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  1. I'm looking for the relation between four important equations which we study in stochastic processes in physics. These equations include Master, Fokker-Planck, Langevin, Kramers-Moyal and Boltzmann.

  2. Also I want to know when can we use each of them?

  3. In other word, what are boundary conditions or limitations of each one?

  4. Do you introduce any paper or book about this?

Since I have recently started studying stochastic field in physics, I hope you can help me for finding answers of my questions.

Qmechanic
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Wisdom
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1 Answers1

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This question is a little too big, entire text books have been written to answer it. A standard reference is van Kampen, Stochastic processes in physics and chemistry.

Roughly speaking, for a Markov process

Master equation -> Kramers-Moyal expansion -> Fokker-Planck equation

where the master equation gives the microscopic probabilistic rule for transitions in some configuration space, and the Fokker-Planck equation is the corresponding equation for single particle probabilities.

The Langevin equation is a simple stochastic model equation designed to give the same FP equation as the master equation. The Boltzmann equuation lives in a larger space (phase space), and is not stochastic, but is again designed so that linearized Boltzmann is equivalent to FP.

[This make it sound as if the Boltzmann equation is just some kind of model equation. This is not correct. There is a separate path to the Boltzmann equation, starting from the classical Liouville or quantum von-Neumann or quantum Kadanoff-Baym equation.]

Thomas
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