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In the non-equilibrium statistical mechanics framework, there are two basic paradigms for defining the dynamics of the system: the Langevin and Fokker-Planck equations for diffusion processes and the discrete and continuous-time master equation for jumping processes. For sure, these two ways of looking at the problem are related and I would like to understand the general procedure of how to go from one framework to the other through a particular example:

Consider the problem of a random walk with left-hopping probability $p$, right-hopping probability $q$ and no-jump probability $r$. My aim is to derive the diffusion equation with proper drift and diffusion constants. The difference equation is $$P(x,t)=pP(x-\Delta x, t - \Delta t) + qP(x+\Delta x, t-\Delta t) +r P(x,t-\Delta t)$$ or, $$P(x,t)-P(x,t-\Delta t)= pP(x-\Delta x, t - \Delta t) + qP(x+\Delta x, t-\Delta t) +r P(x,t-\Delta)-(p+q+r)P(x,t-\Delta t)$$

For asymmetric random walk, $$(p-q)\frac{\Delta x}{\Delta t}=v$$ and $$p+q+r=1$$

From these 2 equations, $$\frac{\Delta x}{\Delta t}=v+(2q+r)\frac{\Delta x}{\Delta t}=(2p+r)\frac{\Delta x}{\Delta t}-v$$

Using this 2nd and 3rd equation, I arrived at $$\frac{\partial P}{\partial t}=-v\frac{\partial P}{\partial x} +(2pq +pr)\frac{\Delta x}{\Delta t}\frac{P(x-\Delta x,t-\Delta t)-P(x,t-\Delta t)}{\Delta x}+(2pq +qr)\frac{\Delta x}{\Delta t}\frac{P(x+\Delta x,t-\Delta t)-P(x,t-\Delta t)}{\Delta x}$$ or, $$\frac{\partial P}{\partial t}=-v\frac{\partial P}{\partial x} +2pq\frac{\Delta x ^2}{\Delta t} \frac{\partial ^2 P}{\partial x^2}+r\frac{\Delta x}{\Delta t}\left( p\frac{P(x-\Delta x,t-\Delta t)-P(x,t-\Delta t)}{\Delta x}+q\frac{P(x+\Delta x,t-\Delta t)-P(x,t-\Delta t)}{\Delta x} \right)$$ or $$\frac{\partial P}{\partial t}=-v\frac{\partial P}{\partial x} +2pq\frac{\Delta x ^2}{\Delta t} \frac{\partial ^2 P}{\partial x^2}+r\frac{\partial P}{\partial t}$$ or, $$\frac{\partial P}{\partial t}=-\frac{(p-q)\frac{\Delta x}{\Delta t}}{p+q}\frac{\partial P}{\partial x}+\frac{2pq\frac{\Delta x^2}{\Delta t}}{p+q}\frac{\partial^2 P}{\partial x^2}$$

Here, the diffusion constant is changed which is not surprising, but the drift term changed, which is wrong, because the origin of drift is the asymmetry in the hopping probability. So, I want to know where I made the mistake.

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Wellcome, @rs_physics. I found it a bit messy how you proceed after the definition of $v$. I would introduce a Taylor expansion in the Chapman-Kolmogorov equation (your first equation).

$P(x,t)=(r+p+q)P(x,t) -\Delta t (r+p+q) \partial_tP(x,t)+(q-p )\Delta x\partial_x P(x,t)+(q+p )\Delta x^2\partial_x^2 P(x,t)$

Using p+q+r=1 and rearranging terms you'll obtain the Fokker-Planck equation:

$\partial_tP(x,t)=(q-p )\frac{\Delta x}{\Delta t}\partial_x P(x,t)+(q+p )\frac{\Delta x^2}{\Delta t}\partial_x^2 P(x,t)$

If you define $v=(q-p )\frac{\Delta x}{\Delta t}$ and $D=(q+p )\frac{\Delta x^2}{\Delta t}$ you obtain the usual biased diffusion equation.

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  • $\begingroup$ Please note that "check my work" questions are off topic on PSE. Supplying worked solutions, even partial solutions, is typically frowned upon here. Thanks for being a part of the PSE community though :) $\endgroup$ Sep 20, 2021 at 13:47

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