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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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  • Thanks for your reply , dear Thomas. I will study your proposed book. What I perceive from your reply is that : we have two starting points: master equation which is a generalized fokker planck eq and has a stochastic approach, and boltzmann eq which has a deterministic approach. is this true? under what approximations or conditions , we can reach from boltzmann eq to fokker planck eq or vice versa? is this possible? – Wisdom May 26 '16 at 18:28
  • I would not call the master equation a generalized FP equation. The master equation is stochastic, while FP is not. I would view master->FP as the usual transition from microscopic to macroscopic equations. 2) The BE is indeed deterministic (like FP), but (like the master equation) it contains more information. You can derive FP from linearized BE, but not the other way around.
    • – Thomas May 27 '16 at 15:41
  • Thanks for your kidness. I have confused a bit! 1- What approximations we should apply to master eq to obtain to fokker planck eq? In other word under what conditions we can use fokker-planck eq instead of master eq. this is my most important question? 2- is Kramers-Moyal generalized form of fokker-planck? 3-If the fokker planck is a transition form, then what is the final eq for treatment of macroscopic state? 4- what are the mother equations for treating stochastic processes in physics? – Wisdom May 28 '16 at 05:08
  • The Kramers-Moyal expansion can be used to derive FP from master equ. 2) Kramers-Moyal provdes a hirarchy of FP-type equations. "The" FP equation for the single particle probability is the first in this hirarchy. 3) FP is already a macroscopic equation. You can go slightly more macroscopic by using a hydrodynamic equation, like the diffusion equation. 4) Not sure what you are asking. If you wish, the starting point is the standard model of particle physics. everything else follows.
    • – Thomas May 30 '16 at 18:29
  • thanks a lot. please tell me, for what physical processes we can use each of these equations (namely boltzmann, fokker-planck and so on) and in which processes we cannot use them. – Wisdom Jul 04 '16 at 13:18