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I'm participating in an introductory course for biophysics. We briefly discussed the derivation of the Fokker-Planck equation and used the so-called Master equation as a starting point.

$$ \frac{\partial P(x)}{\partial t}=\int \mathrm{d}x'\left[W\left(x\vert x'\right)P\left(x'\right)-W\left(x'\vert x\right)P\left(x\right)\right] $$

It seems like the term 'Master equation' simply describes a set of equations which describe the time evolution of a probabilistic system. But why do we use this specific Master equation for the derivation of the Fokker-Planck equation and how was it found?

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The master equation is summarized in words as follows:

The rate of change of the probability of one state occurring is equal to the rate at which transitions occur into that state minus the rate at which transitions occur out of that state.

This is one of those situations where it's useful to think of probability as a fluid: the change in the height of a fluid in a box is equal to the flow into the box minus the flow out of the box.

So, to proceed with the derivation, let's start with the definition of $W(x|x')$. The quantity $W(x|x')$ is the rate at which the system transitions to state $x$ given that it's in state $x'$. This means that the rate at which the system transitions from $x'$ to $x$ is equal to the probability $P(x')$ that it's in state $x'$ in the first place, multiplied by the transition rate $W(x|x')$ from $x'$ to $x$; in other words, the transition rate from $x'$ to $x$ is given by $W(x|x')P(x')$.

The total transition rate from any other state into $x$ is given by the sum of the transition rates for all possible transitions. For systems with a discrete set of states $S$, the total transition rate into $x$ is given by $\sum_{x'\in S}W(x|x')P(x')$. It's easy to extend this to a system with a continuous set of states: just replace the sum with an integral over the entire state space, so the total transition rate into $x$ is given by $\int W(x|x')P(x')dx'$.

Using basically the same argument, the transition rate from $x$ to $x'$ is given by $W(x'|x)P(x)$ (the probability that the system is in state $x$ in the first place, multpilied by the transition rate from $x$ to $x'$ given that the system is in state $x$). Summing/integrating over all possible transitions out of state $x$, we have that the total transition rate from $x$ to any other state is given by $\int W(x'|x)P(x)dx'$.

Now that we have the total transition rate into $x$ and the total transition rate out of $x$, to get the total change in probability density, we just subtract the two (see the fluid analogy: we're saying "change in quantity = flow in - flow out), so that

$$\frac{\partial P(x)}{\partial t}=\int W(x|x')P(x') - W(x'|x)P(x)\;dx'$$

which is precisely the master equation.

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  • $\begingroup$ Okay, I now understand the meaning of the Master equation and its derivation, thanks! But what is the link between this (very abstract) formula and the physical implications of the Fokker-Planck equation, in which we suddenly can identify drift and diffusion? Did somebody just simplify the master equation with the Kramers-Moyal expansion and saw that the result is able to describe the movements of particles? $\endgroup$ – JoKli Jun 5 at 12:42
  • $\begingroup$ @JoKli Yes, the Fokker-Planck equation is the second-order truncation of the master equation (en.wikipedia.org/wiki/Kramers%E2%80%93Moyal_expansion). $\endgroup$ – probably_someone Jun 5 at 13:04
  • $\begingroup$ Sure, we did the calculations in the lecture and I understand the mathematical derivation. However, I don't understand how we can start at an equation which describes an abstract mathematical consideration and then arrive at the Fokker-Planck equation which can describe Brownian motion. There must be some reason why we start at this master equation. $\endgroup$ – JoKli Jun 5 at 13:16
  • $\begingroup$ Couldn't that equation be derived from Liouville's equation in phase space? $$\frac{\partial \rho}{\partial t} = \{\rho, \, H \} = \frac{\partial \rho}{\partial x^k} \, \frac{\partial H}{\partial p_k} - \frac{\partial H}{\partial x^k} \, \frac{\partial \rho}{\partial p_k}.$$ $\endgroup$ – Cham Jun 5 at 13:23
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    $\begingroup$ I just found that the master equation is the Chapman-Kolmogorow equation which describes Markov-chains. So I guess the underlying assumption is, that Brownian motion can be considered as a Markov process and with that assumption we start at the master equation and then derive the FPE. $\endgroup$ – JoKli Jun 5 at 13:39

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