How go from Langevin to Hamilton

  1. 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?
     
  2. jcsd
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