Hopping particles, generating functions and the kernel method

Many paradigmatic model systems in nonequilibrium statistical physics involve particles that hop between sites of a lattice. The master equation that governs the time evolution of the probability distribution for the system takes the form of a set of difference equations, which in turn can (sometimes) be solved using generating functions (essentially, the discrete analogue of a Laplace transform). The kernel method provides one technique to manipulate generating functions and determine the desired probability distribution. I will expose the machinery of the kernel method with reference to a very simple particle hopping model, and then sketch out its application to the more complex problem of interacting run-and-tumble particles.