Solving a differential equation (Lattice Laplacian)

Question

Suppose that $ p_n(t) $ is the probability of finding n particle at a time t. And the dynamics of the particle is described by this equation :

$$ \frac{d}{dt} p_n(t) = \lambda \Delta p_n(t) $$

Defining one - dimensional lattice translation operator $ E_m = e^{mk} $ with $ km - mk = 1 $ and $\Delta = E_1 + E_{-1} - 2 $ is a lattice laplacian.

So, here are my questions:

1. What is the one-dimensional lattice translation operator ? I think this one is a term from statistical physics, can you give me a simple explanation or a reference ?
2. What is the lattice laplacian ? Is it similar to discrete laplacian ? Could you give me a reference for this one ?
3. Is it possible to solve this equation analytically ?

Oh, all of this equation is about random walk with a boundary (random walk of a particle)

Thanks

Answer 1

Write $\large p_{n}\left(t\right) \equiv p_{n}{\rm e}^{-\gamma\mu t\,/a^{2}}$. $\large a$ is a lattice constant. Then you get $\large -\left(\mu/a^{2}\right)p_{n} = \Delta p_{n}$. $\large\tt Assuming$ that $n$ is a lattice site index, you'll get

$$ -\,{\mu \over a^{2}}p_{n} = {p_{n + 1} + p_{n -1} - 2p_{n} \over a^{2}} $$

or/and $$ p_{n + 1} + \left(\mu - 2\right)p_{n} + p_{n -1} = 0 $$

Solutions look like $\large \propto \beta^{n}$.