Derivation of the GLE

The GLE can be derived from the harmonic bath Hamiltonian by simply solving Hamilton's equations of motion, which take the form

This set of equations can also be written as second order differential equation:

In order to derive an equation for q, we solve explicitly for the dynamics of the bath variables and then substitute into the equation for q. The equation for is a second order inhomogeneous differential equation, which can be solved by Laplace transforms. We simply take the Laplace transform of both sides. Denote the Laplace transforms of q and as

and recognizing that

we obtain the following equation for :

or

can be obtained by inverse Laplace transformation, which is equivalent to a contour integral in the complex s-plane around a contour that encloses all the poles of the integrand. This contour is known as the Bromwich contour. To see how this works, consider the first term in the above expression. The inverse Laplace transform is

The integrand has two poles on the imaginary s-axis at . Integration over the contour that encloses these poles picks up both residues from these poles. Since the poles are simple poles, then, from the residue theorem:

By the same method, the second term will give . The last term is the inverse Laplace transform of a product of and . From the convolution theorem of Laplace transforms, the Laplace transform of a convolution gives the product of Laplace transforms:

Thus, the last term will be the convolution of q(t) with . Putting these results together, gives, as the solution for :

The convolution term can be expressed in terms of rather than q by integrating it by parts:

The reasons for preferring this form will be made clear shortly. The bath variables can now be seen to evolve according to

Substituting this into the equation of motion for q, we find

We now introduce the following notation for the sums over bath modes appearing in this equation:

1.

Define a dynamic friction kernel

2.

Define a random force

Using these definitions, the equation of motion for q reads

 

Eq. (1) is known as the generalized Langevin equation. Note that it takes the form of a one-dimensional particle subject to a potential , driven by a forcing function R(t) and with a nonlocal (in time) damping term , which depends, in general, on the entire history of the evolution of q. The GLE is often taken as a phenomenological equation of motion for a coordinate q coupled to a general bath. In this spirit, it is worth taking a moment to discuss the physical meaning of the terms appearing in the equation.