How go from Langevin to Hamilton

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In summary: Alternatively, one can write separate equations for the real and imaginary parts of a, but this will result in two Hamiltonians that cannot be combined into a single Hamiltonian for the system. In these cases, it is unclear which Hamiltonian should be used for calculating quantities such as Gibbs measures.
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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?
 
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In general, it is not possible to obtain a Hamiltonian formulation from a Langevin equation. However, in some cases, such as when the noise is small, it is possible to use techniques such as the Mori-Zwanzig formalism to derive an effective Hamiltonian from the Langevin equation. This effective Hamiltonian will contain additional terms that depend on the noise and will no longer be equivalent to the original F.
 

What is the Langevin equation and how does it relate to Hamiltonian mechanics?

The Langevin equation is a stochastic differential equation that describes the motion of a particle in a dissipative environment. It is used to model systems where random forces and friction play a significant role. In the context of Hamiltonian mechanics, the Langevin equation can be derived from the Hamiltonian equations of motion by considering the effects of a thermal bath on the system.

What is the relationship between the Langevin equation and the Fokker-Planck equation?

The Fokker-Planck equation is a partial differential equation that describes the time evolution of a probability distribution. It is closely related to the Langevin equation as it can be derived from the Langevin equation by considering the probability density of the system. The Fokker-Planck equation is a deterministic equation, while the Langevin equation is a stochastic one.

What is the Hamiltonian and how is it related to the energy of a system?

The Hamiltonian is a mathematical function that represents the total energy of a system. In Hamiltonian mechanics, it is used to describe the dynamics of a system by defining the equations of motion. The Hamiltonian is related to the energy of a system as it is the sum of the kinetic and potential energies of the particles in the system.

What are canonical variables and how are they used in Hamiltonian mechanics?

Canonical variables are pairs of variables that are used to describe the state of a system in Hamiltonian mechanics. They consist of a coordinate and its conjugate momentum. In Hamiltonian mechanics, the equations of motion can be written in terms of these canonical variables, making it easier to analyze the dynamics of a system.

How can one go from the Langevin equation to the Hamiltonian of a system?

To go from the Langevin equation to the Hamiltonian of a system, one must first determine the potential energy function of the system. This can be done by considering the forces acting on the particles in the system and integrating to find the potential energy. Once the potential energy is known, the Hamiltonian can be written in terms of the canonical variables and the Langevin equation can be derived from the Hamiltonian equations of motion.

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