How go from Langevin to Hamilton
I have a system described by the Langevin equation
da/dt = - dF/da* + r
where a are complex amplitudes of electromagnetic modes (and r is the white noise).
How if F was real, it would be the Hamiltonian of the system, but in my case (and in general),
F is complex (because the a are complex themselves).
So F cannot be an Hamiltonian.
How can I obtain a Hamiltonian formulation of this problem?
P.S.: my first idea was to write to separate equation for the real and imaginary parts of a.
But in this case I obtain two Hamiltonian, both depending on both Re[a] and Im[a],
so which is the Hamiltonian of the system?
If e.g. I want to use a Gibbs measure for a, what I must use?