The flow of a Hamilton-Jacobi PDE yields a dynamical system on the space of continuous functions. When the Hamiltonian function is convex in the momentum variable, we may restrict the flow to piecewise $C^1$ functions and give a kinetic description for the solution. We regard the locations of jump discontinuities of the first derivative of solutions as the sites of particles.
These particles interact via collisions and coagulations. When these particles are selected randomly according to certain Gibbs measures initially, then the law of particles remain Gibbsian at later times, and one can derive a Boltzmann/Smoluchowski type PDE for the evolution of these Gibbs measures.