ Y. Specifically, a stratification is usually assumed to satisfy Whitney’s condition B at all y ∈ Y, which is as follows:²⁸ Given X, Y as above, suppose that (i) sequence x₁, x₂, …, ∈ X converges to y, (ii) sequence y₁, y₂, …, ∈ Y converges to y, (iii) the secant lines $li=xiyi¯$ converge to a limiting line l in some local coordinate system near y, and (iv) the tangent planes $TxiX$ converge to a limiting plane τ. Then, l ⊂ τ.

We should really linearize q(x + hv + w), where w is an (unknown) vector in the normal space to M_I at x. The contribution from w vanishes to linear order, since linearizing the equations q_i(x + u) = 0, i ∈ I_eq defining manifold M_I, gives $∑i∈Iequ⋅∇qi=0$⁠. Therefore, any vector u that maintains the constraints to linear order must lie in the tangent space to M_I at x.