In this post I’d like to continue looking at the dynamics of disordered systems. In particular I want to discuss the dynamics of the basic discrete 1D Anderson within the approximation developed in the previous post. As we’ll see, this can be reduced to the solution of a partial integro-differential equation.
1. Dynamics of the discrete Anderson model
Recall that the hamiltonian for the discrete 1D Anderson model is given by
where addition is modulo . (We slightly deviate in notation in this post from the previous by numbering sites on the ring beginning with .)
We are interested in the evolution of the system averaged over the disorder:
This density operator contains all the observable information about the system.
It is convenient, for the approach adopted here, to assume that the system is initialised in an eigenstate of . We write these eigenstates as
We now develop an evolution equation for in the interaction picture via the approach described in the previous post (i.e. via introduction of a fictitious ancilla system per site):
where is the memory kernel. Carrying out the approximation described there yields
encodes the probability distribution for . If we now trace out the fictitious ancillas we arrive at
Writing out the commutators and carrying out the sum over gives:
where (which can be expressed in terms of a bessel function of the first kind, see the previous post). Notice a peculiarity of this evolution equation: if is a function solely of at then it remains so for all time. Thus, eg., if we can write
Although the expansion (8) is not completely general, it will allow us to study the dynamics of the system initialised in an eigenstate of as there exist functions such that
Let’s now go with the expansion (8) and try to derive an evolution equation for :
A change of variable gives us
allows us to write
Thus we arrive at the following evolution equation for
Putting back in the hermitian conjugate terms leaves us with
which can be rewritten as
Such an equation can be expressed in terms of the convolution operation (in the variable) as
This equation is amenable to solution via a fourier transform in and a laplace transform in . I’ll discuss it’s solution in another post.