Degenerate Laplacian describing topologically constrained diffusion: helicity constraint as an alternative to ellipticity

Abstract

We study a second-order non-elliptic partial differential operator that generalizes the classical Laplacian with a degenerate coefficient matrix. The theory is motivated by the statistical mechanics of topologically constrained particles. In the context of diffusion models, the coefficient matrix is given by , with a generalized Poisson matrix that dictates particle dynamics. When is the symplectic matrix of canonical Hamiltonian systems, the operator reduces to the classical Laplacian. However, topological constraints make the matrix degenerate. The ellipticity is broken in the direction parallel to the nullity of . We call such a diffusion operator an orthogonal Laplacian and denote it by . When the nullity foliates the space, reduces into a classical Laplacian on the leaf that is orthogonal to the null space. However, when the nullity of is not integrable (which is the case when has helicity), is akin to elliptic operators having some useful properties of convex analysis. We show that defines an inner product of a Sobolev-like Hilbert space, and is a coercive functional in the L2 topology. Constructing a trace theorem, we discuss a boundary value problem (the orthogonal Poisson equation) for the orthogonal Laplacian, and obtain a unique weak solution by application of Riesz’s representation theorem.