I have a system described by the Langevin equation(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# How go from Langevin to Hamilton

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