The heat equation for the laplacian (self-adjoint, positive semidefinite) is the gradient flow of the Dirichlet energy ,
with solution the heat semigroup. In an orthonormal eigenbasis () it acts diagonally: writing ,
So heat flow is coefficient decay, fast modes first. Its long-time limit is projection onto the harmonic space: every term dies and
the harmonic part cohomology. The approach is exponential at rate the spectral gap (the smallest nonzero eigenvalue): . Thus the flow finds the harmonic representative of its initial data, and the spectrum sets both what survives (the kernel) and how fast (the gap).
This is one construction across scales: on a closed manifold it smooths a function toward its mean; on a graph or a sheaf the same drives an inconsistent family toward the nearest global section, its harmonic cochain.